All practice questions

Question 2161Question →

A square metal plate has an area of $144$ square inches. A rectangular metal plate is cut so that it has the exact same width as the square plate, but its length is twice the width of the square. What is the perimeter of the rectangular plate in inches?

288

72

48

36

Show answer & explanation

Answer: 72

Answer

72

The area of the square is

144

square inches, which means each side of the square is

12

inches. The new rectangular plate has the same width as the square (

12

inches) and a length that is twice that width (

24

inches). To find the perimeter of the rectangle, add the length and width (

12 + 24 = 36

) and multiply by

2

to account for all four sides, resulting in

72

inches.

Step-by-Step Solution

1

Find the side length of the original square metal plate.

Side =

12

inches.

The area of a square is the side length squared (

s^2 = 144

), so the side is the square root of

144

.

2

Determine the dimensions of the rectangular plate.

Width =

12

inches, Length =

24

inches.

The problem states the rectangle's width equals the square's side (

12

), and its length is twice that amount (

2 \times 12 = 24

).

3

Calculate the perimeter of the rectangular plate.

Perimeter =

2 \times (12 + 24) = 72

inches.

The perimeter of a rectangle is found by adding the length and width, then multiplying by

2

.

Key Concept

Perimeter and Area of 2D Figures

Question 2162Question →

Match each electrical measurement scenario with the correct digital multimeter (DMM) lead placement, selector dial setting, and circuit connection method.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Each measurement scenario correctly aligns with its unique combination of meter jack, dial mode, and circuit connection:

12\text{ V}

DC lamp voltage drop matches parallel connection using

\text{V/}\Omega

and

\text{COM}

on

\text{DCV}

;

450\text{ mA}

relay current matches series connection using

\text{mA/}\mu\text{A}

and

\text{COM}

on

\text{DCA}

;

8.2\text{ A}

fan current matches series connection using the

10\text{ A}

jack and

\text{COM}

on

\text{DCA}

;

120\text{ V}

AC outlet voltage matches parallel connection using

\text{V/}\Omega

and

\text{COM}

on

\text{ACV}

.

The correct pairings enforce all foundational meter rules: voltage is measured across a component in parallel using the volt/ohm jack in the appropriate DC or AC dial mode, while current is measured inline in series using the appropriate current jack (milliamp jack for fractions of an amp, and the 10A terminal for high currents) to protect internal fuses and ensure accurate readings.

Step-by-Step Solution

1

Differentiate between voltage and current measurement connection rules.

Voltmeters must always be connected in parallel across components without opening the circuit, whereas ammeters must always be connected in series by breaking the circuit line.

Voltmeters possess extremely high internal impedance to prevent loading the circuit, while ammeters possess near-zero internal impedance to allow current to pass through.

2

Distinguish between direct current (DC) and alternating current (AC) dial configurations.

Automotive DC systems and DC relay lines require the DCV and DCA functions, while standard utility electrical wall receptacles supply sinusoidal alternating voltage requiring the ACV function.

Selecting the incorrect waveform mode causes inaccurate readings, meter errors, or failure to register potential.

3

Determine the proper current input jack based on expected current magnitude.

Low current (

450\text{ mA}

) is safely routed through the standard fused milliamp/microamp (mA/

\mu

A) jack, whereas high current (

8.2\text{ A}

) must be routed through the dedicated

10\text{ A}

high-current terminal.

Placing currents above

200\text{--}500\text{ mA}

into the milliamp jack will instantly blow the internal meter fuse.

Key Concept

Digital Multimeter Voltage and Current Configuration

Question 2163Question →

Match each 3D solid to its corresponding volume formula.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Rectangular Prism matches with

V = l \times w \times h

; Cube matches with

V = s^3

; Cylinder matches with

V = \pi r^2 h

; Sphere matches with

V = \frac{4}{3} \pi r^3

.

Each 3D solid has a distinct volume formula based on its geometry. Matching them correctly demonstrates fundamental knowledge required before solving applied numerical problems.

Step-by-Step Solution

1

Identify the base shape and defining dimensions for each 3D solid.

Rectangular prisms use length, width, and height. Cubes use a single side length. Cylinders use radius and height. Spheres use only radius.

Volume formulas are derived from the foundational properties and base shapes of the solids.

2

Connect the identified dimensions to the standard algebraic formulas.

All solids are successfully matched to their specific volume equations.

Accurate calculation of volume requires applying the correct formula for the specific geometric figure.

Key Concept

Volume Formulas of Basic 3D Solids

Estimated Time:45s

Question 2164Question →

Three geometric puzzle pieces must be assembled in a specific sequence based on their physical connection points. Piece A has a single protruding tab. Piece B has a slot that exactly fits Piece A's tab, and a peg on its opposite side. Piece C has a circular hole that fits Piece B's peg. Arrange the following assembly steps in the logically required physical sequence, starting with Piece A as the foundation.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct logical sequence is to place Piece A first, attach Piece B to Piece A, align Piece C over Piece B, and finally press Piece C into place.

The assembly must strictly follow the physical constraints of the connection points. Piece A provides the initial tab. Piece B must connect to Piece A's tab to integrate properly, which then positions its peg. Piece C must then connect to Piece B's peg. Attempting any other order physically fails because the necessary connection points would not be available.

Step-by-Step Solution

1

Identify the foundation piece.

Piece A is placed first as the base.

The prompt establishes Piece A as the starting component.

2

Determine the next piece that physically connects to Piece A.

Piece B is attached to Piece A.

Piece B has the slot that fits Piece A's tab, making it the only valid next step.

3

Determine the piece that connects to the newly exposed peg on Piece B.

Piece C is aligned with Piece B.

Piece C has the hole that fits Piece B's peg.

4

Finalize the overall assembly.

Piece C is pressed onto Piece B.

This physical action completes the integration of all three components.

Key Concept

Multi-Component Integration

Question 2165Question →

A motor pool dispatcher logs the number of vehicles dispatched each day from Monday to Friday. The daily totals are $11$ , $18$ , $23$ , $13$ , and $15$ . What is the mean number of vehicles dispatched per day?

Show answer & explanation

Answer: 16

Answer

16

The mean is found by adding together all the numbers in the data set and then dividing by the count of numbers. The sum of

11

,

18

,

23

,

13

, and

15

is

80

. Because there are

5

values, dividing

80

by

5

results in a mean of

16

.

Step-by-Step Solution

1

Calculate the sum of the dataset.

11 + 18 + 23 + 13 + 15 = 80

The first step in calculating the mean is finding the total sum of all values.

2

Count the number of values in the dataset.

5

days

The mean formula requires dividing the sum by the number of data points.

3

Divide the sum by the number of values.

80 \div 5 = 16

Dividing the total sum by the count of values yields the mean.

Key Concept

Calculating the arithmetic mean of a dataset

Question 2166Question →

Calculate the specified length for the given shape.

Fill in the blanks below

A student cuts a piece of cardboard into a right triangle. The horizontal base measures

15

cm and the vertical height measures

20

cm. The length of the slanted side (hypotenuse) is cm.

Show answer & explanation

Answer

25

Applying the Pythagorean theorem (

a^2 + b^2 = c^2

), we substitute the legs

a = 15

and

b = 20

. Squaring these gives

225

and

400

. Adding them yields

625

. Taking the square root of

625

gives the hypotenuse length of

25

cm. Alternatively, this can be recognized as a scaled

3-4-5

right triangle, where each side is multiplied by

5

(

3 \times 5 = 15

,

4 \times 5 = 20

, and

5 \times 5 = 25

).

Step-by-Step Solution

1

Identify the known values in the Pythagorean theorem formula,

a^2 + b^2 = c^2

.

The legs are

a = 15

and

b = 20

. The hypotenuse

c

is unknown.

The Pythagorean theorem defines the relationship between the lengths of the two legs of a right triangle and its hypotenuse.

2

Substitute the leg values into the theorem and square them.

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

. Thus,

c^2 = 625

.

This isolates the square of the unknown hypotenuse (

c^2

) on one side of the equation.

3

Take the square root of both sides to find the hypotenuse.

c = \sqrt{625} = 25

.

The square root gives the actual length of the side rather than its squared value.

Key Concept

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs (

a^2 + b^2 = c^2

).

Question 2167Question →

A cross-section of an irrigation canal has the shape of an isosceles trapezoid. The flat bottom of the canal is $10$ meters wide, the open water surface at the top is $20$ meters wide, and the vertical depth of the canal is $12$ meters. To prevent seepage, a waterproof lining is applied only to the flat bottom and the two slanted sides of the canal. What is the total length, in meters, of the canal's cross-section that receives this lining?

Show answer & explanation

Answer: 36

Answer

36

The lined portion consists of the flat bottom and the two slanted sides. By dropping perpendicular altitude lines from the top corners to the bottom plane, two identical right triangles are formed on the left and right edges. The horizontal base of each right triangle is half of the difference between the top and bottom widths:

(20 - 10) / 2 = 5

meters. The vertical height of the canal is given as

12

meters. Using the Pythagorean theorem (

a^2 + b^2 = c^2

), the length of each slanted side is

\sqrt{5^2 + 12^2} = \sqrt{169} = 13

meters. The total length of the waterproof lining is the sum of the bottom width and the two slanted sides:

10 + 13 + 13 = 36

meters.

Step-by-Step Solution

1

Calculate the horizontal difference on each side of the trapezoid.

5 meters

To form a right triangle to calculate the slanted side, we must distribute the difference between the top and bottom widths equally to both sides.

2

Apply the Pythagorean theorem to find the length of the slanted sides.

13 meters

Each slanted side acts as the hypotenuse of a right triangle with legs of 5 meters (horizontal) and 12 meters (vertical height).

3

Sum the lengths of the sections that receive the lining.

36 meters

The lining is only applied to the bottom (10 meters) and the two slanted sides (13 meters each). The top is open and excluded.

Key Concept

Calculating the partial perimeter of a composite/trapezoidal figure using the Pythagorean theorem.

Question 2168Question →

An electronics technician needs to measure the DC voltage drop across an active indicator lamp assembly using a manual-ranging digital multimeter (DMM). What is the correct chronological sequence of steps to perform this measurement safely and obtain the most accurate reading?

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The proper sequence starts with inserting the leads into the COM and V/Ω jacks, setting the dial to the DC voltage mode at the highest range, connecting the probes in parallel across the component, and finally stepping down the range setting for maximum display resolution.

Measuring voltage safely and accurately requires connecting test leads to the COM and V/Ω terminals, selecting the DC voltage mode at the highest range to prevent overload, probing across the component in parallel, and then reducing the range to gain the highest resolution reading.

Step-by-Step Solution

1

Connect the test leads to the meter input jacks (black to COM, red to V/Ω).

The meter is configured for high input impedance measurement instead of current mode.

Placing leads into current jacks by mistake would create an internal short circuit across the load.

2

Set the rotary switch to direct-current voltage (DCV) on the highest scale.

The multimeter's internal attenuation circuits are set to accept large potential differences safely.

Starting at the highest range prevents meter overload and internal surge damage if the circuit voltage is higher than anticipated.

3

Apply probe tips across the load terminals in parallel.

The meter samples the potential difference across the lamp without interrupting normal circuit operation.

Voltmeters possess extremely high internal resistance and must be connected in parallel across components.

4

Decrease the range switch setting progressively to match the observed potential difference.

The display yields the maximum number of significant digits without displaying an overload state.

Lower range scales provide higher decimal resolution and more accurate readings.

Key Concept

Digital Multimeter DC Voltage Measurement Procedure and Range Selection

Estimated Time:1m 0s

Question 2169Question →

A laser cutting machine is programmed to make two straight intersecting cuts on a sheet of acrylic. The measures of a pair of vertical angles created by the intersection are $(9x - 18)^\circ$ and $(5x + 46)^\circ$ . What is the measure of the acute angle formed by these cuts, in degrees?

Show answer & explanation

Answer: 54

Answer

The measure of the acute angle is 54 degrees.

Setting the expressions for the vertical angles equal to each other gives

9x - 18 = 5x + 46

. Solving for

x

yields

x = 16

. Substituting

16

back into the expression

(9x - 18)^\circ

results in a vertical angle of

126^\circ

. Since the intersecting lines form supplementary adjacent angles, the adjacent angle is

180^\circ - 126^\circ = 54^\circ

. This represents the acute angle.

Step-by-Step Solution

1

Set up an equation by equating the two algebraic expressions.

9x - 18 = 5x + 46

Vertical angles formed by intersecting lines are always equal in measure.

2

Solve the equation for the variable

x

.

4x = 64 \Rightarrow x = 16

Isolating

x

provides the numerical value needed to evaluate the angle expressions.

3

Substitute

x = 16

back into either original expression.

9(16) - 18 = 144 - 18 = 126^\circ

Evaluating the expression gives the actual degree measure of the vertical angles.

4

Calculate the measure of the adjacent acute angle.

180^\circ - 126^\circ = 54^\circ

Adjacent angles on a straight line are supplementary (add up to

180^\circ

). Because

54^\circ < 90^\circ

, this is the required acute angle.

Key Concept

Vertical angles are equal, and adjacent angles formed by intersecting lines are supplementary.

Question 2170Question →

A cube-shaped block has an edge length of $5$ centimeters. What is the total surface area of the block in square centimeters?

25

125

150

900

Show answer & explanation

Answer: 150

Answer

The total surface area of the cube is 150 square centimeters.

The total surface area of a cube is calculated using the formula

SA = 6s^2

, where

s

is the edge length. Substituting

s = 5

gives

6(5)^2

. By following the order of operations, square the

5

first to get

25

, and then multiply by

6

to find the total surface area of

150

square centimeters.

Step-by-Step Solution

1

Identify the formula for the total surface area of a cube.

SA = 6s^2

A cube has 6 identical square faces, and the area of each face is the edge length squared (

s^2

).

2

Substitute the given edge length (

s = 5

) into the formula.

SA = 6(5)^2

This sets up the equation with the specific dimensions of the block.

3

Evaluate the exponent first, following the order of operations.

5^2 = 25

Exponents must be calculated before multiplication.

4

Multiply the area of one face by the number of faces.

6 \times 25 = 150

This provides the combined area of all 6 faces, which is the total surface area.

Key Concept

Volume and Surface Area of 3D Figures

Estimated Time:45s

Question 2171Question →

A city park is shaped like a trapezoid with parallel sides measuring $140$ meters and $100$ meters. The perpendicular distance between these parallel sides is $60$ meters. Inside the park, there is a rectangular concrete plaza that measures $40$ meters by $25$ meters. The rest of the park is covered in grass. What is the total area covered in grass, in square meters?

Show answer & explanation

Answer: 6200

Answer

6200

The total area of the trapezoid is found by multiplying the average of the two bases by the height, resulting in

7200

square meters. The rectangular plaza has an area of

40 \times 25 = 1000

square meters. Subtracting the inner plaza's area from the total area yields

6200

square meters for the grass.

Step-by-Step Solution

1

Calculate the total area of the trapezoidal park.

The area of the park is

7200

square meters.

The formula for the area of a trapezoid is

A = \frac{1}{2}(b_1 + b_2)h

. Substituting the given dimensions yields

A = \frac{1}{2}(140 + 100) \times 60 = 7200

.

2

Calculate the area of the rectangular concrete plaza.

The area of the plaza is

1000

square meters.

The area of a rectangle is length times width. Calculating

40 \times 25

yields

1000

.

3

Subtract the area of the plaza from the total area of the park to find the grass area.

The total area covered in grass is

6200

square meters.

The grass covers the portion of the park that is not the plaza. Subtracting

1000

from

7200

leaves

6200

.

Key Concept

Calculating and subtracting areas of composite 2D figures

Question 2172Question →

A structural engineer is designing a roof truss. In the design, two parallel horizontal support beams are intersected by a diagonal brace. The consecutive interior angles (also known as same-side interior angles) formed by this intersection measure $(4x - 15)^\circ$ and $(2x + 45)^\circ$ . What is the measure of the larger of these two angles?

55^\circ

65^\circ

95^\circ

115^\circ

Show answer & explanation

Answer: $95^\circ$

Answer

The measure of the larger angle is

95^\circ

.

Consecutive interior angles formed by two parallel lines and a transversal are supplementary, which means they add up to

180^\circ

. Setting up the equation

(4x - 15) + (2x + 45) = 180

and combining like terms yields

6x + 30 = 180

. Subtracting

30

gives

6x = 150

, so

x = 25

. Substituting

25

back into the expressions gives angles of

85^\circ

and

95^\circ

. Comparing the two, the larger angle is

95^\circ

.

Step-by-Step Solution

1

Identify the geometric relationship between the given angles.

The angles are consecutive (same-side) interior angles.

Consecutive interior angles formed by parallel lines and a transversal are supplementary, meaning their sum is

180^\circ

.

2

Set up an algebraic equation based on the supplementary relationship.

(4x - 15) + (2x + 45) = 180

Adding the expressions for both angles and setting them equal to

180

allows us to solve for

x

.

3

Combine like terms and solve for

x

.

6x + 30 = 180

, so

6x = 150

, giving

x = 25

.

Isolating

x

provides the variable value needed to find the specific angle measures.

4

Substitute

x = 25

back into both original expressions.

First angle:

4(25) - 15 = 100 - 15 = 85^\circ

. Second angle:

2(25) + 45 = 50 + 45 = 95^\circ

.

This step determines the actual degree measures of the two angles.

5

Compare the two calculated angles.

The larger angle is

95^\circ

.

The question asks specifically for the measure of the larger angle.

Key Concept

Same-side (consecutive) interior angles formed by parallel lines cut by a transversal are supplementary (sum to

180^\circ

).

Estimated Time:1m 15s

Question 2173Question →

A military supply depot has $6$ different types of field rations, $4$ different types of hydration packs, and $3$ different types of medical kits. A soldier must build a standardized supply bundle containing exactly $2$ different types of field rations, $1$ hydration pack, and $1$ medical kit. How many different supply bundles can be created?

180

360

72

22

Show answer & explanation

Answer: 180

Answer

180

To find the total number of distinct supply bundles, we must calculate the number of ways to make each independent selection and multiply them together. First, calculating the number of ways to choose

2

rations from

6

requires combinations because order does not matter:

C(6, 2) = 15

. There are

4

ways to choose a hydration pack and

3

ways to choose a medical kit. Multiplying these together using the Fundamental Counting Principle yields

15 \times 4 \times 3 = 180

possible bundles.

Step-by-Step Solution

1

Determine the number of ways to select the field rations.

C(6, 2) = \frac{6 \times 5}{2 \times 1} = 15

ways.

Since the order in which the two rations are selected does not matter, we use the combinations formula.

2

Determine the number of ways to select the hydration pack and the medical kit.

4

ways to select a hydration pack and

3

ways to select a medical kit.

Selecting

1

item from a group of

n

items gives

n

possible choices.

3

Calculate the total number of possible supply bundles.

15 \times 4 \times 3 = 180

different supply bundles.

Apply the Fundamental Counting Principle by multiplying the number of independent choices together.

Key Concept

Combinations and the Fundamental Counting Principle

Question 2174Question →

A technician is preparing to measure the direct current flowing through an active $12\text{ V}$ DC relay coil with an expected load current of approximately $150\text{ mA}$ . Which multimeter setup and circuit connection method is correct?

Insert the black lead into COM, the red lead into the mA terminal, set the selector dial to DC milliamperes, and connect the meter in parallel across the relay coil.

Insert the black lead into COM, the red lead into the mA terminal, set the selector dial to DC milliamperes, and insert the meter in series by opening the circuit line.

Insert the black lead into COM, the red lead into the V/\Omega terminal, set the selector dial to DC volts, and connect the meter in series by opening the circuit line.

Insert the black lead into COM, the red lead into the mA terminal, set the selector dial to AC milliamperes, and insert the meter in series by opening the circuit line.

Show answer & explanation

Answer: Insert the black lead into COM, the red lead into the mA terminal, set the selector dial to DC milliamperes, and insert the meter in series by opening the circuit line.

Answer

Insert the black lead into COM, the red lead into the mA terminal, set the selector dial to DC milliamperes, and insert the meter in series by opening the circuit line.

To measure current accurately and safely, the multimeter must be placed in series with the load by opening the circuit path so that the circuit current passes through the meter. Additionally, the black lead must connect to the common port (COM), the red lead must connect to the appropriate current input port (mA for

150\text{ mA}

), and the selector dial must be set to DC milliamperes.

Step-by-Step Solution

1

Determine input terminal and selector dial requirements based on the measurement type and quantity.

The target is direct current at

150\text{ mA}

, requiring the red probe in the

\text{mA}

jack, black probe in

\text{COM}

, and dial set to

\text{DC mA}

.

Multimeters use dedicated low-resistance fused input jacks for current measurements and separate modes for alternating versus direct current.

2

Determine the required electrical connection configuration within the circuit.

The circuit must be opened so that the ammeter is inserted in series with the relay load.

Current represents the flow of charge through a pathway; all electrons must physically pass through the ammeter to be counted.

Key Concept

Digital Multimeter Current Measurement Hookup and Jack Configuration

Estimated Time:1m 0s

Question 2175Question →

A Coast Guard maritime patrol aircraft is assigned a $720$ -mile patrol route. To meet mission parameters, the aircraft must maintain an overall average speed of $180$ miles per hour from start to finish. The crew completes the first $480$ miles at a cruising speed of $320$ miles per hour, then lands at a coastal airfield for a $30$ -minute refueling stop. What average speed must the aircraft maintain during the final leg of the patrol to achieve the required overall average speed?

40 mph

80 mph

100 mph

120 mph

Show answer & explanation

Answer: 120 mph

Answer

The aircraft must maintain an average speed of

120

mph for the final leg.

The correct answer is found by first determining the total mission time (

720 \div 180 = 4

hours). Next, find the time used for the first leg (

480 \div 320 = 1.5

hours) and add the

30

-minute (

0.5

hour) stop, meaning exactly

2.0

hours have elapsed. Subtracting this from the total time gives

2.0

hours remaining to cover the final

240

miles. Dividing

240

miles by

2.0

hours yields exactly

120

mph.

Step-by-Step Solution

1

Calculate the total time allowed for the entire mission.

720 \text{ miles} \div 180 \text{ mph} = 4 \text{ hours}

Overall average speed equals total distance divided by total time.

2

Calculate the time spent flying the first leg.

480 \text{ miles} \div 320 \text{ mph} = 1.5 \text{ hours}

(or

1

hour and

30

minutes)

Time equals distance divided by speed.

3

Calculate the total time used so far, including the refueling stop.

1.5 \text{ hours} + 0.5 \text{ hours} = 2.0 \text{ hours}

The

30

-minute refueling stop must be added to the flight time to find the total elapsed time.

4

Determine the remaining time available for the final leg.

4.0 \text{ hours} - 2.0 \text{ hours} = 2.0 \text{ hours}

Subtracting the elapsed time from the total allowed time gives the time limit for the remainder of the trip.

5

Determine the remaining distance to travel.

720 \text{ miles} - 480 \text{ miles} = 240 \text{ miles}

Subtracting the distance already covered from the total distance gives the remaining distance.

6

Calculate the required average speed for the final leg.

240 \text{ miles} \div 2.0 \text{ hours} = 120 \text{ mph}

Speed equals distance divided by time.

Key Concept

To find a required average speed for a remaining segment, subtract the distance already traveled from the total distance, subtract the time already spent (including stops) from the total allowed time, and divide the remaining distance by the remaining time.

Question 2176Question →

In an automotive engine with a full-pressure lubrication system, how do the connecting rod bearings receive pressurized motor oil?

Through internal passages drilled through the crankshaft from the main bearing journals

By relying solely on crankshaft counterweights splashing fluid upward from the oil pan

Through coolant passages surrounding the cylinder walls and combustion chambers

By blending motor oil directly into the incoming air-fuel charge during the intake stroke

Show answer & explanation

Answer: Through internal passages drilled through the crankshaft from the main bearing journals

Answer

Pressurized motor oil reaches the connecting rod bearings through internal passages drilled through the crankshaft from the main bearing journals.

In a full-pressure lubrication system, the oil pump delivers pressurized oil to the main oil gallery in the engine block. The oil flows into the main bearing saddles through holes in the upper bearing shells. From the main journals, the oil travels through diagonal passages drilled internally through the crankshaft to lubricate the connecting rod journals and bearings.

Step-by-Step Solution

1

Identify the primary oil distribution pathway in a modern full-pressure automotive engine.

The oil pump forces fluid through the main oil gallery in the cylinder block directly to the crankshaft main bearings.

Main bearings sit directly in the engine block saddles where pressurized oil galleries are located.

2

Trace the path from the stationary block saddles to the rotating connecting rod journals.

Holes drilled at an angle through the crankshaft webs connect each main journal to its adjacent connecting rod journal.

Connecting rod journals rotate in an orbit and cannot receive oil directly from stationary engine block galleries.

Key Concept

Crankshaft oil flow pathways in a full-pressure automotive lubrication system

Estimated Time:1m 0s

Question 2177Question →

A communications center needs to assemble a rapid response team from a pool of $7$ qualified technicians. The team must consist of exactly $1$ team chief and $3$ field technicians. The $3$ field technicians share identical duties. How many different rapid response teams can be formed?

Show answer & explanation

Answer: 140

Answer

140

To find the total number of possible teams, we must account for the distinct roles across two stages of selection. First, we select

1

team chief from the

7

available technicians, which gives

7

options. Next, we select

3

field technicians from the remaining

6

technicians. Because the field technicians have identical roles, the order in which they are selected does not matter. This requires calculating a combination:

C(6,3) = \frac{6!}{3!(6-3)!} = 20

. Finally, using the Fundamental Counting Principle, we multiply the number of choices for the chief by the number of combinations for the field technicians:

7 \times 20 = 140

.

Step-by-Step Solution

1

Select the team chief.

7 possible choices

There are 7 available technicians, and 1 must be chosen for the distinct role of team chief.

2

Select the field technicians.

20 possible combinations

After choosing the chief, 6 technicians remain. We need to select 3 for identical roles. Since order does not matter, this is a combination: C(6,3) = (6 × 5 × 4) / (3 × 2 × 1) = 20.

3

Calculate the total number of unique teams.

140 teams

Using the Fundamental Counting Principle, multiply the independent choices for each stage: 7 × 20 = 140.

Key Concept

Combining the Fundamental Counting Principle with combinations for multi-stage selection.

Question 2178Question →

A solid concrete pillar is built in the shape of a rectangular prism. It has a square base with each side measuring $3$ feet, and a total height of $10$ feet. What is the volume of the pillar in cubic feet?

Show answer & explanation

Answer: 90

Answer

90

The volume of a rectangular prism is calculated by multiplying its length, width, and height. Because the base is a square, both the length and the width are

3

feet. Multiplying

3 \times 3 \times 10

yields a total volume of

90

cubic feet.

Step-by-Step Solution

1

Identify the proper formula to calculate the volume of a rectangular prism.

Volume =

l \times w \times h

This is the standard formula used to find the total 3D space occupied by a box-like object.

2

Determine the values of the length, width, and height from the problem description.

Length (

l

) =

3

feet, Width (

w

) =

3

feet, Height (

h

) =

10

feet.

The base is described as a square with a side measuring 3 feet, meaning both the length and width are 3 feet.

3

Multiply the dimensions to calculate the total volume.

3 \times 3 \times 10 = 90

Multiplying the values provides the final volume in cubic units.

Key Concept

The volume of a rectangular prism is found by multiplying its length, width, and height together.

Alternative Method

You can also think of the volume formula as Volume = Area of the Base × height. First, find the area of the square base (

3 \times 3 = 9

square feet). Then, multiply this base area by the height of the pillar (

9 \times 10 = 90

cubic feet).

Estimated Time:45s

Question 2179Question →

A circular security zone has a circumference of $16\pi$ meters. What is the area of this zone in square meters?

16\pi

64\pi

128\pi

256\pi

Show answer & explanation

Answer: $64\pi$

Answer

64\pi

To find the area, the radius of the circle must first be determined using the circumference formula (

C = 2\pi r

). Setting the given circumference equal to the formula (

16\pi = 2\pi r

) gives a radius of

8

meters. The area is then calculated using the formula

A = \pi r^2

, which results in

\pi(8)^2 = 64\pi

square meters.

Step-by-Step Solution

1

Identify the formula for circumference to solve for the radius.

C = 2\pi r = 16\pi

The radius is required to calculate the area of the circle.

2

Solve for the radius (

r

).

r = 8

meters

Dividing both sides of the equation by

2\pi

isolates the radius.

3

Apply the area formula for a circle using the found radius.

A = \pi r^2 = \pi(8)^2 = 64\pi

square meters

Substituting the radius (

8

) into the area formula yields the final required value.

Key Concept

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

Estimated Time:45s

Question 2180Question →

A municipal park has a flat rectangular section and a semi-circular section attached to one of the rectangle's shorter sides. The overall length of the park, measured from the straight edge of the rectangle to the farthest curved point of the semi-circle, is $200$ meters. The width of the rectangular section is $100$ meters. What is the outer perimeter of the entire park in meters? (Use $\pi \approx 3.14$ )

657

714

557

18,925

Show answer & explanation

Answer: $557$

Answer

The outer perimeter of the park is

557

meters.

The perimeter is the total distance around the outside of the shape. First, find the radius of the semi-circle, which is half of its

100

-meter diameter (

50

m). Subtract this radius from the total

200

-meter length to find the straight edge of the rectangle (

150

m). The outer boundary consists of one

100

-meter straight edge, two

150

-meter straight edges, and the semi-circular arc. The arc length is half the circumference of a full circle (

0.5 \times 3.14 \times 100 = 157

m). Adding these outer edges together gives

100 + 150 + 150 + 157 = 557

meters.

Step-by-Step Solution

1

Identify the diameter and radius of the semi-circle.

Diameter =

100

m, Radius =

50

m

The semi-circle is attached to the

100

-meter wide side of the rectangle, so its diameter matches that width.

2

Determine the length of the rectangular section's straight sides.

200 - 50 = 150

m

The overall length (

200

m) includes both the rectangle's length and the semi-circle's radius (

50

m).

3

Calculate the arc length of the semi-circular section.

\frac{1}{2} \times \pi \times d = 0.5 \times 3.14 \times 100 = 157

m

The outer curved edge is exactly half of a full circle's circumference.

4

Sum the lengths of all outer boundary segments.

100 + 150 + 150 + 157 = 557

m

The total perimeter includes one short straight edge (

100

m), two long straight edges (

150

m each), and the curved arc (

157

m). The internal dashed line is not part of the outer perimeter.

Key Concept

Calculating unknown dimensions and outer perimeters of composite figures.

Estimated Time:1m 0s

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