All practice questions

Question 281Question →

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 282Question →

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 283Question →

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 284Question →

A trapezoidal wooden sign has a height of $18$ inches. Its parallel top and bottom edges measure $24$ inches and $36$ inches, respectively. If a rectangular hole measuring $8$ inches by $6$ inches is cut out of the sign to mount a digital clock, what is the area of the remaining wooden surface in square inches?

Show answer & explanation

Answer: 492

Answer

492

The remaining area is found by subtracting the area of the inner rectangle from the area of the outer trapezoid. The trapezoid area is

\frac{1}{2}(24 + 36) \times 18 = 540

. The rectangle area is

8 \times 6 = 48

. The difference is

540 - 48 = 492

.

Step-by-Step Solution

1

Calculate the area of the full trapezoidal sign.

540 square inches

This gives the total area before the hole is cut, using the formula 0.5 * (base1 + base2) * height.

2

Calculate the area of the rectangular cutout.

48 square inches

This represents the area that is removed from the sign, calculated as length * width.

3

Subtract the cutout area from the total area.

492 square inches

Subtracting the removed area from the initial total area gives the area of the remaining solid surface.

Key Concept

Area of composite 2D figures involving trapezoids and rectangles.

Estimated Time:1m 0s

Question 285Question →

A sail on a small boat is shaped like a right triangle. The horizontal bottom edge of the sail is $7$ feet long, and the vertical edge is $24$ feet high. What is the length, in feet, of the longest edge (the hypotenuse) of the sail?

Show answer & explanation

Answer: 25

Answer

25

The correct answer is found by applying the Pythagorean Theorem (

a^2 + b^2 = c^2

). Substituting the lengths of the two legs (

7

and

24

), the equation becomes

7^2 + 24^2 = c^2

. Squaring the numbers yields

49 + 576

. Adding these together gives

625 = c^2

. Finally, taking the square root of

625

results in

25

, which represents the length of the hypotenuse.

Step-by-Step Solution

1

Identify the known sides of the right triangle.

Leg

a = 7

feet and leg

b = 24

feet.

The horizontal and vertical edges of the sail represent the two perpendicular legs of a right triangle.

2

Set up the Pythagorean Theorem equation.

7^2 + 24^2 = c^2

The Pythagorean Theorem states that the sum of the squares of the legs is equal to the square of the hypotenuse (

c

).

3

Square the lengths of the legs.

49 + 576 = c^2

To evaluate the equation, the individual lengths must be squared before adding them.

4

Add the squared values together.

625 = c^2

Combining the squared leg lengths yields the square of the hypotenuse.

5

Calculate the square root to solve for c.

c = 25

Taking the square root of

625

gives the final length of the hypotenuse.

Key Concept

Pythagorean Theorem

Question 286Question →

A rectangular landing pad is designed so that its length is exactly three times its width. If the total area of the landing pad is $147$ square meters, what is the perimeter of the landing pad in meters?

Show answer & explanation

Answer: 56

Answer

56

By setting the width as

w

and length as

3w

, the area equation is

3w^2 = 147

. Solving for

w

yields a width of

7

meters, making the length

21

meters. The perimeter is the sum of all four sides:

2(21 + 7) = 56

meters.

Step-by-Step Solution

1

Define variables and set up the area equation.

w \times 3w = 147

The area of a rectangle is width times length, and the length is given as three times the width.

2

Solve for the width (

w

).

w^2 = 49 \Rightarrow w = 7

Dividing both sides by

3

isolates

w^2

, and taking the square root gives the width.

3

Calculate the length and the perimeter.

Length =

21

, Perimeter =

56

Length is

3 \times 7

. Perimeter is

2(21 + 7)

.

Key Concept

Calculating the perimeter of a rectangle when given its area and the proportional relationship between its sides.

Question 287Question →

In the provided technical drawing, two horizontal parallel lines are intersected by a diagonal transversal line. The measures of two same-side interior angles are given by the expressions $(8x - 14)^\circ$ and $(6x + 26)^\circ$ . What is the numerical value of $x$ ?

Show answer & explanation

Answer: 12

Answer

12

The two angles described are same-side interior angles. Because the lines are parallel, these angles are supplementary and their sum is exactly 180 degrees. Setting up the equation

(8x - 14) + (6x + 26) = 180

and solving for

x

correctly yields exactly 12.

Step-by-Step Solution

1

Identify the relationship between the two given angles.

The angles are supplementary.

When two parallel lines are cut by a transversal, same-side interior angles are supplementary, meaning their sum is exactly 180 degrees.

2

Set up an algebraic equation.

(8x - 14) + (6x + 26) = 180

The sum of the measures of the two supplementary angles must equal 180.

3

Combine the like terms on the left side of the equation.

14x + 12 = 180

Adding the variable terms (

8x + 6x = 14x

) and the constant terms (

-14 + 26 = 12

) simplifies the equation.

4

Isolate the variable term.

14x = 168

Subtracting 12 from both sides of the equation leaves only the variable term on the left side.

5

Solve for x.

x = 12

Dividing both sides by 14 gives the final numerical value of x.

Key Concept

Same-side interior angles formed by parallel lines and a transversal are supplementary.

Question 288Question →

A passenger train leaves West Station heading east toward East Station, which is $270$ miles away, traveling at a constant speed of $60$ mph. Exactly $1$ hour later, a freight train leaves East Station heading west toward West Station on a parallel track at a constant speed of $45$ mph. How many hours after the passenger train's departure will the two trains pass each other?

Show answer & explanation

Answer: 3

Answer

3 hours

The passenger train travels 60 miles in its first hour, leaving 210 miles between the trains. Traveling toward each other, their combined relative speed is 105 mph. It takes them 2 hours to cover the 210 miles together. Adding the initial 1 hour, the total time since the passenger train departed is 3 hours.

Step-by-Step Solution

1

Calculate the distance covered by the passenger train during its 1-hour head start.

60 \text{ mph} \times 1 \text{ hour} = 60 \text{ miles}

This establishes how much distance is covered before the second train begins moving.

2

Find the remaining distance between the two trains.

270 \text{ miles} - 60 \text{ miles} = 210 \text{ miles}

This is the remaining gap the two trains must close together once both are moving.

3

Calculate the combined (relative) speed of the two trains.

60 \text{ mph} + 45 \text{ mph} = 105 \text{ mph}

Because they are traveling toward each other from opposite directions, their speeds add up to close the distance.

4

Determine the time it takes to close the remaining 210 miles.

210 \text{ miles} \div 105 \text{ mph} = 2 \text{ hours}

Using the formula Time = Distance / Rate.

5

Calculate the total time since the passenger train's departure.

1 \text{ hour} + 2 \text{ hours} = 3 \text{ hours}

The question specifically asks for the time after the passenger train's departure, which includes its initial 1-hour head start.

Key Concept

Relative motion and converging objects with staggered start times

Question 289Question →

A flat circular roof of a silo has a circumference of $44$ meters. A sealant must be applied to the entire top surface. If one bucket of sealant covers exactly $30$ square meters, what is the minimum number of buckets that must be purchased to seal the roof? (Use $\pi = \frac{22}{7}$ )

Show answer & explanation

Answer: 6

Answer

6

To find the correct number of buckets, we first derived the radius (

7

meters) from the given circumference (

44

meters). Next, we calculated the total area (

154

square meters). Finally, dividing the total area by the coverage per bucket (

30

square meters) gives a value of

5.133

. Because we cannot purchase a fraction of a bucket and need to cover the entire roof, we must round up to

6

full buckets.

Step-by-Step Solution

1

Set up the circumference formula

C = 2\pi r

using the given values to solve for the radius.

44 = 2 \times \frac{22}{7} \times r

, which simplifies to

44 = \frac{44}{7} \times r

. Multiplying both sides by

\frac{7}{44}

gives

r = 7

meters.

The radius is required to calculate the surface area of the circular roof.

2

Calculate the area of the circular roof using the formula

A = \pi r^2

.

A = \frac{22}{7} \times (7)^2 = \frac{22}{7} \times 49 = 154

square meters.

The total area determines the total amount of sealant that is needed.

3

Determine the number of buckets needed by dividing the total area by the coverage per bucket.

154 \div 30 = 5

with a remainder of

4

(or

5.133...

). Since partial buckets cannot be purchased, you must round up to

6

buckets.

To seal the entire surface area without leaving any sections exposed, you must purchase enough full buckets to cover the total area.

Key Concept

Calculating the area of a circle from its circumference and applying the result to a real-world scenario requiring discrete quantities.

Question 290Question →

A supply cabinet contains $10$ identical safety helmets, of which $2$ are yellow and $8$ are white. A construction worker randomly selects a helmet, records its color, returns it to the cabinet, and then randomly selects a second helmet. What is the probability of selecting a yellow helmet both times? Express your answer as a decimal.

Show answer & explanation

Answer: 0.04

Answer

0.04

Since the helmet is replaced after the first draw, the two events are independent. The probability of drawing a yellow helmet is

\frac{2}{10}

(or

0.2

). To find the probability of drawing a yellow helmet twice in a row, multiply the probabilities of the two independent events:

0.2 \times 0.2 = 0.04

.

Step-by-Step Solution

1

Find the probability of the first event.

\frac{2}{10}

or

0.2

There are

2

yellow helmets out of

10

total.

2

Find the probability of the second event.

\frac{2}{10}

or

0.2

Because the helmet was replaced, the total number and the number of yellow helmets remain the same.

3

Calculate the compound probability.

0.04

For independent events, multiply the individual probabilities together:

0.2 \times 0.2 = 0.04

.

Key Concept

The probability of two independent events occurring is the product of their individual probabilities:

P(A \text{ and } B) = P(A) \times P(B)

.

Question 291Question →

A mechanic measures the diameter of an engine valve stem using a standard 0 to 1-inch outside micrometer. On the sleeve, the edge of the thimble has uncovered the major graduation mark 3 and 2 additional minor graduation lines. On the thimble, graduation line 18 aligns precisely with the sleeve index line. What is the total measured diameter of the valve stem in inches?

Show answer & explanation

Answer: 0.368

Answer

The total measured diameter of the valve stem is

0.368\text{ inches}

.

A standard imperial micrometer sleeve displays major marks at increments of

0.100\text{ in}

and minor subdivision marks at increments of

0.025\text{ in}

. Uncovering the major mark 3 gives

0.300\text{ in}

, and uncovering 2 minor marks adds

0.050\text{ in}

, creating a sleeve subtotal of

0.350\text{ in}

. Adding the aligned thimble reading of 18 thousandths (

0.018\text{ in}

) results in a total diameter of

0.300 + 0.050 + 0.018 = 0.368\text{ inches}

.

Step-by-Step Solution

1

Determine the value of the major divisions uncovered on the sleeve.

The number 3 represents

3 \times 0.100\text{ in} = 0.300\text{ in}

.

Each numbered graduation line along the sleeve index line represents one-tenth of an inch (

0.100\text{ in}

).

2

Determine the value of the visible subdivision lines on the sleeve.

Two minor lines represent

2 \times 0.025\text{ in} = 0.050\text{ in}

.

Each space between major tenths is subdivided into four increments, each representing twenty-five thousandths of an inch (

0.025\text{ in}

).

3

Determine the value of the graduation on the thimble aligned with the sleeve index line.

Graduation 18 represents

18 \times 0.001\text{ in} = 0.018\text{ in}

.

The circumference of the thimble is divided into 25 equal increments, each representing one-thousandth of an inch (

0.001\text{ in}

).

4

Add the sleeve major reading, sleeve minor reading, and thimble reading together.

0.300\text{ in} + 0.050\text{ in} + 0.018\text{ in} = 0.368\text{ in}

.

The complete measurement is the sum of all three distinct scale components.

Key Concept

Reading a standard imperial outside micrometer by summing sleeve tenths (

0.100\text{ in}

), sleeve quarter-tenths (

0.025\text{ in}

), and thimble thousandths (

0.001\text{ in}

).

Estimated Time:1m 0s

Question 292Question →

A rectangular garden has a perimeter of $56$ meters. The length of the garden is $4$ meters greater than its width. If a $3$ -meter wide paved path is built entirely inside the garden along all four of its borders, what is the area, in square meters, of the remaining unpaved section of the garden?

Show answer & explanation

Answer: 60

Answer

The area of the remaining unpaved section of the garden is 60 square meters.

The original dimensions of the garden are found to be 12 meters and 16 meters based on the given perimeter and side relationships. The inner unpaved section has dimensions of 12 - 6 = 6 meters and 16 - 6 = 10 meters because the 3-meter path cuts into both sides of each dimension. Multiplying these inner dimensions yields a final area of 60 square meters.

Step-by-Step Solution

1

Use the perimeter formula to find the outer length and width of the garden.

Width = 12 meters, Length = 16 meters.

The relationship between length and width (L = W + 4) and the total perimeter formula

2L + 2W = 56

allows us to find the exact dimensions.

2

Calculate the dimensions of the inner unpaved area by subtracting the path's width.

Inner width = 6 meters, Inner length = 10 meters.

The 3-meter path runs along all four borders, meaning the path's width must be subtracted twice (3 + 3 = 6 meters) from both the outer length and the outer width.

3

Calculate the final area of the inner unpaved section.

60 square meters.

The area of a rectangle is found by multiplying its length by its width (

10 \times 6

).

Key Concept

Solving for unknown dimensions using perimeter, and determining the area of a smaller inscribed rectangle by correctly subtracting border widths.

Question 293Question →

During a naval training exercise, a submarine travels submerged in a straight line at a constant speed of $15$ knots. Exactly $2$ hours later, a destroyer departs from the same location and pursues the submarine along the exact same path at a constant speed of $25$ knots. How many hours will it take for the destroyer to overtake the submarine from the time the destroyer begins moving?

Show answer & explanation

Answer: 3

Answer

It will take 3 hours for the destroyer to catch up to the submarine.

To solve a catch-up scenario, first determine the initial distance between the two objects. The submarine travels at

15

knots for

2

hours, creating a

30

nautical mile gap. Next, find the relative speed by subtracting the slower speed from the faster speed (

25 - 15 = 10

knots) because they are moving in the same direction. Finally, divide the initial gap by the relative speed (

30 \div 10 = 3

) to find the time it takes the destroyer to close the gap.

Step-by-Step Solution

1

Calculate the head-start distance of the submarine.

30

nautical miles

The submarine travels at

15

knots for

2

hours before the destroyer starts.

2

Calculate the relative catch-up speed.

10

knots

The destroyer travels in the same direction, so the relative speed is the difference between their speeds (

25 - 15

).

3

Calculate the time required to close the distance gap.

3

hours

Time equals distance divided by relative rate. Dividing the

30

nautical mile gap by the relative speed of

10

knots gives

3

hours.

Key Concept

Relative Motion and Catch-Up Scenarios

Question 294Question →

A landscaping company is designing a parallelogram-shaped flower bed that will cover exactly $288$ square feet. If the design requires the base of the flower bed to be twice its height, what must the length of the base be in feet?

Show answer & explanation

Answer: 24

Answer

24

The area of a parallelogram is the product of its base and height (

A = b \times h

). By substituting

2h

for the base, the equation becomes

288 = 2h^2

. Solving for

h

yields

12

. The base is twice the height, so

2 \times 12 = 24

.

Step-by-Step Solution

1

Set up the area formula

A = b \times h

To relate the given area to the dimensions of the parallelogram.

2

Express the base in terms of height

b = 2h

The problem states the base is twice the height.

3

Substitute the expressions into the area equation

288 = (2h) \times h

To create an equation with only one variable.

4

Solve for the height (

h

)

2h^2 = 288 \Rightarrow h^2 = 144 \Rightarrow h = 12

To find the numerical value of the height.

5

Calculate the base (

b

)

b = 2 \times 12 = 24

The question asks for the length of the base, not the height.

Key Concept

Area of a parallelogram and algebraic substitution

Estimated Time:1m 0s

Question 295Question →

A carpenter is using $8\text{d}$ (8-penny) common nails to fasten a subfloor board with an actual thickness of $1.0\text{ inch}$ to underlying framing joists. For standard nails between $2\text{d}$ and $10\text{d}$ , the total nail length $L$ in inches is determined by the formula $L = \frac{d}{4} + 0.5$ , where $d$ is the penny size. Assuming the nail is driven perpendicular and flush with the top surface of the subfloor, what is the depth of penetration (in inches) of the nail into the framing joist?

Show answer & explanation

Answer: 1.5

Answer

1.5 inches

An 8d common nail has a total length of 2.5 inches based on the standard pennyweight formula (8 / 4 + 0.5 = 2.5). When driven completely flush through a 1.0-inch board into the joist below, the remaining 1.5 inches penetrates into the framing joist.

Step-by-Step Solution

1

Determine the total length of the 8d nail using the penny conversion formula

Total length = 2.5 inches

Applying the formula L = (d / 4) + 0.5 with d = 8 gives L = 2.0 + 0.5 = 2.5 inches.

2

Subtract the thickness of the top material (subfloor board) from the total nail length

Penetration depth = 1.5 inches

When driven flush, the portion of the nail passing through the 1.0-inch board does not enter the joist, leaving 2.5 - 1.0 = 1.5 inches of penetration into the joist.

Key Concept

Nail pennyweight sizing conversion and framing penetration calculation

Alternative Method

Recall standard trade penny sizes directly from memory: an 8d nail is 2-1/2 inches long. Subtracting the 1-inch board thickness directly gives 1-1/2 (1.5) inches.

Estimated Time:1m 0s

Question 296Question →

A technician is testing a mechanical linkage that uses a straight lever pinned at one end, which serves as the fulcrum. A control rod is attached to the lever at a distance of $4\text{ inches}$ from the fulcrum and creates a resistance of $150\text{ lbs}$ . An actuator applies an effort force at the free end of the lever, located $12\text{ inches}$ from the fulcrum. If all forces are applied perpendicular to the lever, what effort force (in pounds) is required to perfectly balance the control rod's resistance?

Show answer & explanation

Answer: 50

Answer

50

In a balanced lever system, the torque produced by the effort force must equal the torque produced by the resistance force. Torque is calculated by multiplying the force by its distance from the fulcrum. Therefore,

F_E \times 12 = 150 \times 4

, which simplifies to

12F_E = 600

, resulting in an effort force of

50\text{ lbs}

.

Step-by-Step Solution

1

Identify the known variables and their distances from the fulcrum.

Resistance Force (

F_R

) =

150\text{ lbs}

, Resistance Arm (

d_R

) =

4\text{ inches}

, Effort Arm (

d_E

) =

12\text{ inches}

.

Correctly measuring distances from the pivot point (fulcrum) is necessary to calculate torque.

2

Set up the torque balance equation for the lever.

F_E \times d_E = F_R \times d_R

For the lever system to be in equilibrium, the opposing torques must be equal.

3

Substitute the values and calculate the required effort force.

F_E \times 12 = 150 \times 4 \Rightarrow 12F_E = 600 \Rightarrow F_E = 50\text{ lbs}

.

Solving the algebraic equation provides the necessary balancing force.

Key Concept

Torque balance and lever mechanical advantage

Alternative Method

Calculate the mechanical advantage (MA) first. MA = Effort Arm

\div

Resistance Arm =

12 \div 4 = 3

. Since the lever provides a mechanical advantage of

3

, the required effort force is exactly one-third of the resistance force (

150 \div 3 = 50\text{ lbs}

).

Estimated Time:1m 0s

Question 297Question →

An automated loading mechanism pushes a $40 \text{ kg}$ projectile along a horizontal guide rail into a weapon system. The mechanism applies a constant forward force of $700 \text{ N}$ . If the sliding friction in the guide rail opposes the motion with a force of $100 \text{ N}$ , what is the acceleration of the projectile?

Show answer & explanation

Answer: 15

Answer

15

The correct acceleration is found by first determining the net force acting on the projectile. Subtracting the

100 \text{ N}

of friction from the

700 \text{ N}

applied force yields a net force of

600 \text{ N}

. According to Newton's Second Law (

F = ma

), dividing this net force by the

40 \text{ kg}

mass results in an acceleration of

15 \text{ m/s}^2

.

Step-by-Step Solution

1

Calculate the net horizontal force acting on the projectile.

700 \text{ N} - 100 \text{ N} = 600 \text{ N}

The net force is the vector sum of all forces; friction directly opposes the forward applied force.

2

Use Newton's Second Law to set up the equation for acceleration.

a = \frac{F_{\text{net}}}{m}

Acceleration equals the net force divided by the mass of the object.

3

Substitute the known values into the equation.

a = \frac{600}{40} = 15 \text{ m/s}^2

Dividing the computed net force by the given mass provides the exact acceleration.

Key Concept

Newton's Second Law (F=ma) and Net Force

Question 298Question →

During a routine weight and balance check, a small aircraft is parked horizontally on weighing scales. The scale under the nose wheel reads $500$ pounds, and the combined reading from the scales under the two main wheels is $1,500$ pounds. The nose wheel is located exactly $100$ inches forward of the main landing gear axis. How many inches forward of the main landing gear axis is the aircraft's center of gravity?

Show answer & explanation

Answer: 25

Answer

25

In static equilibrium, the sum of the moments around any point must be zero. By using the main landing gear axis as the reference point, the moment created by the total weight of the aircraft (

2,000

pounds) acting at the center of gravity must equal the moment created by the nose wheel's load. The nose wheel is

100

inches away and supports

500

pounds, creating a moment of

50,000

pound-inches. Dividing this total moment by the total weight (

50,000 \div 2,000

) gives a center of gravity located

25

inches forward of the main gear.

Step-by-Step Solution

1

Determine the total weight of the aircraft.

500 + 1,500 = 2,000

pounds

The total weight is needed to find the proportion of the load carried by each support.

2

Set up the balance equation (sum of moments) around the main landing gear axis.

(Total\ Weight) \times (CG\ Distance) = (Nose\ Wheel\ Weight) \times (Distance\ to\ Nose\ Wheel)

In static equilibrium, the moment generated by the total weight at the CG equals the sum of the moments of individual components.

3

Substitute the known values into the equation and solve for the CG distance.

2,000 \times d = 500 \times 100 \Rightarrow 2,000d = 50,000 \Rightarrow d = 25

inches

Solving the algebraic equation isolates the distance from the main gear to the center of gravity.

Key Concept

Center of Gravity and Static Equilibrium

Alternative Method

Instead of using the main gear as the pivot, you can use the nose wheel as the pivot. The moment of the main gear is

1,500 \times 100 = 150,000

. Dividing by the total weight (

2,000

) gives

75

inches from the nose wheel. Subtracting this from the total wheelbase (

100 - 75

) gives

25

inches from the main gear.

Estimated Time:1m 0s

Question 299Question →

A $60\text{-pound}$ crate is placed on one end of a first-class lever, exactly $4\text{ feet}$ from the fulcrum. What downward effort force must be applied at a distance of $8\text{ feet}$ on the opposite side of the fulcrum to perfectly balance the lever?

Show answer & explanation

Answer: 30

Answer

30\text{ lbs}

A lever is perfectly balanced when the turning force (torque) on one side of the fulcrum equals the torque on the other side. Torque is calculated by multiplying the force by its distance from the fulcrum. The crate creates a torque of

240\text{ ft-lbs}

(

60\text{ lbs} \times 4\text{ ft}

). Because the effort is being applied twice as far from the fulcrum (

8\text{ ft}

), it only needs to be half the force to generate the same

240\text{ ft-lbs}

of torque. Therefore,

240 \div 8 = 30\text{ lbs}

.

Step-by-Step Solution

1

Set up the torque balance equation.

F_1 \times d_1 = F_2 \times d_2

For a lever to be balanced, the torque (force times distance) on the left side must equal the torque on the right side.

2

Substitute the known values into the equation.

60 \times 4 = F_2 \times 8

The crate applies a

60\text{ lb}

force at a distance of

4\text{ ft}

, and the effort force is applied at a distance of

8\text{ ft}

.

3

Solve for the unknown effort force (

F_2

).

240 = 8 \times F_2 \rightarrow F_2 = 30

Dividing both sides of the equation by

8

isolates

F_2

, revealing the required balancing force.

Key Concept

Torque equilibrium and mechanical advantage in a first-class lever

Estimated Time:45s

Question 300Question →

A warehouse worker uses a first-class lever to lift a machinery part weighing $200\text{ lbs}$ . The fulcrum is placed exactly $2\text{ feet}$ from the machinery part. If the worker applies a downward effort force at a distance of $8\text{ feet}$ from the fulcrum on the opposite side, how much effort force is required to balance the load? (Assume the lever itself is weightless.)

Show answer & explanation

Answer: 50

Answer

50

A first-class lever is balanced when the torque on one side of the fulcrum equals the torque on the other side. Torque is calculated by multiplying the force by its distance from the fulcrum. The machinery part creates a torque of

200\text{ lbs} \times 2\text{ ft} = 400\text{ ft-lbs}

. To balance this, the effort force applied at

8\text{ ft}

must also create

400\text{ ft-lbs}

of torque. Therefore, the required effort force is

\frac{400}{8} = 50\text{ lbs}

.

Step-by-Step Solution

1

Set up the torque balance equation for a lever in equilibrium.

\text{Load} \times \text{Load Arm} = \text{Effort Force} \times \text{Effort Arm}

For a lever to balance, the rotational forces (torques) on both sides of the fulcrum must be equal.

2

Substitute the given values into the equation.

200 \times 2 = \text{Effort Force} \times 8

The load is

200\text{ lbs}

located

2\text{ ft}

from the fulcrum, and the effort is applied

8\text{ ft}

from the fulcrum.

3

Calculate the unknown effort force.

400 = \text{Effort Force} \times 8 \implies \text{Effort Force} = \frac{400}{8} = 50

Dividing the total load torque by the effort arm distance gives the required effort force.

Key Concept

The principle of moments (torque balance) dictates that a lever is in equilibrium when the clockwise torque equals the counterclockwise torque.

Answer

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