All practice questions

Question 261Question →

During a training exercise, an army jeep travels at a constant fuel consumption rate of 14 miles per gallon. If the jeep starts with exactly 8 gallons of fuel in its tank, what is the maximum number of miles the jeep can travel before running out of gas?

Show answer & explanation

Answer: 112

Answer

112

The correct answer is found by multiplying the fuel efficiency rate (14 miles per gallon) by the amount of fuel in the tank (8 gallons). 14 * 8 equals 112 miles.

Step-by-Step Solution

1

Identify the given values

Fuel efficiency = 14 miles per gallon, Fuel available = 8 gallons

These are the necessary components to calculate total distance.

2

Calculate the total distance

14 * 8 = 112 miles

Distance is the product of the rate of fuel consumption (miles per gallon) and the total amount of fuel (gallons).

Key Concept

Calculating maximum travel range based on fuel efficiency and fuel volume.

Question 262Question →

A base landscaping crew needs to purchase concentrated herbicide. Vendor A sells a $5$ -gallon drum of the herbicide for $\$140.00$ . Vendor B sells the same herbicide in $2$ -quart bottles for $\$15.00$ each. What is the difference in price per gallon, in dollars, between the two vendors?

Show answer & explanation

Answer: 2

Answer

The difference in price per gallon between the two vendors is 2 dollars.

By converting both options to a common unit price of dollars per gallon, we find Vendor A charges

28.00 per gallon and Vendor B charges

30.00 per gallon. The difference between these unit prices is

30.00 -

28.00 = $2.00.

Step-by-Step Solution

1

Calculate the price per gallon for Vendor A.

\$140.00 \div 5 \text{ gallons} = \$28.00 \text{ per gallon}

To compare prices accurately, both must be expressed as a unit rate (price per gallon).

2

Convert the volume of Vendor B's bottles into gallons to find how many bottles equal one gallon.

Since there are

4 \text{ quarts}

in

1 \text{ gallon}

, a

2

-quart bottle is

0.5 \text{ gallons}

. Therefore, it takes

2

bottles to make

1 \text{ gallon}

.

Unit prices must be in the same measurement (gallons) for a valid comparison.

3

Calculate the price per gallon for Vendor B.

\$15.00 \times 2 \text{ bottles} = \$30.00 \text{ per gallon}

This establishes the unit rate for Vendor B.

4

Find the difference between the two unit prices.

\$30.00 - \$28.00 = \$2.00

The question asks for the numeric difference in price per gallon.

Key Concept

Comparing unit prices by converting to a common unit of measure.

Question 263Question →

A straight road intersects two parallel train tracks. One of the acute angles formed at an intersection measures $72^\circ$ . What is the measure of any of the obtuse angles formed by these parallel tracks and the intersecting road, in degrees?

Show answer & explanation

Answer: 108

Answer

108

When a transversal intersects parallel lines, the sum of any acute angle and any obtuse angle formed is always

180^\circ

. Subtracting

72^\circ

from

180^\circ

gives the correct obtuse angle of

108^\circ

.

Step-by-Step Solution

1

Identify the relationship between the angles.

The acute and obtuse angles are supplementary.

When a transversal intersects parallel lines, consecutive interior angles (or any adjacent pair of acute and obtuse angles) add up to

180^\circ

.

2

Calculate the measure of the obtuse angle.

180^\circ - 72^\circ = 108^\circ

Subtracting the given acute angle from

180^\circ

yields the supplementary obtuse angle.

Key Concept

Parallel Lines and Transversals

Estimated Time:30s

Question 264Question →

An engineering battalion is tasked with laying $4,200$ tons of asphalt to repair a section of an airfield runway. By the end of the first week, they have completed $65\%$ of the total required amount. How many tons of asphalt do they still need to lay to complete the repair?

Show answer & explanation

Answer: 1470

Answer

1470

The correct answer is found by determining the remaining percentage of the total project. If

65\%

of the work is completed, then

100\% - 65\% = 35\%

remains. To find the exact tonnage, calculate

35\%

of

4,200

tons by converting the percentage to a decimal and multiplying:

4,200 \times 0.35 = 1,470

.

Step-by-Step Solution

1

Find the percentage of the asphalt that still needs to be laid.

100\% - 65\% = 35\%

The total project represents

100\%

. Subtracting the completed percentage gives the remaining percentage.

2

Calculate

35\%

of the total

4,200

tons.

4,200 \times 0.35 = 1,470

To find a percentage of a quantity, convert the percentage to a decimal and multiply it by the total quantity.

Key Concept

Calculating the remaining percentage of a whole quantity

Estimated Time:1m 0s

Question 265Question →

A commercial photographer is purchasing a new studio lighting kit. The original retail price of the kit is $\$1,200$ . The camera supply store is currently offering a $25\%$ seasonal discount on all studio equipment. In addition, the photographer receives a $10\%$ professional discount, which is applied to the already reduced price. What is the final price of the lighting kit before sales tax?

Show answer & explanation

Answer: 810

Answer

The final price of the lighting kit is

\$810

.

The correct calculation requires applying the first discount to find a new subtotal, and then applying the second discount to that new subtotal. Applying

25\%

off

\$1,200

yields

\$900

. Applying the

10\%

professional discount to the new base of

\$900

reduces the price by another

\$90

, resulting in a final price of

\$810

.

Step-by-Step Solution

1

Calculate the first discount amount.

25\%

of

\$1,200

is

\$300

.

The seasonal discount applies to the original retail price.

2

Find the intermediate reduced price.

\$1,200 - \$300 = \$900

.

Subtracting the first discount gives the new base price before the second discount.

3

Calculate the second discount amount.

10\%

of

\$900

is

\$90

.

The professional discount applies to the already reduced price, not the original price.

4

Determine the final price.

\$900 - \$90 = \$810

.

Subtracting the second discount from the intermediate price gives the final amount paid.

Key Concept

Calculating successive discounts requires applying the second percentage to the new, already discounted price, rather than the original amount.

Question 266Question →

An isosceles triangle has a perimeter of $36$ centimeters. The base of the triangle measures $10$ centimeters. What is the area of the triangle in square centimeters?

Show answer & explanation

Answer: 60

Answer

60

To find the area, you must first know the perpendicular height. By splitting the isosceles triangle in half via its altitude, a right triangle is formed with a hypotenuse of

13

(the side length) and a base of

5

(half of the full base). The Pythagorean theorem (

5^2 + h^2 = 13^2

) reveals the height is

12

. The full area is then calculated as

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

, which is

\frac{1}{2} \times 10 \times 12 = 60

.

Step-by-Step Solution

1

Calculate the length of the two equal sides.

The two equal sides each measure

13

centimeters.

The perimeter is

36

cm and the base is

10

cm. The remaining

26

cm must be divided equally between the two identical legs:

26 \div 2 = 13

cm.

2

Determine the base of the right triangle formed by the altitude.

The base of the right triangle is

5

centimeters.

The altitude of an isosceles triangle bisects the base, splitting the

10

-centimeter base into two equal

5

-centimeter segments.

3

Calculate the height using the Pythagorean theorem.

The height is

12

centimeters.

In the right triangle formed, the hypotenuse is

13

and one leg is

5

. Let

h

be the height.

5^2 + h^2 = 13^2 \Rightarrow 25 + h^2 = 169 \Rightarrow h^2 = 144 \Rightarrow h = 12

.

4

Calculate the total area of the triangle.

The area is

60

square centimeters.

The area formula is

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

. Thus,

\frac{1}{2} \times 10 \times 12 = 60

.

Key Concept

Using the Pythagorean theorem to find missing dimensions in isosceles triangles to calculate area.

Question 267Question →

An armored cavalry troop conducts a training movement. They travel $45$ miles over rough terrain at a speed of $15$ miles per hour. After a $90$ -minute halt for vehicle maintenance, they proceed $75$ miles along a highway at $50$ miles per hour. What is their overall average speed, in miles per hour, for the entire operation?

Show answer & explanation

Answer: 20

Answer

The overall average speed is

20

miles per hour.

The correct answer is

20

because average speed is the ratio of total distance to total time. The total distance is

120

miles (

45 + 75

). The total time includes the travel times (

3

hours and

1.5

hours) plus the maintenance halt (

1.5

hours), which equals

6

hours. Dividing

120

miles by

6

hours gives an overall average speed of

20

miles per hour.

Step-by-Step Solution

1

Calculate the time taken for the first travel segment.

\frac{45}{15} = 3

hours

Time is required to determine the total elapsed time of the operation.

2

Convert the maintenance halt time from minutes to hours.

90

minutes

= 1.5

hours

All time units must be consistent (hours) before they can be added together.

3

Calculate the time taken for the second travel segment.

\frac{75}{50} = 1.5

hours

This gives the travel time for the final leg of the route.

4

Calculate the total distance traveled.

45 + 75 = 120

miles

Total distance is the numerator in the average speed formula.

5

Calculate the total elapsed time for the entire operation.

3 + 1.5 + 1.5 = 6

hours

Total time is the denominator in the average speed formula, and it must include the maintenance halt.

6

Calculate the overall average speed.

\frac{120}{6} = 20

miles per hour

Average speed is defined as the total distance divided by the total time.

Key Concept

Average speed is calculated by dividing the total distance traveled by the total time elapsed, including any rest stops or delays.

Question 268Question →

A hiker walks $8$ miles due north, then turns and walks $15$ miles due east. What is the straight-line distance, in miles, from the hiker's starting point to their current location?

Show answer & explanation

Answer: 17

Answer

17

Because the hiker's path consists of two perpendicular segments (north and east), they form the legs of a right triangle measuring

8

miles and

15

miles. The straight-line distance from the start to the end is the hypotenuse of this triangle. Applying the Pythagorean theorem (

a^2 + b^2 = c^2

), we get

8^2 + 15^2 = c^2

, which simplifies to

64 + 225 = 289

. The square root of

289

is

17

. Therefore, the straight-line distance is

17

miles.

Step-by-Step Solution

1

Visualize the geometrical shape formed by the hiker's path.

A right triangle is formed with legs of

8

miles and

15

miles.

The directions 'due north' and 'due east' are perpendicular, creating a

90^\circ

angle.

2

Set up the Pythagorean theorem equation.

8^2 + 15^2 = c^2

The straight-line distance corresponds to the hypotenuse (

c

) of the right triangle.

3

Square the values of the legs and add them together.

64 + 225 = 289

To evaluate the left side of the equation.

4

Calculate the square root of the sum.

c = \sqrt{289} = 17

To solve for

c

and find the final straight-line distance.

Key Concept

Calculating distance using the Pythagorean theorem

Question 269Question →

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 270Question →

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 271Question →

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 272Question →

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 273Question →

A rectangular garden gate needs a diagonal wooden brace for support. The gate is $4$ feet tall and $3$ feet wide. What is the length, in feet, of the diagonal brace?

Show answer & explanation

Answer: 5

Answer

5

The diagonal brace forms the hypotenuse of a right triangle where the width (

3

feet) and height (

4

feet) are the legs. According to the Pythagorean theorem (

a^2 + b^2 = c^2

), we substitute the given lengths:

3^2 + 4^2 = c^2

. This simplifies to

9 + 16 = 25

. Taking the square root of

25

gives a diagonal length of

5

feet. This forms a common

3-4-5

Pythagorean triple.

Step-by-Step Solution

1

Identify the right triangle formed by the gate

The legs of the right triangle are

3

and

4

, and the hypotenuse is the unknown diagonal brace.

The corners of a rectangular gate form right angles, so drawing a diagonal creates a right triangle.

2

Apply the Pythagorean theorem

3^2 + 4^2 = c^2

The Pythagorean theorem (

a^2 + b^2 = c^2

) is used to find the missing side length of a right triangle.

3

Calculate the squares of the legs

9 + 16 = c^2

Squaring the given lengths prepares the equation for addition.

4

Solve for the hypotenuse

25 = c^2

, which means

c = 5

Taking the square root of the sum yields the length of the diagonal brace.

Key Concept

Using the Pythagorean theorem to find the hypotenuse of a right triangle.

Question 274Question →

A rectangular shipping crate has a length of $6$ feet, a width of $4$ feet, and a height of $3$ feet. What is the volume of the crate in cubic feet?

Show answer & explanation

Answer: 72

Answer

72

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

6

feet by

4

feet by

3

feet results in exactly

72

cubic feet.

Step-by-Step Solution

1

Recall the volume formula for a rectangular prism.

V = l \times w \times h

To find the volume, we must multiply the length, width, and height of the 3D figure.

2

Substitute the given dimensions into the formula.

V = 6 \times 4 \times 3

The problem states the length is

6

feet, the width is

4

feet, and the height is

3

feet.

3

Multiply the values to find the total volume.

V = 24 \times 3 = 72

Completing the arithmetic gives the final volume in cubic feet.

Key Concept

Volume of a Rectangular Prism

Question 275Question →

A base defense unit receives a shipment of $7,500$ sandbags to reinforce security perimeters. If $18\%$ of the sandbags are deployed to the north gate and $34\%$ are deployed to the east gate, how many sandbags remain in the reserve depot?

Show answer & explanation

Answer: 3600

Answer

There are 3,600 sandbags remaining in the reserve depot.

The correct answer is found by first determining the total percentage of sandbags allocated to the gates, which is

18\% + 34\% = 52\%

. Subtracting this from the total

100\%

leaves

48\%

for the reserve depot. Multiplying

0.48

by the initial

7,500

sandbags gives the final remaining amount of

3,600

sandbags.

Step-by-Step Solution

1

Calculate the total percentage of deployed sandbags.

18\% + 34\% = 52\%

This consolidates the parts of the whole that have already been allocated.

2

Determine the percentage of sandbags that remain.

100\% - 52\% = 48\%

Subtracting the allocated portion from the total whole (100%) gives the remaining fraction.

3

Calculate 48% of the initial 7,500 sandbags.

0.48 \times 7500 = 3600

This translates the remaining percentage back into a specific physical quantity.

Key Concept

Calculating the remaining percentage of a quantity by subtracting the used percentages from

100\%

, then multiplying by the original total.

Question 276Question →

A kite designer is creating a kite in the shape of a rhombus. The two crossing wooden dowels that form the internal diagonals of the kite measure $16$ inches and $30$ inches. What is the total perimeter of the kite's outer edge, in inches?

Show answer & explanation

Answer: 68

Answer

68

The correct answer is found by recognizing that a rhombus's diagonals bisect each other perpendicularly. This allows the use of the Pythagorean theorem with half-diagonal lengths (

8

and

15

) to find a side length of

17

. Multiplying this single side length by the four equal sides of the rhombus yields a total perimeter of

68

inches.

Step-by-Step Solution

1

Determine the properties of the diagonals in a rhombus.

The diagonals bisect each other at right angles, forming four identical right triangles.

This breaks down the rhombus into simpler geometric shapes (right triangles), allowing for side length calculation.

2

Calculate the lengths of the legs for one of the right triangles.

The legs are

8

inches (

16 \div 2

) and

15

inches (

30 \div 2

).

The diagonals cut each other exactly in half at the intersection.

3

Use the Pythagorean theorem to find the length of the rhombus's outer edge.

The side length is

\sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17

inches.

The outer edge forms the hypotenuse of the right triangle created by the half-diagonals.

4

Calculate the total perimeter.

17 \times 4 = 68

inches.

A rhombus has four equal sides, so the total perimeter is four times the length of one side.

Key Concept

Perimeter of a Rhombus and Pythagorean Theorem

Question 277Question →

A military supply convoy is tasked with delivering equipment to an outpost $210$ miles away. The convoy commander is ordered to maintain an overall average speed of $35$ miles per hour for the entire journey. If the convoy completes the first $90$ miles in $2$ hours, what average speed, in miles per hour, must they maintain for the remainder of the trip to exactly meet their overall time target?

Show answer & explanation

Answer: 30

Answer

The convoy must maintain a speed of 30 mph for the remainder of the trip.

The overall target requires the

210

-mile trip to be completed in

6

hours (

210 \div 35

). Because the first

90

miles took

2

hours, there are

120

miles left to travel and

4

hours remaining to do so. Therefore, the required speed is

120

miles

\div

4

hours =

30

mph.

Step-by-Step Solution

1

Calculate the total allowable time for the convoy's trip.

Total time =

210 \div 35 = 6

hours.

Overall average speed is the total distance divided by total time. Knowing the total distance and target average speed allows us to find the total time allowed.

2

Calculate the remaining distance for the second part of the trip.

Remaining distance =

210 - 90 = 120

miles.

We need to find out how much further the convoy has to travel.

3

Calculate the remaining time available to complete the journey.

Remaining time =

6 - 2 = 4

hours.

Subtracting the time already spent from the total time gives the time left to cover the remaining distance.

4

Calculate the required average speed for the final leg.

Required speed =

120 \div 4 = 30

mph.

Average speed for any leg is the distance of that leg divided by the time spent on that leg.

Key Concept

Average speed is calculated by dividing the total distance by the total time, and the properties of individual legs (distance and time) sum to the total distance and total time.

Question 278Question →

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 279Question →

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 280Question →

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.

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