All practice questions

Question 321Question →

A manual meat grinder is operated by turning a crank handle. The crank arm has a length of $8.0 \text{ inches}$ (acting as the radius of the wheel). The internal grinding shaft, which acts as the axle, has a diameter of $1.6 \text{ inches}$ . If a user applies a steady perpendicular force of $15 \text{ pounds}$ to the crank handle, what is the resulting theoretical output force (in pounds) exerted at the outer surface of the grinding shaft? Assume $100\%$ efficiency.

Show answer & explanation

Answer: 150

Answer

150

The machine multiplies the input force based on the ratio of the crank arm radius to the grinding shaft radius. Converting the

1.6 \text{ inch}

axle diameter to a

0.8 \text{ inch}

radius gives an Ideal Mechanical Advantage (IMA) of

10

(

8.0 \div 0.8

). Multiplying the

15 \text{ pound}

input force by this IMA results in a

150 \text{ pound}

theoretical output force.

Step-by-Step Solution

1

Determine the radius of the axle.

Axle radius =

0.8 \text{ inches}

The Ideal Mechanical Advantage formula requires comparing equivalent radial measurements. Since the wheel dimension is given as a radius, converting the axle's diameter to a radius ensures units are consistent.

2

Calculate the Ideal Mechanical Advantage (IMA).

IMA = 10

Dividing the wheel radius (

8.0 \text{ inches}

) by the axle radius (

0.8 \text{ inches}

) determines the mechanical advantage multiplier of the machine.

3

Calculate the theoretical output force.

Output Force =

150 \text{ pounds}

Multiplying the input force (

15 \text{ pounds}

) by the mechanical advantage (

10

) yields the final output force exerted at the axle surface.

Key Concept

The Ideal Mechanical Advantage (IMA) of a wheel and axle is the ratio of the wheel's radius to the axle's radius. The resulting output force is equal to the input force multiplied by this mechanical advantage.

Question 322Question →

A rapid transit commuter train leaves a station, accelerating uniformly along a straight track. It reaches a cruising velocity of $27.0 \text{ m/s}$ from a complete stop. If the acceleration phase lasts exactly $0.75 \text{ minutes}$ , what is the train's average acceleration in $\text{m/s}^2$ ?

Show answer & explanation

Answer: 0.6

Answer

The train's average acceleration is

0.6 \text{ m/s}^2

.

Acceleration is defined as the change in velocity divided by the time it takes for that change to occur. By converting the time unit to seconds and substituting the initial velocity, final velocity, and time into the standard acceleration formula, the correct magnitude of

0.6 \text{ m/s}^2

is determined.

Step-by-Step Solution

1

Identify the known kinematic variables.

Initial velocity

v_i = 0 \text{ m/s}

, final velocity

v_f = 27.0 \text{ m/s}

, and time

t = 0.75 \text{ minutes}

.

Extracting the given values is necessary before selecting the appropriate formula.

2

Convert the time from minutes to standard SI units (seconds).

t = 0.75 \times 60 = 45 \text{ seconds}

.

Acceleration in

\text{m/s}^2

requires the time unit to match the time component of the velocity unit (seconds).

3

Calculate the average acceleration using the formula

a = \frac{v_f - v_i}{t}

.

a = \frac{27.0 - 0}{45} = 0.6 \text{ m/s}^2

.

This formula directly relates the change in an object's velocity to the time interval over which the change occurred.

Key Concept

Kinematics: Uniform Acceleration and Unit Conversion

Question 323Question →

An industrial sheet metal shear operates as a second-class lever. The pivot point (fulcrum) is located at the far end of the cutting arm. A mechanic applies a downward effort force of $150 \text{ N}$ at the handle, which is located $80 \text{ cm}$ from the fulcrum. The metal sheet is positioned $16 \text{ cm}$ from the fulcrum. Assuming the lever is rigid and frictionless, what is the downward cutting force, in newtons, applied to the metal sheet?

Show answer & explanation

Answer: 750

Answer

750

A second-class lever always places the resistance between the effort and the fulcrum, providing a mechanical advantage greater than 1. The torque applied by the effort (

150 \text{ N} \times 80 \text{ cm}

) must equal the opposing torque exerted on the resistance (

F_r \times 16 \text{ cm}

). Solving this equilibrium equation yields a cutting force of

750 \text{ N}

.

Step-by-Step Solution

1

Identify the lever parameters and equation

F_e = 150 \text{ N}

,

d_e = 80 \text{ cm}

,

d_r = 16 \text{ cm}

Establishing the values needed for the principle of moments (torque balance).

2

Apply the law of the lever

150 \times 80 = F_r \times 16

In equilibrium, the input torque equals the output torque.

3

Calculate the unknown resistance force

F_r = \frac{12000}{16} = 750 \text{ N}

Isolating

F_r

gives the mechanical force applied to the metal.

Key Concept

Mechanical Advantage and Torque Balance in Second-Class Levers

Estimated Time:1m 0s

Question 324Question →

The circumference of a circular stained glass window is $18\pi$ inches. What is the area of the window in square inches?

(Enter only the numerical value that multiplies $\pi$ . For example, if the area is $10\pi$ , enter 10.)

Show answer & explanation

Answer: 81

Answer

81

The correct answer is 81. First, the radius is found by dividing the circumference (

18\pi

) by

2\pi

, which gives a radius of 9 inches. Then, the area is calculated by squaring this radius and multiplying by

\pi

(

9^2 = 81

), resulting in a total area of

81\pi

. Since the prompt asks for just the numerical value multiplying

\pi

, the final response is 81.

Step-by-Step Solution

1

Find the radius of the circular window using the circumference formula

C = 2\pi r

.

18\pi = 2\pi r \Rightarrow r = 9

inches.

The area formula requires the radius, so we must work backward from the given circumference to find it.

2

Calculate the area of the window using the area formula

A = \pi r^2

.

A = \pi(9)^2 = 81\pi

square inches.

Squaring the radius and multiplying by

\pi

yields the total area of the circle.

3

Identify the requested numerical value.

81

The question specifically asks for the numerical coefficient that multiplies

\pi

.

Key Concept

Calculating the area of a circle by first determining the radius from the circumference.

Question 325Question →

Two angles are supplementary. The measure of the larger angle is $24^\circ$ more than three times the measure of the smaller angle. What is the measure, in degrees, of the larger angle?

Show answer & explanation

Answer: 141

Answer

141

Supplementary angles have a total sum of

180^\circ

. By letting the smaller angle equal

x

, the larger angle becomes

3x + 24

. Adding these gives

x + (3x + 24) = 180

. Simplifying this results in

4x = 156

, so

x = 39

. Since the question asks for the larger angle, we must substitute

x = 39

into our expression for the larger angle:

3(39) + 24 = 117 + 24 = 141

. Therefore, the larger angle measures

141^\circ

.

Step-by-Step Solution

1

Assign algebraic expressions to the unknown angles.

Smaller angle =

x

, Larger angle =

3x + 24

This translates the given text relationships into mathematical variables that can be used to set up an equation.

2

Set up an equation based on the geometric property of supplementary angles.

x + (3x + 24) = 180

Supplementary angles, by definition, always add up to exactly 180 degrees.

3

Simplify and solve the linear equation for

x

.

4x = 156

, meaning

x = 39

Isolating

x

provides the measure of the smaller angle.

4

Calculate the measure of the larger angle.

3(39) + 24 = 141

The question specifically asks for the measure of the larger angle, so we must plug

x

back into the original expression.

Key Concept

Solving algebraic equations involving supplementary angles

Question 326Question →

A heavy equipment transporter has a maximum fuel capacity of $250$ gallons and operates with a fuel efficiency of $4$ miles per gallon. The vehicle is dispatched to deliver cargo to a remote site and immediately return to its base along the same route. Military regulations require the transporter to arrive back at base with a minimum of $50$ gallons of fuel in reserve. What is the maximum one-way distance, in miles, the transporter can travel to the remote site without refueling?

Show answer & explanation

Answer: 400

Answer

The maximum one-way distance the transporter can travel is 400 miles.

The correct answer is derived by first identifying the usable fuel (250 - 50 = 200 gallons). Next, the total round-trip distance is calculated by multiplying the usable fuel by the fuel efficiency (200 gallons * 4 miles/gallon = 800 miles). Finally, since the vehicle must travel to the site and return, the total range is divided by 2, resulting in a maximum one-way distance of 400 miles.

Step-by-Step Solution

1

Determine the amount of usable fuel available for the mission.

200 gallons

The transporter has a 250-gallon tank but must retain a 50-gallon reserve upon return, meaning only 200 gallons can be consumed during the trip.

2

Calculate the total distance the vehicle can travel using the usable fuel.

800 miles

Multiplying the 200 gallons of usable fuel by the vehicle's fuel efficiency of 4 miles per gallon gives the total range.

3

Determine the maximum one-way distance.

400 miles

Because the mission is a round trip along the same route, the total range must cover the distance to the site and back. Therefore, the one-way distance is half of the total range.

Key Concept

Calculating one-way distance using fuel efficiency, total fuel capacity, required reserves, and accounting for round-trip constraints.

Question 327Question →

A technician uses a pair of long industrial tweezers to securely hold a sensitive electronic component. The tweezers pivot at the joined top end and have a total length of $40 \text{ cm}$ from the pivot to the gripping tips. The technician's fingers apply pressure at a point $8 \text{ cm}$ from the pivot. If the component requires a holding force of $50 \text{ N}$ at the tips, what effort force (in Newtons) must the technician apply?

Show answer & explanation

Answer: 250

Answer

The technician must apply an effort force of 250 N.

In a third-class lever like tweezers, the effort is applied between the fulcrum and the load. Since the effort arm (

8 \text{ cm}

) is shorter than the load arm (

40 \text{ cm}

), the mechanical advantage is less than 1. This means a much larger effort force is required to generate a smaller load force at the tips. Using the torque balance equation

F_e \times d_e = F_l \times d_l

, we get

F_e \times 8 = 50 \times 40

. Solving this gives

250 \text{ N}

.

Step-by-Step Solution

1

Identify the lever configuration and its parameters.

The tweezers are a third-class lever (fulcrum at one end, effort in the middle). Effort arm

d_e = 8 \text{ cm}

, load arm

d_l = 40 \text{ cm}

, load force

F_l = 50 \text{ N}

.

Correctly identifying the lever class ensures the correct distances are used relative to the fulcrum.

2

Set up the torque equilibrium equation.

F_e \times d_e = F_l \times d_l

For the lever to remain stable and provide the required grip, the effort torque must balance the load torque.

3

Calculate the required effort force.

F_e \times 8 = 50 \times 40 \Rightarrow 8 F_e = 2000 \Rightarrow F_e = 250 \text{ N}

To find the unknown variable by substituting the given values.

Key Concept

Torque Balance in Third-Class Levers

Question 328Question →

A flat metal washer is manufactured by stamping a circular hole with a diameter of $4 \text{ mm}$ out of the exact center of a solid circular disk with a diameter of $12 \text{ mm}$ . What is the area of the remaining flat face of the washer in square millimeters? (Enter only the numerical coefficient of $\pi$ in your answer. For example, if the area is $50\pi$ , enter 50).

Show answer & explanation

Answer: 32

Answer

32

The area of the remaining metal washer face is found by calculating the area of the large outer circle (

36\pi

) and subtracting the area of the smaller inner circle hole (

4\pi

), which leaves a remaining area of

32\pi

. Since the question asks for the coefficient of

\pi

, the correct numerical entry is 32.

Step-by-Step Solution

1

Calculate the radius of the outer disk.

R = 12 \div 2 = 6 \text{ mm}

The area formula for a circle requires the radius, which is half of the diameter.

2

Calculate the area of the outer disk.

\text{Area}_{\text{outer}} = \pi \times 6^2 = 36\pi \text{ mm}^2

This represents the total surface area before the center hole is removed.

3

Calculate the radius of the inner hole.

r = 4 \div 2 = 2 \text{ mm}

The radius of the removed section is needed to determine how much area to subtract.

4

Calculate the area of the inner hole.

\text{Area}_{\text{inner}} = \pi \times 2^2 = 4\pi \text{ mm}^2

This is the area of the metal that is stamped out.

5

Subtract the inner area from the outer area.

36\pi - 4\pi = 32\pi \text{ mm}^2

The remaining face area of the washer is the difference between the full disk and the removed hole.

6

Identify the coefficient of

\pi

.

32

The stem specifically asks for the numerical coefficient only.

Key Concept

Calculating the area of an annulus (the region between two concentric circles) by subtracting the area of the inner circle from the area of the outer circle.

Question 329Question →

In carpentry and construction, nail lengths are designated using the penny system, indicated by the lowercase letter $\text{d}$ . According to standard pennyweight sizing rules for common nails up to $10\text{d}$ , what is the total length in inches of a $6\text{d}$ (6-penny) nail?

Show answer & explanation

Answer: 2

Answer

A

6\text{d}

nail is exactly

2.0\text{ inches}

long.

For standard common, box, and finishing nails from

2\text{d}

through

10\text{d}

, the length in inches increases by

\frac{1}{4}\text{ inch}

for each penny step:

\text{Length} = \frac{d}{4} + 0.5\text{ inches}

. Substituting

d = 6

yields

\frac{6}{4} + 0.5 = 1.5 + 0.5 = 2.0\text{ inches}

.

Step-by-Step Solution

1

Identify the penny size number (

d

) given in the problem.

d = 6

The question specifies a

6\text{d}

(6-penny) nail.

2

Apply the standard formula for converting penny size (

d

) to length in inches for nails up to

10\text{d}

:

\text{Length} = \frac{d}{4} + \frac{1}{2}

.

\text{Length} = \frac{6}{4} + \frac{1}{2} = 1.5 + 0.5 = 2.0\text{ inches}

Each penny increase represents a

\frac{1}{4}\text{-inch}

increase in length starting from a

2\text{d}

base of

1\text{ inch}

.

Key Concept

Penny (

d

) size conversion to physical nail length in inches

Estimated Time:45s

Question 330Question →

A custom corner desk is shaped like a right triangle so it fits perfectly against two perpendicular walls. The sections of the desk that touch the walls measure $24\text{ inches}$ and $10\text{ inches}$ in length. What is the length of the front edge of the desk in inches?

Show answer & explanation

Answer: 26

Answer

26

Because the walls meet at a perpendicular right angle, the corner desk forms a right triangle. The edges touching the walls act as the legs (

a = 24

,

b = 10

), and the front edge is the hypotenuse (

c

). Using the Pythagorean Theorem (

a^2 + b^2 = c^2

), we square both legs to get

576

and

100

. Adding these together gives

676

. The square root of

676

is

26

, which is the exact length of the front edge in inches.

Step-by-Step Solution

1

Identify the known sides of the right triangle

Legs are

a = 24

and

b = 10

.

The walls are perpendicular, forming a

90^{\circ}

angle. Therefore, the edges touching the walls are the legs of a right triangle.

2

Set up the Pythagorean Theorem

24^2 + 10^2 = c^2

The Pythagorean Theorem (

a^2 + b^2 = c^2

) is the standard formula used to find the missing hypotenuse (

c

) when both legs are known.

3

Square the lengths of the legs

576 + 100 = c^2

Evaluating the exponents is the required next step in the order of operations.

4

Add the squared values

676 = c^2

Combining the squares of the legs gives the square of the hypotenuse.

5

Find the square root of the sum

c = 26

Taking the square root of

676

isolates

c

, revealing the true length of the desk's front edge.

Key Concept

Using the Pythagorean Theorem (

a^2 + b^2 = c^2

) to calculate the length of a hypotenuse.

Alternative Method

You can solve this rapidly without large calculations by recognizing special Pythagorean triples. This triangle is a scaled-up version of the fundamental

5-12-13

right triangle. Multiplying each side of a

5-12-13

triangle by

2

gives side lengths of

10

,

24

, and

26

. Since the legs are

10

and

24

, the hypotenuse must be

26

.

Estimated Time:45s

Question 331Question →

The circular foundation of a radar dome has a circumference of $36\pi \text{ feet}$ . What is the area of this circular foundation, in square feet?

(Enter only the numerical coefficient of $\pi$ in your answer. For example, if the area is $100\pi$ , enter 100.)

Show answer & explanation

Answer: 324

Answer

324

The circumference of a circle is calculated using

C = 2\pi r

. Setting the given circumference

36\pi = 2\pi r

gives a radius of

18 \text{ feet}

. The area is found using

A = \pi r^2

, which becomes

A = \pi(18)^2 = 324\pi \text{ square feet}

. Since the question asks for the numerical coefficient of

\pi

, the correct value to enter is 324.

Step-by-Step Solution

1

Find the radius of the circular foundation from its given circumference.

r = 18

The circumference formula is

C = 2\pi r

. Setting

2\pi r = 36\pi

allows us to solve for the radius.

2

Calculate the area of the foundation using the radius.

A = 324\pi

The area formula is

A = \pi r^2

. Substituting

r = 18

gives

A = \pi(18)^2 = 324\pi

.

3

Extract the numerical coefficient of

\pi

for the final answer.

324

The question explicitly asks to enter only the numerical coefficient of

\pi

.

Key Concept

Calculating the area of a circle by first deriving the radius from its circumference

Question 332Question →

The total surface area of a cube is $150$ square centimeters. What is the volume of the cube, in cubic centimeters?

Show answer & explanation

Answer: 125

Answer

125

The correct answer is derived by first identifying the side length of the cube. Given the total surface area is

150

square centimeters, we divide by

6

to find the area of a single face (

25

square centimeters). The square root of

25

gives a side length of

5

centimeters. Finally, cubing this side length (

5 \times 5 \times 5

) results in a total volume of

125

cubic centimeters.

Step-by-Step Solution

1

Use the total surface area to find the area of a single face of the cube.

150 \div 6 = 25

square centimeters.

A cube has exactly

6

identical square faces. Dividing the total surface area by

6

isolates the area of just one face (

s^2

).

2

Determine the length of one edge of the cube.

s = \sqrt{25} = 5

centimeters.

Because the area of a square face is the side length squared (

s^2

), taking the square root gives the side length.

3

Calculate the total volume of the cube.

V = 5^3 = 125

cubic centimeters.

The formula for the volume of a cube is the side length cubed (

V = s^3

).

Key Concept

Converting between the surface area and volume of a 3D figure by finding the missing dimension.

Estimated Time:45s

Question 333Question →

In general carpentry and light framing, a standard rule of thumb requires a fastener to penetrate into the receiving structural member by at least twice the thickness of the board being fastened. If a worker is securing a $0.75\text{-inch}$ -thick wooden board directly to framing lumber, what is the minimum total nail length, in inches, required to satisfy this standard?

Show answer & explanation

Answer: 2.25

Answer

The minimum required nail length is

2.25\text{ inches}

(

2\frac{1}{4}\text{ inches}

, equivalent to a

7\text{d}

nail).

To satisfy the standard carpentry rule of thumb that a nail must penetrate into the supporting timber at least twice the thickness of the piece being fastened, the penetration must be

2 \times 0.75\text{ in.} = 1.50\text{ inches}

. Adding the

0.75\text{-inch}

thickness of the surface board gives a minimum total fastener length of

0.75\text{ in.} + 1.50\text{ in.} = 2.25\text{ inches}

.

Step-by-Step Solution

1

Calculate the required penetration depth into the base framing lumber.

Penetration depth =

2 \times 0.75\text{ in.} = 1.50\text{ inches}

.

Standard carpentry practice dictates that the fastener must embed into the receiving member at least twice the thickness of the fastened material to ensure adequate withdrawal and shear resistance.

2

Calculate the total minimum nail length by combining the board thickness and the penetration depth.

Total minimum length =

0.75\text{ in.} + 1.50\text{ in.} = 2.25\text{ inches}

.

The total nail length must pass completely through the attached board before penetrating into the receiving member.

Key Concept

Fastener length determination using the two-thirds total length / two-times thickness penetration rule of thumb in carpentry.

Estimated Time:1m 0s

Question 334Question →

A painter leans a $25\text{-foot}$ ladder against a vertical wall. The base of the ladder is placed exactly $7\text{ feet}$ away from the base of the wall on level ground. How high up the wall, in feet, does the top of the ladder reach?

Show answer & explanation

Answer: 24

Answer

The top of the ladder reaches 24 feet up the wall.

The correct answer is derived by modeling the scenario as a right triangle. The ladder acts as the hypotenuse (

c = 25

) and the distance along the ground acts as one leg (

a = 7

). Using the Pythagorean theorem (

a^2 + b^2 = c^2

), we solve for the missing height (

b

). Substituting the values gives

7^2 + b^2 = 25^2

, which simplifies to

49 + b^2 = 625

. Subtracting 49 yields

b^2 = 576

. The square root of 576 is 24, meaning the ladder reaches 24 feet up the wall.

Step-by-Step Solution

1

Set up the right triangle relationship.

a^2 + b^2 = c^2

, where

c = 25

and

a = 7

The wall is perpendicular to the ground, forming a right triangle with the ladder acting as the hypotenuse.

2

Substitute known values into the Pythagorean theorem.

7^2 + b^2 = 25^2

We know the hypotenuse (

25

) and one leg (

7

), and we need to find the other leg (

b

).

3

Square the known numbers.

49 + b^2 = 625

Evaluating the exponents is required before isolating the variable term.

4

Isolate the variable term.

b^2 = 625 - 49 = 576

Subtracting

49

from both sides leaves

b^2

by itself.

5

Solve for the height.

b = \sqrt{576} = 24

Taking the square root of both sides gives the length of the missing side.

Key Concept

Using the Pythagorean theorem to find a missing leg of a right triangle in a real-world scenario.

Question 335Question →

A builder is fastening a $0.5\text{-inch}$ -thick plywood panel and a $0.75\text{-inch}$ -thick wooden furring strip to wall framing studs using standard $16\text{d}$ common nails ( $3.5\text{ inches}$ long). If each nail is driven straight and flush with the outer face of the furring strip, how many inches of the nail will penetrate into the framing stud?

Show answer & explanation

Answer: 2.25

Answer

2.25 inches

The nail passes through both the

0.75\text{-inch}

furring strip and the

0.5\text{-inch}

plywood panel, which take up a combined thickness of

1.25\text{ inches}

. Subtracting this

1.25\text{ inches}

from the total

3.5\text{-inch}

nail length leaves exactly

2.25\text{ inches}

of penetration into the framing stud.

Step-by-Step Solution

1

Calculate the total thickness of the layers through which the fastener passes before entering the stud.

0.75\text{ in} + 0.50\text{ in} = 1.25\text{ in}

The nail must pass completely through both the outer furring strip and the intermediate plywood panel.

2

Determine the effective length of the fastener driven flush.

3.50\text{ in}

A standard

16\text{d}

(16-penny) common nail has an overall physical length of

3.5\text{ inches}

.

3

Subtract the combined surface thickness from the total length of the nail.

3.50\text{ in} - 1.25\text{ in} = 2.25\text{ in}

The remaining portion of the nail shank embeds directly into the receiving wooden stud.

Key Concept

Fastener penetration depth and standard pennyweight length calculation

Estimated Time:1m 0s

Question 336Question →

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 337Question →

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 338Question →

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 339Question →

A city is installing a new fence around a rectangular dog park. The park has a length of $80$ feet and a width of $60$ feet. How many feet of fencing are required to completely enclose the park?

Show answer & explanation

Answer: 280

Answer

280

The fencing must cover the entire outer edge of the rectangular park. The perimeter is found by adding the length and width together, then multiplying the sum by 2 to account for all four sides. Thus, 2 × (80 + 60) = 280 feet.

Step-by-Step Solution

1

Identify the given dimensions of the rectangular park.

Length = 80 feet, Width = 60 feet.

To determine the values needed for the perimeter formula.

2

Apply the formula for the perimeter of a rectangle.

Perimeter = 2 × (80 + 60)

The total fencing required is equal to the perimeter of the rectangle.

3

Perform the addition inside the parentheses.

80 + 60 = 140

Following the order of operations.

4

Multiply the sum by 2.

2 × 140 = 280

To account for all four sides of the rectangular boundary.

Key Concept

The perimeter of a rectangle is the total distance around the outside edge, calculated using the formula 2 × (Length + Width).

Question 340Question →

If $5m + 12 = 2m + 27$ , what is the value of $m$ ?

Show answer & explanation

Answer: 5

Answer

The correct value of the variable is 5.

By using inverse operations to collect the variable terms on one side and the constant terms on the other, the equation simplifies to

3m = 15

, which directly yields a result of

5

.

Step-by-Step Solution

1

Move all terms containing the variable to one side of the equation.

3m + 12 = 27

Subtracting

2m

from both sides eliminates the variable from the right side and groups it on the left.

2

Move all constant terms to the opposite side of the equation.

3m = 15

Subtracting

12

from both sides isolates the term containing the variable.

3

Solve for the variable.

m = 5

Dividing both sides by the coefficient

3

provides the final value of

m

.

Key Concept

Solving Linear Equations with Variables on Both Sides

Estimated Time:45s

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