All practice questions

Question 2121Question →

A base dining facility prepares $1,200$ meals for a holiday event. If $65\%$ of the meals feature a beef entree, $20\%$ feature a chicken entree, and the remaining meals are vegetarian, how many vegetarian meals were prepared?

Show answer & explanation

Answer: 180

Answer

180

The problem requires finding a remaining portion after accounting for two given percentages. Since beef and chicken account for

85\%

of the total meals, the vegetarian meals must make up the remaining

15\%

. Multiplying the total

1,200

meals by

0.15

yields exactly

180

vegetarian meals.

Step-by-Step Solution

1

Calculate the total percentage of meat entrees.

65\% + 20\% = 85\%

Adding the given percentages determines what portion of the total is already accounted for.

2

Determine the percentage of vegetarian meals.

100\% - 85\% = 15\%

The entire batch of meals represents

100\%

. Subtracting the known meat percentage leaves the vegetarian portion.

3

Calculate

15\%

of

1,200

.

1,200 \times 0.15 = 180

To find a percentage of a quantity, multiply the total quantity by the percentage expressed as a decimal.

Key Concept

Calculating Percentages of Quantities

Question 2122Question →

A graphic artist is constructing a composite logo by stacking four clear square acetate slides perfectly on top of each other. Each slide features exactly one opaque geometric element. By observing how the shapes overlap in the final logo, determine the correct order in which the slides were stacked.

The final composite logo reveals the following depth cues:
- A thick light gray diagonal line spans the graphic, but its central portion is completely blocked from view by a large black circle.
- A medium white square sits perfectly centered inside the black circle, fully visible against the black background.
- A thin dark gray horizontal line stretches straight across the logo, passing visibly over the white square, the black circle, and the exposed ends of the diagonal line without interruption.

Arrange the four acetate slides in their correct stacking sequence from the BOTTOM layer (placed first) to the TOP layer (placed last).

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct stacking order from bottom to top is: the slide with the light gray diagonal line, followed by the slide with the black circle, then the slide with the white square, and finally the slide with the dark gray horizontal line on top.

In superimposition problems involving opaque shapes, visual occlusion determines the stacking sequence. A shape that is partially hidden by another must be positioned underneath it. The diagonal line is hidden by the circle, placing it at the bottom. The circle is partially obscured by the square, placing it second. The square is overlaid by the horizontal line, placing it third. The horizontal line is completely uninterrupted, making it the top layer.

Step-by-Step Solution

1

Identify the bottommost layer by finding the shape that is hidden by others.

The thick light gray diagonal line is hidden by the large black circle, meaning it must be beneath it.

An opaque element covering another indicates it is layered on top, leaving the obscured element underneath.

2

Determine the relative position of the black circle and the white square.

The white square is fully visible against the black circle, meaning the square is placed above the circle.

If the circle were above the square, the square would be completely hidden by the opaque black circle.

3

Identify the topmost layer that covers all other elements.

The thin dark gray horizontal line crosses over the square, circle, and diagonal line without interruption.

A continuous, unbroken line passing over all other shapes must be the final layer placed on top.

Key Concept

Visual occlusion and depth order in superimposed geometric layers

Question 2123Question →

A medical supply drone is programmed to complete a $120$ -mile delivery route with a required overall average speed of $40$ miles per hour. During the first segment, the drone flies $60$ miles at an average speed of $30$ miles per hour. It then lands for a $20$ -minute battery swap. At what average speed must the drone fly for the remainder of the route to achieve its required overall average speed?

50

mph

60

mph

90

mph

150

mph

Show answer & explanation

Answer: $90$ mph

Answer

The drone must fly at

90

mph for the remainder of the route.

To solve for a missing speed when an overall average is required, you must first find the total time allowed. For a

120

-mile trip at

40

mph, the drone has exactly

3

hours. The first leg took

2

hours (

60

miles

\div

30

mph), and the battery swap took

20

minutes. This leaves only

40

minutes (which is

2/3

of an hour) to travel the remaining

60

miles. Dividing

60

miles by

2/3

hours yields a required speed of

90

mph.

Step-by-Step Solution

1

Calculate the total time allowed for the entire trip.

T_{total} = 120 \text{ miles} \div 40 \text{ mph} = 3 \text{ hours}

.

Overall average speed equals total distance divided by total time.

2

Calculate the time spent flying the first segment.

T_1 = 60 \text{ miles} \div 30 \text{ mph} = 2 \text{ hours}

.

Time equals distance divided by speed.

3

Determine the total time used so far, including the battery swap.

2 \text{ hours} + 20 \text{ minutes} = 2 \text{ hours and } 20 \text{ minutes}

.

Rest stops and delays must be included in total transit time.

4

Calculate the remaining time available.

3 \text{ hours} - 2 \text{ hours } 20 \text{ minutes} = 40 \text{ minutes}

.

Subtract the time already used from the total time allowed.

5

Convert the remaining time into hours.

40 \text{ minutes} = \frac{40}{60} \text{ hours} = \frac{2}{3} \text{ hours}

.

Speed is required in miles per hour, so time must be in hours.

6

Calculate the remaining distance.

120 \text{ miles} - 60 \text{ miles} = 60 \text{ miles}

.

Subtract the distance already covered from the total distance.

7

Calculate the required average speed for the final segment.

V_2 = 60 \text{ miles} \div \frac{2}{3} \text{ hours} = 60 \times \frac{3}{2} = 90 \text{ mph}

.

Speed equals remaining distance divided by remaining time.

Key Concept

Average speed over a multi-segment journey is defined solely as the total distance divided by the total time. All stops, breaks, and varying speeds must be accounted for by combining their respective times.

Question 2124Question →

A restaurant manager orders a bulk supply of premium cooking oil. The invoice shows a final cost of $\$510$ after a $15\%$ wholesale discount was applied to the standard price. What was the original standard price of the bulk supply before the discount?

\$586.50

\$600.00

\$433.50

\$525.00

Show answer & explanation

Answer: $\$600.00$

Answer

The original standard price of the bulk supply was $600.00.

The final cost of

510 represents 85% of the original price, because a 15% discount was subtracted (100% - 15% = 85%). To find the original whole amount, divide the part by the percentage as a decimal:

510 \div 0.85 = \

600

.

Step-by-Step Solution

1

Determine the percentage of the original price that the final cost represents.

Since a

15\%

discount was applied, the manager paid

100\% - 15\% = 85\%

of the original price.

The final discounted price represents the remainder of the original cost after the discount is subtracted.

2

Set up an equation and solve for the original price.

Let

x

be the original price. The equation is

0.85x = 510

. Solving for

x

gives

x = 510 \div 0.85 = 600

.

Dividing the part (the discounted price) by its equivalent percentage (expressed as a decimal) gives the whole original amount.

Key Concept

To find the original price before a discount, divide the final discounted price by the complement of the discount percentage (100% minus the discount rate).

Question 2125Question →

The hypotenuse of a right triangle measures $13$ inches, and one of its legs measures $5$ inches. What is the length of the other leg?

12

inches

144

inches

194

inches

8

inches

Show answer & explanation

Answer: $12$ inches

Answer

The length of the other leg is

12

inches.

The correct answer is found by properly substituting the hypotenuse (

13

) and the known leg (

5

) into the Pythagorean theorem (

a^2 + b^2 = c^2

). Rearranging to solve for the missing leg gives

a^2 = 13^2 - 5^2

, which simplifies to

a^2 = 169 - 25 = 144

. Taking the square root of

144

yields

12

.

Step-by-Step Solution

1

Set up the Pythagorean theorem equation (

a^2 + b^2 = c^2

) with the given values.

a^2 + 5^2 = 13^2

The Pythagorean theorem relates the two legs (

a

and

b

) to the hypotenuse (

c

) of a right triangle.

2

Square the known numerical values.

a^2 + 25 = 169

Evaluating the exponents is the first operational step in solving the equation.

3

Isolate the variable term

a^2

by subtracting

25

from both sides.

a^2 = 144

This leaves the squared variable isolated on one side of the equation.

4

Take the square root of both sides to find the length of the leg.

a = 12

The square root is the inverse operation of squaring, yielding the final length.

Key Concept

The Pythagorean theorem (

a^2 + b^2 = c^2

) can be rearranged to find a missing leg by subtracting the square of the known leg from the square of the hypotenuse (

a^2 = c^2 - b^2

).

Question 2126Question →

A computer monitor is shaped like a rectangle. It has a diagonal measurement of $26$ inches and a height of $10$ inches. What is the perimeter of the computer monitor?

68

inches

240

inches

52

inches

72

inches

Show answer & explanation

Answer: $68$ inches

Answer

The correct perimeter of the monitor is

68

inches.

The correct answer is found by first recognizing that the diagonal, height, and width form a right triangle. Using the Pythagorean theorem, the missing width is calculated as the square root of (

26^2 - 10^2

), which equals

24

inches. The perimeter of the rectangular monitor is then

2

times the height plus

2

times the width:

2(10) + 2(24) = 68

inches.

Step-by-Step Solution

1

Identify the right triangle formed by the monitor's dimensions.

The diagonal (

26

inches) is the hypotenuse, and the height (

10

inches) is one leg.

A rectangle's diagonal divides it into two identical right triangles.

2

Use the Pythagorean theorem (

a^2 + b^2 = c^2

) to find the missing width.

10^2 + b^2 = 26^2

→

100 + b^2 = 676

→

b^2 = 576

→

b = 24

inches.

The width is required to calculate the perimeter of the rectangle.

3

Calculate the perimeter of the rectangle.

P = 2l + 2w = 2(10) + 2(24) = 20 + 48 = 68

inches.

The perimeter is the total distance around the outside of the monitor.

Key Concept

Triangles and Pythagorean Theorem

Question 2127Question →

A logistics team is packing a shipment containing standard crates and heavy crates. A load of $4$ standard crates and $3$ heavy crates weighs a total of $170$ pounds. Another load consisting of $2$ standard crates and $5$ heavy crates weighs a total of $190$ pounds. Based on this information, what is the weight, in pounds, of a single heavy crate?

Show answer & explanation

Answer: 30

Answer

30

The correct answer is found by translating the word problem into the system of equations

4x + 3y = 170

and

2x + 5y = 190

. By multiplying the second equation by

-2

, the

x

terms are eliminated when the equations are added, yielding

-7y = -210

, which simplifies to

y = 30

.

Step-by-Step Solution

1

Define variables for the unknown quantities.

Let

x

be the weight of a standard crate and

y

be the weight of a heavy crate.

To translate the word problem into a mathematical system of equations.

2

Set up a system of linear equations based on the given information.

Equation 1:

4x + 3y = 170

Equation 2:

2x + 5y = 190

To mathematically represent the total weights of the two different loads.

3

Use the elimination method to cancel out the

x

variable. Multiply Equation 2 by

-2

.

-2(2x + 5y) = -2(190) \Rightarrow -4x - 10y = -380

Creating a

-4x

term allows the

x

variables to be eliminated when added to the

4x

in Equation 1.

4

Add the new Equation 2 to Equation 1.

(4x + 3y) + (-4x - 10y) = 170 - 380 \Rightarrow -7y = -210

This leaves a single-variable equation that can be directly solved.

5

Solve for

y

.

y = \frac{-210}{-7} = 30

To isolate

y

, which represents the target value: the weight of the heavy crate.

Key Concept

Solving Systems of Linear Equations

Question 2128Question →

During a field training exercise, an armored vehicle consumes $45\%$ of its total fuel capacity on the first day and $30\%$ of its total fuel capacity on the second day. If the vehicle has a total fuel capacity of $400$ gallons, how many gallons of fuel remain at the end of the second day?

154

325

100

300

Show answer & explanation

Answer: 100

Answer

100

The vehicle consumed a total of 75% of its fuel capacity (45% + 30%). This means 25% of its original capacity remains. Multiplying the total capacity of 400 gallons by 0.25 gives 100 gallons remaining.

Step-by-Step Solution

1

Calculate the total percentage of fuel consumed over the two days.

45\% + 30\% = 75\%

Both percentages are based on the total fuel capacity, so they can be added together.

2

Determine the percentage of fuel remaining.

100\% - 75\% = 25\%

The total capacity represents 100%. Subtracting the consumed percentage gives the remaining percentage.

3

Calculate the remaining gallons by finding 25% of the total capacity.

0.25 \times 400 = 100

gallons

This converts the remaining percentage into the actual remaining volume of fuel.

Key Concept

Calculating percentages of a whole and finding the remaining amount after multiple percentage reductions.

Question 2129Question →

A circular helipad has a diameter of $20$ meters. What is the area of the helipad in square meters?

20\pi

400\pi

10\pi

100\pi

Show answer & explanation

Answer: $100\pi$

Answer

The correct area is

100\pi

square meters.

The area of a circle is found using the formula

A = \pi r^2

. Since the given diameter is

20

meters, the radius is half of that amount, which is

10

meters. Squaring the radius gives

100

, so the area is

100\pi

square meters.

Step-by-Step Solution

1

Identify the given information and what needs to be found.

The diameter of the helipad is

20

meters, and the area needs to be calculated.

Understanding the starting values is required to choose the correct geometric formulas.

2

Calculate the radius of the helipad.

The radius is half of the diameter:

r = 20 \div 2 = 10

meters.

The standard formula for the area of a circle requires the radius, not the diameter.

3

Apply the area formula for a circle.

A = \pi r^2 = \pi (10)^2 = 100\pi

square meters.

Squaring the radius and multiplying by

\pi

yields the total area.

Key Concept

Calculating the area of a circle from its diameter.

Question 2130Question →

A municipal park features a garden in the shape of a trapezoid. The parallel sides of the garden measure $18$ meters and $12$ meters. If the total area of the garden is $150$ square meters, what is the perpendicular distance between the parallel sides?

5

meters

10

meters

30

meters

50

meters

Show answer & explanation

Answer: $10$ meters

Answer

The perpendicular distance between the parallel sides is

10

meters.

The correct answer accurately applies the trapezoid area formula. By setting

150 = \frac{1}{2}(18 + 12)h

, the sum of the bases is calculated as

30

. Multiplying by

\frac{1}{2}

results in

15

. Dividing the total area (

150

) by

15

yields a perpendicular height of

10

meters.

Step-by-Step Solution

1

Identify the formula for the area of a trapezoid.

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

This formula relates the known area and bases to the unknown height.

2

Substitute the known values into the formula.

150 = \frac{1}{2}(18 + 12)h

The given area is

150

, and the bases are

18

and

12

.

3

Simplify the sum inside the parentheses.

150 = \frac{1}{2}(30)h

Adding the lengths of the bases gives

30

.

4

Multiply

30

by

\frac{1}{2}

.

150 = 15h

This isolates the variable term

h

.

5

Divide both sides by

15

to solve for

h

.

h = 10

Dividing by

15

determines the perpendicular height.

Key Concept

Using the area formula of a trapezoid to solve for an unknown dimension.

Question 2131Question →

A rectangular parking lot has a perimeter of $200$ yards. The length of the parking lot is $40$ yards greater than its width. What is the area of the parking lot, in square yards?

Show answer & explanation

Answer: 2100

Answer

2100

The correct answer is derived by first finding the individual dimensions using the perimeter formula and the given relationship (

l = w + 40

), which gives a width of 30 yards and a length of 70 yards. Multiplying these dimensions yields an area of 2100 square yards.

Step-by-Step Solution

1

Write the perimeter formula for a rectangle.

P = 2l + 2w

The perimeter is the total distance around the rectangle.

2

Translate the given information into equations.

2l + 2w = 200

and

l = w + 40

The problem states the perimeter is 200 yards and the length is 40 yards greater than the width.

3

Substitute the expression for length into the perimeter equation and solve for width.

2(w + 40) + 2w = 200 \Rightarrow 2w + 80 + 2w = 200 \Rightarrow 4w = 120 \Rightarrow w = 30

Substitution allows us to create an equation with only one variable.

4

Calculate the length.

l = 30 + 40 = 70

Now that we know the width is 30 yards, we can add 40 to find the length.

5

Calculate the area of the rectangle.

Area = l \times w = 70 \times 30 = 2100

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

Key Concept

Using algebraic equations to find the dimensions of a rectangle from its perimeter, and then calculating its area.

Question 2132Question →

Calculate the required average speed to complete the mission as specified. Enter your numerical answer in the blank.

Fill in the blanks below

A surveillance aircraft is assigned a

420

-mile mission route. The aircraft completes the first

240

miles of the route in

3

hours. To achieve an overall average speed of

60

miles per hour for the entire mission, the aircraft must complete the remaining distance at an average speed of miles per hour.

Show answer & explanation

Answer

45

To find the required speed for the second leg, you must first determine the total time allowed for the trip. Since the total distance is

420

miles and the target average speed is

60

mph, the total time must be

7

hours. The first leg took

3

hours, leaving

4

hours for the remainder of the trip. The remaining distance is

180

miles (

420 - 240

). Dividing the remaining

180

miles by the remaining

4

hours gives the required speed of

45

mph.

Step-by-Step Solution

1

Calculate the total time allowed for the entire mission.

Total Time =

420 \text{ miles} \div 60 \text{ mph} = 7 \text{ hours}

.

Overall average speed requires knowing the total allowable time, which is found by dividing the total distance by the target average speed.

2

Determine the remaining distance to be covered.

Remaining Distance =

420 \text{ miles} - 240 \text{ miles} = 180 \text{ miles}

.

Subtract the distance traveled during the first leg from the total mission distance.

3

Determine the remaining time available.

Remaining Time =

7 \text{ hours} - 3 \text{ hours} = 4 \text{ hours}

.

Subtract the time spent on the first leg from the total allowable mission time.

4

Calculate the required average speed for the remaining distance.

Required Speed =

180 \text{ miles} \div 4 \text{ hours} = 45 \text{ mph}

.

Speed is calculated by dividing the remaining distance by the remaining time.

Key Concept

Average Speed

Question 2133Question →

An IT director is upgrading a company's data center and orders 4 new enterprise servers. The original retail price for each server is $\$4,500$ . The hardware supplier offers a $20\%$ corporate discount on the total order. Additionally, because the company pays the invoice in full within 10 days, a $5\%$ early-payment discount is applied to the already reduced price. What is the final total price, in dollars, paid for the servers?

Show answer & explanation

Answer: 13680

Answer

13680

The correct total is found by first determining the initial total order value (

\$18,000

), reducing it by the initial

20\%

discount to find the new subtotal (

\$14,400

), and then reducing that subtotal by the additional

5\%

discount to arrive at the final cost (

\$13,680

).

Step-by-Step Solution

1

Calculate the total original price of the 4 servers.

4 \times \$4,500 = \$18,000

The discounts apply to the total order, so the total initial cost must be determined first.

2

Apply the 20% corporate discount to the total original price.

\$18,000 \times 0.20 = \$3,600 \text{ discount; } \$18,000 - \$3,600 = \$14,400

The first discount reduces the total cost before any other discounts are considered.

3

Apply the 5% early-payment discount to the new, reduced price.

\$14,400 \times 0.05 = \$720 \text{ discount; } \$14,400 - \$720 = \$13,680

Successive discounts are applied sequentially to the most recently calculated price, not the original price.

Key Concept

Calculating successive discounts on a total order amount.

Estimated Time:1m 30s

Question 2134Question →

A circle is perfectly inscribed within a square that has a perimeter of $48$ centimeters. What is the area of the region located inside the square but outside the circle, in square centimeters?

144\pi - 144

48 - 36\pi

144 - 36\pi

144 - 12\pi

Show answer & explanation

Answer: $144 - 36\pi$

Answer

The correct answer is

144 - 36\pi

.

To find the area of the specified region, the area of the inscribed circle must be subtracted from the area of the square. A perimeter of 48 gives the square a side length of 12 and an area of 144. The inscribed circle's diameter matches the square's side length (12), meaning its radius is 6. Its area is calculated as pi times the radius squared, which is 36pi. Subtracting these gives the final area of 144 - 36pi.

Step-by-Step Solution

1

Find the side length of the square.

The side length is

48 \div 4 = 12

cm.

The perimeter of a square is equal to

4

times its side length.

2

Calculate the area of the square.

The area is

12 \times 12 = 144

cm

^2

.

The area of a square is the side length squared.

3

Determine the radius of the inscribed circle.

The radius is

12 \div 2 = 6

cm.

Because the circle is perfectly inscribed, its diameter equals the side length of the square (

12

cm). The radius is half of the diameter.

4

Calculate the area of the circle.

The area is

\pi \times 6^2 = 36\pi

cm

^2

.

The formula for the area of a circle is

A = \pi r^2

.

5

Find the area of the region outside the circle but inside the square.

The area is

144 - 36\pi

cm

^2

.

Subtracting the circle's area from the total square's area leaves the remaining corner regions.

Key Concept

Area of composite figures and inscribed shapes

Question 2135Question →

An engineer is evaluating the specifications of a circular pipe and recording various measurements. Match each measurement description with the correct geometric term.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The boundary length matches to Circumference, the center-to-edge measurement to Radius, the edge-to-edge measurement through the center to Diameter, and the enclosed space to Area.

Each practical measurement perfectly aligns with a fundamental geometric property of a circle. The continuous outer boundary corresponds to the circumference. The distance from the center point to the edge is the radius. The line segment completely crossing the circle through its center is the diameter. Finally, the total two-dimensional region contained inside is the area.

Step-by-Step Solution

1

Analyze the description for the boundary of the circular pipe.

Matches with Circumference.

By definition, the perimeter or outer boundary of a circle is called its circumference.

2

Analyze the description for the measurement from the center to the outer edge.

Matches with Radius.

The radius is the line segment connecting the center point to any point on the outer edge of the circle.

3

Analyze the description for the full width passing through the center.

Matches with Diameter.

The diameter is the longest chord of a circle, spanning from one edge to the other directly through the center.

4

Analyze the description for the enclosed two-dimensional space.

Matches with Area.

Area is the mathematical term for the amount of surface space enclosed by a two-dimensional shape.

Key Concept

Basic terminology and properties of circles

Question 2136Question →

The two linear equations below form a system.

2x + 3y = 12

2x - y = 4

Based on the solution to this system, what is the value of the expression $x + 2y$ ?

5

7

10

12

Show answer & explanation

Answer: 7

Answer

The correct answer is 7.

The correct answer is found by first solving the system of equations. Subtracting the second equation from the first eliminates the x variable, giving 3y - (-y) = 12 - 4, which simplifies to 4y = 8 and y = 2. Substituting y = 2 into the second equation yields 2x - 2 = 4, meaning 2x = 6 and x = 3. Finally, evaluating the expression x + 2y with these values gives 3 + 2(2) = 3 + 4 = 7.

Step-by-Step Solution

1

Eliminate the x variable by subtracting the second equation from the first.

(2x - 2x) + (3y - (-y)) = 12 - 4 \Rightarrow 4y = 8

Because the coefficient of x is identical in both equations, subtracting them cancels out the x terms entirely, leaving an equation with only y.

2

Solve for y.

y = 2

Dividing both sides of the equation by 4 isolates the y variable.

3

Substitute the value of y back into the second equation to solve for x.

2x - (2) = 4 \Rightarrow 2x = 6 \Rightarrow x = 3

Using the known value of y allows us to find the corresponding x value for the system's ordered pair.

4

Evaluate the target expression

x + 2y

using the found values.

3 + 2(2) = 3 + 4 = 7

This answers the specific question asked in the stem by applying the solution to the given expression.

Key Concept

Solving Systems of Linear Equations by Elimination

Question 2137Question →

Read the problem carefully and calculate the remaining budget.

Fill in the blanks below

A base civil engineering squadron has an annual budget of \$

60,000

for facility maintenance. During the first quarter, they spend

30\%

of the total budget. During the second quarter, an unexpected repair requires them to spend

25\%

of the *remaining* budget. The amount of money left in the budget for the rest of the year is \$.

Show answer & explanation

Answer

The remaining budget is $31,500.

To find the final remaining amount, calculate the expenses sequentially. First, find

30\%

of the initial

\$60,000

, which is

\$18,000

. Subtracting this leaves

\$42,000

. Next, find

25\%

of the remaining

\$42,000

, which is

\$10,500

. Subtracting this from

\$42,000

leaves a final balance of

\$31,500

.

Step-by-Step Solution

1

Calculate the amount spent in the first quarter.

30\%

of

\$60,000 = 0.30 \times 60,000 = \$18,000

The squadron spends 30% of their initial total budget.

2

Calculate the budget remaining after the first quarter.

\$60,000 - \$18,000 = \$42,000

Subtract the first quarter expenses from the total budget to find the new balance.

3

Calculate the amount spent in the second quarter.

25\%

of

\$42,000 = 0.25 \times 42,000 = \$10,500

The second quarter expenses are based on the remaining budget, not the original total.

4

Calculate the final remaining amount.

\$42,000 - \$10,500 = \$31,500

Subtract the second quarter expenses from the balance to find the final remaining funds.

Key Concept

Calculating sequential percentages of changing quantities.

Question 2138Question →

A circular crop field has a diameter of $30$ yards. A dirt road that is $5$ yards wide is built along the entire outer edge of the field. What is the area of the dirt road in square yards?

175\pi

325\pi

25\pi

40\pi

Show answer & explanation

Answer: $175\pi$

Answer

The area of the dirt road is

175\pi

square yards.

The dirt road forms a ring (an annulus) around the circular field. To find this area, calculate the area of the total circular space (field plus road) and subtract the area of the field itself. The field's radius is

15

yards (

30 \div 2

), making the total outer radius

20

yards (

15 + 5

). Subtracting the inner area (

225\pi

) from the outer area (

400\pi

) gives the road's area:

175\pi

square yards.

Step-by-Step Solution

1

Find the radius of the inner circular field.

r_{inner} = \frac{30}{2} = 15

yards

The area formula for a circle requires the radius, but the diameter of the field is given.

2

Determine the total radius of the entire area, including the road.

r_{outer} = 15 + 5 = 20

yards

The dirt road adds

5

yards to the outer edge of the field's radius.

3

Calculate the total area enclosed by the outer edge of the road.

Area_{outer} = \pi (20)^2 = 400\pi

square yards

This represents the total area of the field and the road combined.

4

Calculate the area of just the inner crop field.

Area_{inner} = \pi (15)^2 = 225\pi

square yards

This area must be subtracted to isolate the area of the road.

5

Subtract the inner area from the total outer area.

400\pi - 225\pi = 175\pi

square yards

The area of the ring (annulus) is the difference between the two concentric circular areas.

Key Concept

Area of an annulus (a ring formed by two concentric circles)

Question 2139Question →

A rectangular community garden is $3$ times as long as it is wide. If the total area of the garden is $108$ square feet, what is the perimeter of the garden, in feet?

48

108

42

24

Show answer & explanation

Answer: $48$

Answer

The perimeter of the garden is

48

feet.

The correct answer is found by first determining the dimensions of the garden. Since the length is

3

times the width, the area can be represented as

(3w)(w) = 3w^2

. Setting this equal to the given area of

108

square feet gives

3w^2 = 108

. Dividing by

3

yields

w^2 = 36

, so the width is

6

feet. The length is

3 \times 6 = 18

feet. The perimeter formula for a rectangle is

P = 2(length + width)

. Substituting the dimensions gives

P = 2(18 + 6) = 2(24) = 48

feet.

Step-by-Step Solution

1

Let the width of the garden be

w

. The length is

3

times the width, so the length is

3w

.

Width =

w

, Length =

3w

Translating the word problem into algebraic expressions allows us to set up an equation.

2

Set up an equation using the formula for the area of a rectangle (

Area = length \times width

).

3w \times w = 108

We are given the total area, which links our variable expressions to a known value.

3

Solve the equation for

w

.

3w^2 = 108 \Rightarrow w^2 = 36 \Rightarrow w = 6

Isolating the variable determines the numerical value for the width.

4

Substitute

w = 6

back into the expression for length (

3w

).

Length =

3(6) = 18

feet

Both dimensions are needed to calculate the perimeter.

5

Calculate the perimeter using the formula

P = 2(length + width)

.

P = 2(18 + 6) = 2(24) = 48

feet

The perimeter represents the total distance around the outside of the rectangular garden.

Key Concept

Perimeter and Area of 2D Figures

Estimated Time:1m 0s

Question 2140Question →

A motor pool contains a total of 24 vehicles, consisting exclusively of transport trucks and utility vehicles. The number of transport trucks is 6 less than twice the number of utility vehicles. How many utility vehicles are in the motor pool?

Show answer & explanation

Answer: 10; ten; 10 utility vehicles

Answer

10

By defining

t

as transport trucks and

u

as utility vehicles, we can create the system

t + u = 24

and

t = 2u - 6

. Substituting the value of

t

into the first equation yields

(2u - 6) + u = 24

. Combining the terms gives

3u - 6 = 24

. Adding 6 to both sides results in

3u = 30

, and dividing by 3 shows there are

10

utility vehicles.

Step-by-Step Solution

1

Define variables for the unknown quantities.

Let

t

be the number of transport trucks and

u

be the number of utility vehicles.

This allows us to translate the word problem into a system of algebraic equations.

2

Set up the system of equations based on the problem statement.

Equation 1:

t + u = 24

(total vehicles).
Equation 2:

t = 2u - 6

(transport trucks are 6 less than twice the utility vehicles).

These equations accurately represent the given conditions in mathematical terms.

3

Substitute the expression for

t

from Equation 2 into Equation 1.

(2u - 6) + u = 24

Substitution eliminates the variable

t

, leaving a single-variable equation that can be solved directly for

u

.

4

Combine like terms and solve for

u

.

3u - 6 = 24 \Rightarrow 3u = 30 \Rightarrow u = 10

Isolating

u

gives the exact number of utility vehicles requested by the problem.

Key Concept

Solving Systems of Linear Equations Using Substitution

Alternative Method

This can also be solved by elimination if we rearrange the second equation to

-t + 2u = 6

. Adding this to

t + u = 24

yields

3u = 30

, which gives

u = 10

.

Estimated Time:1m 0s

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