All practice questions

Question 2641Question →

A farmer needs to completely fill a cylindrical grain silo with wheat. If the silo has a diameter of $12$ meters and a height of $10$ meters, what is the maximum capacity of the silo in cubic meters?

60\pi

120\pi

360\pi

1440\pi

Show answer & explanation

Answer: $360\pi$

Answer

The maximum capacity of the silo is

360\pi

cubic meters.

The question asks for the maximum capacity to completely fill the silo, which means finding its volume. The volume of a cylinder is found using the formula

V = \pi r^2 h

. First, the diameter of

12

meters must be halved to find the radius of

6

meters. Substituting the radius and the height (

10

meters) into the formula gives

V = \pi(6)^2(10) = \pi(36)(10) = 360\pi

cubic meters.

Step-by-Step Solution

1

Determine the required measurement and the appropriate formula.

The problem asks for the capacity to 'fill' a cylinder, which requires the volume formula:

V = \pi r^2 h

.

Capacity refers to the interior space (volume) of a 3D figure.

2

Find the radius of the circular base.

The radius is half of the diameter:

12 \div 2 = 6

meters.

The volume formula requires the radius, but the problem provides the diameter.

3

Substitute the values into the formula and solve.

V = \pi(6)^2(10) = \pi(36)(10) = 360\pi

cubic meters.

Executing the calculation according to the order of operations yields the final volume.

Key Concept

Volume of a Cylinder

Estimated Time:1m 0s

Question 2642Question →

Determine the missing measurement to complete the statement below.

Fill in the blanks below

A museum exhibit features a custom glass panel shaped exactly like a right triangle. The longest edge of the glass panel measures

29\text{ inches}

, and the shortest edge measures

20\text{ inches}

. The length of the third edge is inches.

Show answer & explanation

Answer

21

The correct answer is

21

. According to the Pythagorean theorem (

a^2 + b^2 = c^2

), the square of the hypotenuse (

29^2 = 841

) minus the square of the known leg (

20^2 = 400

) equals the square of the unknown leg (

441

). The square root of

441

is

21

.

Step-by-Step Solution

1

Identify the given parts of the right triangle.

Hypotenuse

c = 29

, and leg

a = 20

.

The longest edge of a right triangle is always the hypotenuse.

2

Set up the Pythagorean theorem equation.

20^2 + b^2 = 29^2

The Pythagorean theorem (

a^2 + b^2 = c^2

) connects the lengths of the legs to the hypotenuse.

3

Square the known side lengths.

400 + b^2 = 841

20 \times 20 = 400

and

29 \times 29 = 841

.

4

Isolate the unknown variable

b^2

.

b^2 = 841 - 400 = 441

Subtract

400

from both sides to find the square of the missing leg.

5

Solve for

b

by taking the square root.

b = \sqrt{441} = 21

The square root of

441

gives the length of the third edge in inches.

Key Concept

The Pythagorean Theorem (

a^2 + b^2 = c^2

) allows you to find an unknown leg of a right triangle by subtracting the square of the known leg from the square of the hypotenuse.

Question 2643Question →

A shop technician is selecting fasteners for different fabrication and assembly tasks. Which pairings correctly match each screw head style on the left to its defining seating profile and primary application on the right?

Click a left item, then click its matching right item

Show answer & explanation

Answer

Flat Head pairs with the

82^\circ

conical countersunk underside for flush mounting; Pan Head pairs with the flat bearing shoulder that sits proud on workpiece surfaces; Bugle Head pairs with the curved transition that compresses drywall paper without tearing; Truss Head pairs with the extra-wide dome profile that distributes clamping load across thin sheet metal.

Each screw head style is matched to its unique physical profile and mechanical purpose: Flat Head screws feature an

82^\circ

countersunk profile for flush seating in pre-beveled holes; Pan Head screws feature a flat underside for general-purpose surface mounting; Bugle Head screws have a curved flare engineered to dimple drywall paper without tearing; and Truss Head screws feature an oversized diameter profile to prevent pull-through in thin sheet metal.

Step-by-Step Solution

1

Analyze countersunk vs. non-countersunk head geometries.

Flat and bugle heads are countersinking styles, whereas pan and truss heads are surface-bearing styles.

Countersinking screws taper inward beneath the head to sit flush or sub-surface, while surface-bearing screws have flat undersides to bear against the workpiece.

2

Differentiate between the countersinking styles.

Flat heads have a sharp

82^\circ

conical taper requiring a countersunk hole, while bugle heads use a curved, flared taper specifically engineered for drywall installation.

The bugle contour depresses drywall paper without slicing it, preserving the structural holding strength of the paper face.

3

Differentiate between the surface-bearing styles.

Pan heads provide standard general-purpose surface bearing, while truss heads offer an extra-wide diameter to prevent tear-out in sheet metal.

The broader flange of a truss head spreads mechanical clamping force across a larger surface area on light-gauge materials.

Key Concept

Screw head geometry determines whether a fastener mounts flush or sits proud on the workpiece surface, as well as how clamping load is distributed across different material types.

Estimated Time:1m 30s

Question 2644Question →

In a mechanical shop, a technician selects a machine screw marked with the specification $1/4\text{-}20 \times 1\text{-}1/2\text{ in.}$ What does the number $20$ represent in this fastener designation?

The recommended screwdriver tip size required to install the screw

The tensile strength rating grade of the screw material

The number of threads per inch along the fastener shank

The length of the threaded portion measured in sixteenths of an inch

Show answer & explanation

Answer: The number of threads per inch along the fastener shank

Answer

The number 20 represents the thread pitch expressed as the number of threads per inch (TPI) along the fastener shank.

In standard imperial fastener notation (Unified National series), specifications follow the pattern [Nominal Diameter]-[Threads Per Inch]

\times

[Length]. In the designation

1/4\text{-}20 \times 1\text{-}1/2\text{ in.}

, the number 20 indicates that the screw has 20 threads per linear inch (TPI), classifying it as Unified National Coarse (UNC).

Step-by-Step Solution

1

Analyze the standard Unified National imperial thread designation format.

The standard format is structured as: [Nominal Diameter] - [Threads Per Inch]

\times

[Fastener Length].

Imperial machine screws and bolts use this standardized notation to define thread geometry and physical sizing.

2

Map the specific values from

1/4\text{-}20 \times 1\text{-}1/2\text{ in.}

to the standard notation components.

1/4\text{ in.}

is the nominal major diameter,

20

is the threads per inch (TPI, indicating Unified National Coarse / UNC), and

1\text{-}1/2\text{ in.}

is the overall length.

The second parameter specifically identifies thread density per linear inch.

Key Concept

Unified National machine screw thread specification notation

Estimated Time:1m 0s

Question 2645Question →

A community center is repainting the inside of an empty rectangular swimming pool. The pool is $25$ meters long, $10$ meters wide, and has a uniform depth of $2$ meters. If the entire bottom and all four interior walls are to be painted, what is the total area, in square meters, that needs to be painted?

Show answer & explanation

Answer: 390

Answer

390

The correct total area is found by adding the area of the bottom (

250

m

^2

) and the areas of the four vertical walls (

100

m

^2

and

40

m

^2

). Because it is a swimming pool, there is no top to paint, making the total area

390

square meters.

Step-by-Step Solution

1

Determine which surfaces of the rectangular pool need to be painted.

One bottom surface and four side walls (two long and two short). The top is open.

A pool is shaped like an open-top rectangular prism, so we only need to calculate the area of 5 faces instead of 6.

2

Calculate the area of the bottom surface.

Area =

25 \times 10 = 250

square meters.

The bottom is a rectangle bounded by the length and width of the pool.

3

Calculate the combined area of the four side walls.

Long walls =

2 \times (25 \times 2) = 100

square meters. Short walls =

2 \times (10 \times 2) = 40

square meters.

There are two identical walls along the length and two identical walls along the width. The area of each wall is its base dimension multiplied by the depth.

4

Sum the areas of all the painted surfaces.

250 + 100 + 40 = 390

square meters.

The total area to be painted is the sum of the areas of the individual painted faces.

Key Concept

Surface area of an open rectangular prism

Question 2646Question →

A rectangular television screen has a horizontal width of $20\text{ inches}$ and a vertical height of $15\text{ inches}$ . What is the length of the diagonal of the screen?

5\text{ inches}

\sqrt{175}\text{ inches}

25\text{ inches}

35\text{ inches}

Show answer & explanation

Answer: $25\text{ inches}$

Answer

The correct length of the diagonal is

25\text{ inches}

.

The diagonal of the screen forms the hypotenuse of a right triangle with the width and height acting as the two legs. By applying the Pythagorean theorem (

a^2 + b^2 = c^2

), we calculate

20^2 + 15^2 = 400 + 225 = 625

. Taking the square root of 625 gives a diagonal length of 25 inches.

Step-by-Step Solution

1

Identify the shape and the relationship between the given measurements.

The width (

20\text{ inches}

) and height (

15\text{ inches}

) form the legs of a right triangle, and the diagonal represents the hypotenuse.

A rectangle's corners are right angles, so drawing a diagonal line creates two identical right triangles.

2

Apply the Pythagorean theorem (

a^2 + b^2 = c^2

) to find the missing hypotenuse.

15^2 + 20^2 = c^2

The Pythagorean theorem relates the lengths of the legs (

a

and

b

) to the hypotenuse (

c

) in any right triangle.

3

Square the lengths of the legs and add them together.

225 + 400 = 625

Following the order of operations, exponents must be evaluated before addition.

4

Take the square root of the sum to find the length of the diagonal.

\sqrt{625} = 25\text{ inches}

To solve for

c

from

c^2

, we must find the principal square root.

Key Concept

Using the Pythagorean theorem to find the hypotenuse (diagonal) of a right triangle formed within a rectangle.

Estimated Time:1m 0s

Question 2647Question →

The entrance of a camping tent is shaped like an isosceles triangle. The base of the tent resting on the ground is $16\text{ feet}$ wide, and the two slanted sides are each $17\text{ feet}$ long. What is the maximum height of the tent, in feet, from the ground to the peak?

Show answer & explanation

Answer: 15

Answer

15

By dividing the isosceles triangle in half, we form a right triangle with a hypotenuse of 17 and a base leg of 8. Applying the Pythagorean theorem (

8^2 + h^2 = 17^2

) gives

64 + h^2 = 289

, resulting in

h^2 = 225

. The square root of 225 is 15, which is the maximum height of the tent.

Step-by-Step Solution

1

Divide the base of the isosceles triangle in half to find the base of the inner right triangle.

16 \div 2 = 8\text{ feet}

The height (altitude) of an isosceles triangle bisects the base, creating a right angle necessary for the Pythagorean theorem.

2

Set up the Pythagorean theorem equation (

a^2 + b^2 = c^2

).

h^2 + 8^2 = 17^2

The vertical height (

h

) and half of the base (

8

) are the legs of the right triangle, while the slanted side (

17

) is the hypotenuse.

3

Square the known values.

h^2 + 64 = 289

Simplifying the terms makes it possible to solve for the unknown.

4

Subtract 64 from both sides to isolate

h^2

.

h^2 = 225

Standard algebraic process for solving a one-variable equation.

5

Take the square root of both sides to find

h

.

h = 15

The inverse operation of squaring gives the final linear dimension of the height.

Key Concept

Using the properties of isosceles triangles alongside the Pythagorean theorem to find an unknown altitude

Question 2648Question →

When preparing to drive a wood screw into dense hardwood, what specific diameter of the screw's threaded portion should the pilot hole drill bit match to prevent wood splitting while allowing the threads to achieve maximum holding power?

Show answer & explanation

Answer: root diameter; minor diameter; core diameter; root; minor; core

Answer

Root diameter (or minor diameter / core diameter)

The pilot hole drilled into the receiving piece of wood should match the root diameter (also called the minor or core diameter) of the screw's threaded section. This removes material corresponding to the solid metal body of the fastener, preventing hardwood from splitting, while leaving all surrounding wood intact for the threads to bite into and achieve maximum withdrawal resistance.

Step-by-Step Solution

1

Identify the two distinct diameters present across the threaded portion of a standard screw.

The major diameter (outermost diameter measured across the crests of the threads) and the root or minor diameter (innermost diameter measured across the solid central core at the base of the thread grooves).

Understanding screw thread anatomy is essential to sizing pilot and clearance holes correctly.

2

Determine the functional purpose of drilling a pilot hole in dense materials like hardwood.

The pilot hole relieves internal hydraulic and mechanical pressure by removing solid wood equal in volume to the solid core of the screw.

Driving a screw into dense hardwood without a pilot hole creates excessive outward radial stress that splits the workpiece or shears off the screw head.

3

Correlate pilot hole sizing to the thread engagement mechanics.

Sizing the pilot hole to the root (minor) diameter clears room for the central core while leaving full wood material for the outer threads to cut into and grip.

If the hole is drilled to the major diameter, the threads cannot engage any wood; if drilled smaller than the root diameter in hardwood, the core will bind and split the wood.

Key Concept

Pilot hole sizing relative to screw root (minor) diameter versus major diameter

Estimated Time:1m 0s

Question 2649Question →

A technician is servicing marine hardware and needs to install a cross-recess screw that features sharp, $45^\circ$ flutes meeting at a pointed apex rather than a blunt tip. Which type of screw drive is specifically designed with this geometry so that one screwdriver bit fits all standard screw sizes?

Phillips

Frearson (Reed and Prince)

Torx

Robertson

Show answer & explanation

Answer: Frearson (Reed and Prince)

Answer

Frearson (Reed and Prince)

The Frearson (also known as Reed and Prince) screw drive features a four-point cruciform recess with sharp, true

45^\circ

flutes meeting at a sharp point. Because the angle is identical regardless of fastener diameter, a single Frearson screwdriver bit can drive all standard screw sizes while delivering high torque resistance without camming out.

Step-by-Step Solution

1

Analyze the drive recess geometry described in the scenario

The screw features a cross-recess with sharp

45^\circ

flutes that form a pointed V-shape at the bottom rather than a blunt center.

Cruciform screw drives differ primarily in flute angle, corner radius, and center depth.

2

Match the identified mechanical characteristics to the appropriate drive standard

The Frearson (or Reed and Prince) design utilizes a true

45^\circ

angle across all sizes, enabling one standard bit to fit any Frearson fastener.

Unlike Phillips drives, which require separate driver sizes (#1 through #4) and have rounded corners designed to cam out, Frearson slots remain consistent in angle.

Key Concept

Cross-recess screw drive identification and geometry (Frearson vs. Phillips)

Alternative Method

Visually inspect the center of the cross: a sharp, deep point indicates a Frearson recess, whereas a blunt, rounded center indicates a Phillips recess.

Estimated Time:45s

Question 2650Question →

A shop technician must select the appropriate specialized screw for various fabrication and installation tasks. Match each specialized screw type on the left with its defining design feature or application on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Self-drilling screw matches with the integrated drill-bit point that bores its own pilot hole in sheet metal; Bugle-head drywall screw matches with the smooth, concave countersink transition designed to depress surface paper without tearing; Deck screw matches with the fluted cutting tip and corrosion-resistant coating for exterior lumber; Thread-rolling screw matches with the lobed, non-circular shank that cold-forms threads without creating chips.

Each screw type is engineered for a specific substrate and mechanical challenge: self-drilling screws integrate a fluted drill point to eliminate pilot hole drilling in sheet metal; bugle heads distribute clamping force across drywall paper to prevent punch-through; deck screws provide weather resistance and anti-split tips for treated wood; and thread-rolling screws plastically deform metal to form tight, chip-free internal threads.

Step-by-Step Solution

1

Identify the distinct tip design of self-drilling screws

Correlate the Tek screw to the integrated drill-bit point used for penetrating sheet metal without pre-drilling.

Self-drilling screws eliminate the need for a separate pilot drill bit by combining drilling and tapping in one fastener.

2

Analyze the head geometry of the bugle-head drywall fastener

Connect the bugle head to the curved countersink that depresses gypsum paper facing without tearing it.

Standard flat-head countersunk screws have sharp edges that slice through paper facing, whereas bugle profiles compress the paper evenly.

3

Evaluate exterior wood fastening characteristics for deck screws

Match deck screws with corrosion-resistant coatings and fluted anti-splitting tips.

Pressure-treated lumber contains copper compounds that corrode untreated steel, and outdoor decking requires fluted points to avoid board splitting.

4

Differentiate thread-forming/rolling mechanisms from thread-cutting fasteners

Pair thread-rolling screws with the lobed shank cross-section that cold-forms threads without generating chips.

Unlike thread-cutting screws that remove material, thread-rolling screws plastically displace ductile metal, eliminating loose swarf in electronic or mechanical assemblies.

Key Concept

Specialized screw point geometries, head profiles, and thread-forming mechanisms

Estimated Time:1m 0s

Question 2651Question →

A right-angled triangular warning sign is being designed for a factory floor. The longest side of the sign measures $17\text{ inches}$ , and one of its perpendicular sides measures $8\text{ inches}$ . What is the total area of the front face of the sign?

36\text{ square inches}

40\text{ square inches}

60\text{ square inches}

68\text{ square inches}

Show answer & explanation

Answer: $60\text{ square inches}$

Answer

60\text{ square inches}

To find the area of a right triangle, the lengths of both perpendicular legs must be known. Using the Pythagorean theorem, the missing leg is calculated as

15\text{ inches}

(since

\sqrt{17^2 - 8^2} = 15

). The area is then found by multiplying one half by the product of the two legs (

0.5 \times 8 \times 15 = 60

).

Step-by-Step Solution

1

Identify the given sides of the right triangle.

The hypotenuse (longest side,

c

) is

17\text{ inches}

and one leg (

a

) is

8\text{ inches}

.

The area formula for a right triangle requires the lengths of both perpendicular legs, not the hypotenuse.

2

Use the Pythagorean theorem (

a^2 + b^2 = c^2

) to find the missing leg (

b

).

8^2 + b^2 = 17^2

, which simplifies to

64 + b^2 = 289

. Solving for

b^2

gives

225

, so

b = 15\text{ inches}

.

Both perpendicular legs are needed to establish the base and height of the triangle.

3

Calculate the area of the triangle using the formula Area =

0.5 \times \text{base} \times \text{height}

.

Area =

0.5 \times 8 \times 15 = 60\text{ square inches}

.

The two legs of a right triangle serve as its base and height.

Key Concept

Pythagorean Theorem and Triangle Area

Question 2652Question →

Fill in the blanks with the correct numerical values based on the geometric properties described.

Fill in the blanks below

A city park features a right-angled triangular garden. The longest side of the garden measures

34\text{ yards}

, and one of the shorter sides is

16\text{ yards}

. The length of the other shorter side is yards, and the area of the garden is square yards.

Show answer & explanation

Answer

The missing shorter side is 30 yards, and the area of the garden is 240 square yards.

By applying the Pythagorean theorem (

a^2 + b^2 = c^2

) with a hypotenuse of

34

and a leg of

16

, the missing leg is

\sqrt{34^2 - 16^2} = \sqrt{1156 - 256} = \sqrt{900} = 30\text{ yards}

. The area is found by multiplying half of the base by the height:

\frac{1}{2} \times 16 \times 30 = 240\text{ square yards}

.

Step-by-Step Solution

1

Identify the given sides in relation to a right-angled triangle.

The hypotenuse (longest side)

c = 34\text{ yards}

, and one leg

a = 16\text{ yards}

. The other leg

b

is unknown.

Correctly identifying the hypotenuse is necessary to set up the Pythagorean theorem.

2

Use the Pythagorean theorem to find the length of the unknown leg.

a^2 + b^2 = c^2 \Rightarrow 16^2 + b^2 = 34^2 \Rightarrow 256 + b^2 = 1156 \Rightarrow b^2 = 900 \Rightarrow b = 30

.

The Pythagorean theorem relates the lengths of all three sides in a right triangle.

3

Calculate the area of the triangular garden.

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 16 \times 30 = 240

.

The area of a right triangle is half the product of the lengths of its two perpendicular legs.

Key Concept

Using the Pythagorean theorem to find a missing leg, and applying the triangle area formula.

Question 2653Question →

When fastening softwoods such as pine or cedar in construction and shop framing, why are coarse-thread screws preferred over fine-thread screws?

They prevent splitting by allowing dense hardwoods to compress around the shank without drilling a pilot hole.

They have wider, deeper threads that bite securely into porous wood fibers to maximize holding power.

They prevent screwdriver slippage by matching the tapered cross-slot geometry of common drivers.

They indicate a higher structural hardness rating and tensile strength certification for heavy loads.

Show answer & explanation

Answer: They have wider, deeper threads that bite securely into porous wood fibers to maximize holding power.

Answer

Coarse-thread screws have wider, deeper threads that bite securely into porous wood fibers to maximize holding power.

Softwoods have low density and porous grain structures. Coarse-thread screws possess a larger thread pitch and deeper thread valleys, allowing more wood fibers to sit between the threads. This creates substantial mechanical interlock and high withdrawal resistance without stripping out the softwood.

Step-by-Step Solution

1

Analyze the workpiece material properties

Softwoods (such as pine, fir, and cedar) have a less dense, fibrous, and porous grain structure compared to hardwoods or metals.

Fastener thread design must match the physical resistance and density of the base material.

2

Evaluate the mechanical interaction of coarse versus fine screw threads

Coarse threads have fewer threads per inch and deeper flutes, packing more wood fiber between thread crests. Fine threads have shallower cuts that easily strip the soft grain under torque.

Deeper thread engagement in soft fibers delivers superior withdrawal resistance and prevents spinning.

3

Identify the primary functional reason for selecting coarse-thread screws in softwoods

Coarse threads maximize grip and holding strength in softer woods by biting deeply between the loose grain fibers.

Matching thread pitch to material density ensures a rigid joint without stripping the hole.

Key Concept

Screw thread pitch selection based on material density and grain structure

Estimated Time:1m 0s

Question 2654Question →

A mechanic is securing a suspension component using a threaded stud, a castellated nut, and a cotter pin. Place the following installation and locking steps in the correct chronological sequence from first to last.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct sequence is: (1) Thread and torque the castellated nut to the initial specification, (2) Inspect the alignment between the nut slots and the stud hole, (3) Tighten slightly further to align the nearest slot if needed, and (4) Insert the cotter pin and bend the prongs to lock the fastener.

Proper installation of a castellated nut requires seating the joint to base torque first, checking hole-to-slot alignment, advancing the nut only in the tightening direction to achieve alignment, and finally installing and spreading the cotter pin prongs to positively lock the assembly.

Step-by-Step Solution

1

Torque the castellated nut to the lower end of the specified torque range.

The joint is properly seated under clamping force.

Fasteners must achieve sufficient clamping load before locking devices are installed.

2

Inspect the position of the nut slots relative to the drilled hole in the bolt or stud.

Determine if direct alignment exists or if further rotation is required.

A cotter pin cannot pass through unless a slot and the hole form an unobstructed passageway.

3

Rotate the nut clockwise slightly to bring the nearest slot into alignment with the hole.

The slot aligns with the hole while maintaining or slightly increasing proper torque.

Shop safety rules dictate that castellated nuts should never be loosened to align cotter pin holes, as loosening compromises joint preload.

4

Insert the cotter pin completely through the assembly and bend one prong over the end of the stud and the other around the side of the nut.

The nut is positively locked against rotational movement.

Positive mechanical locking prevents fastener loosening caused by vibration and dynamic operational loads.

Key Concept

Proper installation procedure and positive mechanical locking of castellated nuts using cotter pins.

Estimated Time:1m 0s

Question 2655Question →

A naval patrol vessel leaves port and sails due South for $10\text{ nautical miles}$ . It then changes course and sails due West. After completing this leg of the journey, the vessel's radar indicates it is exactly $26\text{ nautical miles}$ in a straight line from the port. How many nautical miles did the vessel travel due West?

16

24

36

576

Show answer & explanation

Answer: $24$

Answer

The vessel traveled

24

nautical miles due West.

The path of the vessel forms a right triangle. The straight-line distance from the port is the hypotenuse (

26\text{ nautical miles}

), and the southward distance is one leg (

10\text{ nautical miles}

). Using the Pythagorean theorem (

a^2 + b^2 = c^2

), we substitute the known values:

10^2 + b^2 = 26^2

. This simplifies to

100 + b^2 = 676

. Subtracting

100

from both sides gives

b^2 = 576

. Taking the square root of

576

yields

24\text{ nautical miles}

.

Step-by-Step Solution

1

Identify that the vessel's path forms a right triangle.

The southward path (

10

) and westward path (

x

) are the legs, and the straight-line distance (

26

) is the hypotenuse.

South and West directions are perpendicular, creating a 90-degree angle.

2

Set up the Pythagorean theorem equation (

a^2 + b^2 = c^2

).

10^2 + x^2 = 26^2

This formula relates the lengths of the legs to the hypotenuse in a right triangle.

3

Square the known values.

100 + x^2 = 676

To begin isolating the unknown variable

x

.

4

Subtract

100

from both sides of the equation.

x^2 = 576

To completely isolate the

x^2

term.

5

Take the square root of both sides.

x = 24

To find the final distance traveled due West.

Key Concept

Applying the Pythagorean theorem to find the missing leg of a right triangle.

Estimated Time:1m 0s

Question 2656Question →

The floor of a rectangular storage room has a diagonal measurement of $39$ feet and a width of $15$ feet. What is the total area of the floor in square feet?

Show answer & explanation

Answer: 540

Answer

540

The storage room floor forms a right triangle with the 39-foot diagonal as the hypotenuse and the 15-foot width as a leg. The missing length is calculated to be 36 feet using the Pythagorean theorem. Multiplying this 36-foot length by the 15-foot width yields a total area of 540 square feet.

Step-by-Step Solution

1

Identify the right triangle formed by the floor dimensions

A right triangle is formed with the diagonal (39 feet) as the hypotenuse and the width (15 feet) as one leg.

The diagonal of any rectangle splits it into two identical right triangles.

2

Find the missing length of the floor

The length is 36 feet.

Using the Pythagorean theorem (

a^2 + b^2 = c^2

),

15^2 + b^2 = 39^2

. This simplifies to

225 + b^2 = 1521

, so

b^2 = 1296

and

b = 36

. Alternatively, recognize the 5-12-13 Pythagorean triple multiplied by 3.

3

Calculate the area of the rectangle

Area = 540 square feet.

The area of a rectangle is its length multiplied by its width (

36 \times 15 = 540

).

Key Concept

Using the Pythagorean theorem to find unknown dimensions for area calculations.

Alternative Method

Instead of calculating

39^2

and

15^2

manually, recognize that the given sides are multiples of the common 5-12-13 Pythagorean triple (

3 \times 5 = 15

and

3 \times 13 = 39

). This immediately gives the missing length as

3 \times 12 = 36

.

Estimated Time:1m 0s

Question 2657Question →

An automotive technician installing an alternator mounting bracket consults the service manual, which specifies an $\text{M}10 \times 1.5 \times 40$ hex cap screw. What does the value $1.5$ represent in this standard metric fastener callout?

The total number of threads per inch along the bolt shank

The distance in millimeters between adjacent thread crests

The tensile strength property class rating stamped on the fastener

The nominal diameter of the unthreaded shank in centimeters

Show answer & explanation

Answer: The distance in millimeters between adjacent thread crests

Answer

The value 1.5 represents the thread pitch, defined as the distance in millimeters between adjacent thread crests.

In standard metric fastener notation (M10 x 1.5 x 40), the letter 'M' indicates metric ISO thread profile, the first number (10) represents the nominal major diameter in millimeters, the second number (1.5) represents the thread pitch in millimeters between adjacent crests, and the final number (40) is the length of the bolt in millimeters.

Step-by-Step Solution

1

Deconstruct the metric fastener callout notation M10 x 1.5 x 40 into its standard components.

'M' indicates metric ISO thread; '10' is nominal major diameter in millimeters; '1.5' is thread pitch; '40' is fastener shank length in millimeters.

Metric fastener specifications follow the standard format: Metric Designation + Major Diameter (mm) x Thread Pitch (mm) x Length (mm).

2

Determine the physical definition of thread pitch in metric fastener standards.

A thread pitch of 1.5 specifies that there is a 1.5 mm linear distance from the peak of one thread to the peak of the adjacent thread.

Unlike Imperial threads that count threads per inch (TPI), metric fasteners directly measure the axial distance between individual crests.

Key Concept

Metric fastener callout structure and thread pitch specification

Estimated Time:1m 0s

Question 2658Question →

A canvas sail is shaped like a right triangle. The longest side of the sail measures $15\text{ feet}$ , and the bottom edge measures $9\text{ feet}$ . What is the total area of the canvas sail?

54\text{ square feet}

67.5\text{ square feet}

108\text{ square feet}

27\text{ square feet}

Show answer & explanation

Answer: $54\text{ square feet}$

Answer

54\text{ square feet}

The area of a right triangle is half the product of its legs. First, the Pythagorean theorem (

a^2 + b^2 = c^2

) must be used to find the missing perpendicular height (the second leg). The longest side is the hypotenuse (

15

), and the bottom edge is a leg (

9

). Solving for the missing height gives

\sqrt{15^2 - 9^2} = \sqrt{225 - 81} = \sqrt{144} = 12\text{ feet}

. The area is then found by using the formula

\frac{1}{2} \times \text{base} \times \text{height}

, which equals

\frac{1}{2} \times 9 \times 12 = 54\text{ square feet}

.

Step-by-Step Solution

1

Identify the known sides of the right triangle.

The hypotenuse (longest side) is

15\text{ feet}

and the base (one leg) is

9\text{ feet}

.

The Pythagorean theorem requires knowing two sides of a right triangle to find the third.

2

Use the Pythagorean theorem (

a^2 + b^2 = c^2

) to find the missing height (the other leg).

Substituting the known values gives

9^2 + b^2 = 15^2

, which simplifies to

81 + b^2 = 225

. Solving for

b^2

gives

144

, meaning the height is

\sqrt{144} = 12\text{ feet}

.

The height of the right triangle is necessary to calculate its total area.

3

Calculate the area of the right triangle.

Area =

\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 9 \times 12 = 54\text{ square feet}

.

The standard triangle area formula provides the final answer.

Key Concept

Using the Pythagorean theorem to find a missing side length before calculating the area of a right triangle.

Question 2659Question →

A mechanic is securing two steel plates together using a through-bolt, two flat washers, a split-ring lock washer, and a standard hex nut. In what sequential order should these hardware components and materials be assembled from the bolt head side to the nut side?

Drag items to arrange them in the correct order

Show answer & explanation

Answer

First flat washer under bolt head, passed through the joined plates, followed by the second flat washer, the split-ring lock washer, and finally the hex nut.

In standard mechanical assembly, a through-bolted joint follows a specific hardware order: the bolt head holds a flat washer to distribute pressure; the bolt passes through the workpiece; the opposite side receives a second flat washer for load distribution, followed by the split-ring lock washer to maintain tension, and finishes with the hex nut.

Step-by-Step Solution

1

Place the initial flat washer onto the bolt shank up to the bolt head.

Protects the front workpiece surface and distributes clamping force under the head.

Prevents head embedment into the material during tightening.

2

Insert the bolt assembly through the workpiece clearance holes.

Hardware spans the joint plates.

Aligns and joins the two structural components.

3

Slide the second flat washer over the threaded end against the workpiece.

Provides a smooth, wide bearing surface on the nut side.

Prevents the lock washer from digging into the softer workpiece material.

4

Slide the split-ring lock washer over the threads against the flat washer.

Prepares spring-tension retention.

Maintains constant tension against the nut to resist vibrational loosening.

5

Thread and torque the hex nut onto the bolt shank.

Compresses the lock washer and clamps the joint.

Develops the required clamping force across the entire fastener stack.

Key Concept

Through-bolt fastener assembly stack-up sequence and washer placement rules

Estimated Time:1m 0s

Question 2660Question →

Read the illustrative examples provided in each sentence to determine the meaning of the underlined word. Match each vocabulary word to its contextual definition. Which pairings correctly connect each term to its intended meaning?

Click a left item, then click its matching right item

Show answer & explanation

Answer

The term 'quandary' pairs with 'A perplexing dilemma or state of uncertainty between challenging alternatives'; 'deleterious' pairs with 'Producing harmful, damaging, or destructive consequences'; 'alleviate' pairs with 'To lessen, reduce, or relieve the severity of pain or distress'; and 'rudimentary' pairs with 'Basic, primitive, or developed to an early and unsophisticated level'.

Each match is directly supported by the explicit examples provided in the stem sentences: deciding between competing operational risks demonstrates that a 'quandary' is a dilemma; water contamination and habitat damage confirm that 'deleterious' means harmful; supportive first-aid actions establish that 'alleviate' means to reduce pain; and simple hand mirrors and horns show that 'rudimentary' means basic or primitive.

Step-by-Step Solution

1

Locate signal phrases introducing examples in each sentence.

Identified 'such as' in the first sentence, 'including' in the second, 'for example' in the third, and 'like' in the fourth.

Example signal words directly point to concrete illustrations of the target vocabulary terms.

2

Analyze the listed examples to deduce the category and functional meaning of each target word.

The trade-off between grounding and flying denotes a dilemma; ecosystem harms indicate destructive effects; cold compresses and splints indicate symptom relief; hand mirrors and lanterns indicate primitive equipment.

Synthesizing specific examples reveals the shared conceptual meaning of the unknown word.

3

Match each analyzed term to its corresponding definition on the right.

Matched 'quandary' to dilemma, 'deleterious' to harmful, 'alleviate' to relieve, and 'rudimentary' to primitive/basic.

Ensures complete alignment between contextual evidence and definitions.

Key Concept

Using signal words such as 'like', 'such as', 'for example', and 'including' to extract the shared traits of listed examples and define target vocabulary in context.

Estimated Time:1m 0s

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