All practice questions

Question 441Question →

A newly constructed rectangular parking lot is $40$ yards long and $25$ yards wide. What is the total perimeter of the parking lot, in yards?

65

105

130

1,000

Show answer & explanation

Answer: $130$

Answer

The total perimeter of the parking lot is

130

yards.

To find the perimeter of a rectangle, you must add the lengths of all four sides. Using the formula

P = 2(l + w)

, we substitute the given length (

40

) and width (

25

). Adding those gives

65

, and multiplying by

2

results in a total perimeter of

130

yards.

Step-by-Step Solution

1

Identify the shape and the given dimensions.

Shape: Rectangle. Length (

l

) =

40

yards, Width (

w

) =

25

yards.

Understanding the shape and dimensions is necessary to select the correct formula.

2

Recall the formula for the perimeter of a rectangle.

P = 2l + 2w

or

P = 2(l + w)

Perimeter is the total distance around the outside of the shape.

3

Substitute the dimensions into the formula and solve.

P = 2(40 + 25) = 2(65) = 130

Calculating the expression yields the final perimeter in yards.

Key Concept

Perimeter of Rectangles

Alternative Method

Instead of using the formula

P = 2(l + w)

, you can simply add all four sides together:

40 + 40 + 25 + 25 = 130

.

Estimated Time:45s

Question 442Question →

An inside sales representative is compensated with a fixed weekly base pay of $\$450$ . Furthermore, for any weekly sales volume surpassing an $\$8,000$ threshold, they receive a $5\%$ commission on the excess amount. If the representative generated $\$14,500$ in sales last week, what was their total gross pay for that week, in dollars?

Show answer & explanation

Answer: 775

Answer

775

The correct answer is determined by first finding the sales amount eligible for commission (

14,500 -

8,000 =

6,500). Next, calculate the 5% commission on this excess amount (

6,500 \times 0.05 =

325). Finally, add this commission to the fixed weekly base pay to find the total gross pay (

450 +

325 =

775).

Step-by-Step Solution

1

Determine the portion of sales subject to the commission rate.

\$14,500 - \$8,000 = \$6,500

The commission is only applied to the sales volume that surpasses the $8,000 threshold.

2

Calculate the commission earned on the excess sales.

\$6,500 \times 0.05 = \$325

The representative earns 5% on the eligible sales amount.

3

Calculate the total gross pay.

\$450 + \$325 = \$775

Total gross pay is the sum of the fixed base pay and the earned commission.

Key Concept

Calculating total gross pay using a base salary and a commission rate applied to sales above a specified threshold.

Question 443Question →

A combat engineer battalion is clearing debris from a damaged supply route. They are able to clear $1,800$ square feet of the route in $45$ minutes. Operating at this constant rate, what is the battalion's clearance rate in square feet per hour?

40

1,350

2,400

4,000

Show answer & explanation

Answer: $2,400$

Answer

The correct clearance rate is

2,400

square feet per hour.

To find the hourly clearance rate, we can first determine the rate per minute:

1,800

square feet divided by

45

minutes gives

40

square feet per minute. Since there are

60

minutes in an hour, we multiply

40

by

60

to get

2,400

square feet per hour. Alternatively,

45

minutes is

0.75

hours, and

1,800 \div 0.75 = 2,400

.

Step-by-Step Solution

1

Find the unit rate per minute by dividing the total area cleared by the total minutes.

1,800 \div 45 = 40

square feet per minute

This establishes the base clearance rate in the provided time unit.

2

Convert the per-minute rate to a per-hour rate by multiplying by

60

.

40 \times 60 = 2,400

square feet per hour

There are

60

minutes in one hour, so multiplying the minute rate by

60

gives the hourly rate.

Key Concept

Calculating unit rates and performing necessary unit conversions.

Estimated Time:1m 0s

Question 444Question →

Solve the following word problem by applying properties of right triangles.

Fill in the blanks below

A 25-foot ladder is leaning against a vertical wall, with its base placed 7 feet away from the bottom of the wall. If the top of the ladder slides exactly 4 feet straight down the wall, the base of the ladder will slide feet further away from the wall.

Show answer & explanation

Answer

8

By applying the Pythagorean theorem to the initial position, we establish that the ladder reached a height of 24 feet (

25^2 - 7^2 = 24^2

). When it slips 4 feet down, the new height becomes 20 feet. Using the Pythagorean theorem again for the new position with a constant hypotenuse of 25, the new base distance is 15 feet (

25^2 - 20^2 = 15^2

). The ladder started 7 feet away, so it moved an additional 8 feet (

15 - 7 = 8

).

Step-by-Step Solution

1

Find the initial height of the top of the ladder on the wall using the Pythagorean theorem.

a^2 + 7^2 = 25^2

, which simplifies to

a^2 + 49 = 625

. Solving for

a^2

gives

576

, so

a = 24

feet.

The ladder, the wall, and the ground form a right triangle where the ladder is the hypotenuse.

2

Calculate the new height of the top of the ladder after it slides down the wall.

24 - 4 = 20

feet.

The top of the ladder slid exactly 4 feet downwards, reducing the vertical leg of our right triangle.

3

Find the new distance of the base of the ladder from the wall.

b^2 + 20^2 = 25^2

, which simplifies to

b^2 + 400 = 625

. Solving for

b^2

gives

225

, so

b = 15

feet.

The ladder's overall length does not change, creating a new right triangle with a height of 20 and a hypotenuse of 25.

4

Calculate the difference between the new base distance and the original base distance.

15 - 7 = 8

feet.

The question specifically asks how much further the base slid away from the wall, not the final total distance.

Key Concept

Applying the Pythagorean theorem iteratively to linked right triangles that share a common hypotenuse length

Question 445Question →

Complete the factored form of the polynomial by finding the missing values.

Fill in the blanks below

The polynomial

6x^2 - 6x - 72

can be factored completely into the form

6(x - a)(x + b)

. The missing positive integers are

a =

and

b =

.

Show answer & explanation

Answer

The missing values are a = 4 and b = 3. The completely factored form is 6(x - 4)(x + 3).

To factor the polynomial

6x^2 - 6x - 72

completely, we first factor out the greatest common factor (GCF), which is

6

. This leaves us with the expression

6(x^2 - x - 12)

. Next, we factor the quadratic trinomial

x^2 - x - 12

. We need to find two integers that multiply to give

-12

and add up to

-1

. Those integers are

-4

and

3

. This means the trinomial factors as

(x - 4)(x + 3)

. Combining these, the complete factorization is

6(x - 4)(x + 3)

. By comparing this result to the given format

6(x - a)(x + b)

, we can see that the first missing value is

a = 4

and the second missing value is

b = 3

.

Step-by-Step Solution

1

Identify and factor out the greatest common factor (GCF) from the polynomial.

The GCF of the terms

6x^2

,

-6x

, and

-72

is

6

. Factoring this out gives:

6(x^2 - x - 12)

.

Factoring out the GCF simplifies the remaining quadratic expression, making it easier to factor into binomials.

2

Factor the remaining quadratic trinomial

x^2 - x - 12

.

We look for two numbers that multiply to

-12

and add up to

-1

(the coefficient of the middle term). These numbers are

-4

and

3

. Thus,

x^2 - x - 12

factors to

(x - 4)(x + 3)

.

A monic quadratic trinomial

x^2 + px + q

factors into

(x + m)(x + n)

where

m \times n = q

and

m + n = p

.

3

Write the complete factorization and match it to the given format.

The completely factored expression is

6(x - 4)(x + 3)

. Matching this to the given form

6(x - a)(x + b)

, we determine that

a = 4

and

b = 3

.

This directly identifies the specific positive integers required to complete the given algebraic expression.

Key Concept

Factoring out a Greatest Common Factor (GCF) followed by factoring a standard quadratic trinomial.

Question 446Question →

An architect is designing a room in the shape of a convex pentagon. The measures of the interior angles of the room are defined algebraically as $(x + 20)^\circ$ , $(x + 10)^\circ$ , $(x + 5)^\circ$ , $(x - 5)^\circ$ , and $(2x)^\circ$ . What is the measure of the smallest interior angle of this room?

10^\circ

50^\circ

80^\circ

90^\circ

Show answer & explanation

Answer: $80^\circ$

Answer

The smallest interior angle measures

80^\circ

.

A pentagon has 5 sides, so the sum of its interior angles is

(5 - 2) \times 180^\circ = 540^\circ

. Setting the sum of the algebraic expressions equal to

540^\circ

yields

6x + 30 = 540

. Subtracting

30

leaves

6x = 510

, so

x = 85

. Substituting

85

into the smallest expression,

(x - 5)^\circ

, gives

85 - 5 = 80^\circ

.

Step-by-Step Solution

1

Determine the sum of the interior angles for a pentagon.

Using the formula

(n - 2) \times 180^\circ

for a 5-sided polygon (

n = 5

), the sum is

(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ

.

Before solving for

x

, the total sum that the algebraic expressions add up to must be known.

2

Set up an equation by adding all the given angle expressions.

(x + 20) + (x + 10) + (x + 5) + (x - 5) + 2x = 540

The sum of the individual interior angles must equal the total calculated sum of the polygon's angles.

3

Combine like terms and solve for

x

.

6x + 30 = 540 \Rightarrow 6x = 510 \Rightarrow x = 85

Simplifying the algebraic equation isolates the variable

x

.

4

Substitute

x = 85

back into each expression to find all angle measures and identify the smallest one.

The angles are

105^\circ

,

95^\circ

,

90^\circ

,

80^\circ

, and

170^\circ

. The smallest of these is

80^\circ

.

The variable

x

alone is not the answer; it must be substituted back into the expressions to find the actual angle measures.

Key Concept

The sum of the interior angles of a convex polygon is

(n-2) \times 180^\circ

. This geometric principle can be combined with algebraic equations to solve for unknown angle measures.

Question 447Question →

A civil engineer is analyzing the intersection of two straight structural beams, Beam 1 and Beam 2. Angle $X$ and angle $Y$ form a linear pair along Beam 1. Angle $Y$ and angle $Z$ are vertical angles formed by the intersection of the two beams. The measure of angle $X$ is given by the expression $(3x - 10)^\circ$ and the measure of angle $Z$ is represented by $(2x + 15)^\circ$ . What is the measure of angle $Y$ in degrees?

49

65

85

89

Show answer & explanation

Answer: 85

Answer

85

The correct answer is found by recognizing that since angle X and angle Y form a linear pair, they sum to 180 degrees. Because angle Y and angle Z are vertical angles, they are equal. Substituting Z for Y means angle X and angle Z must also sum to 180 degrees. Setting up the equation

(3x - 10) + (2x + 15) = 180

yields

x = 35

. Substituting this back into the expression for angle Z (which is the same measure as angle Y) gives

2(35) + 15 = 85^\circ

.

Step-by-Step Solution

1

Identify the relationship between angle X and angle Z.

Since angle X and angle Y form a linear pair, they are supplementary (

X + Y = 180^\circ

). Angle Y and angle Z are vertical angles, so they are equal (

Y = Z

). Therefore, angle X and angle Z must also be supplementary (

X + Z = 180^\circ

).

Establishing the geometric relationship allows us to set up an algebraic equation based on the given expressions.

2

Set up and solve the algebraic equation for x.

(3x - 10) + (2x + 15) = 180 \rightarrow 5x + 5 = 180 \rightarrow 5x = 175 \rightarrow x = 35

.

Finding the numerical value of the variable is necessary to determine the specific angle measures.

3

Substitute the value of x back into the expression for angle Z (which equals angle Y).

Angle

Y = \text{Angle } Z = 2(35) + 15 = 70 + 15 = 85^\circ

.

The question specifically asks for the measure of angle Y in degrees.

Key Concept

Applying algebraic equations to solve for unknown values in supplementary and vertical angle relationships.

Question 448Question →

A retiree wants to earn exactly $1,800$ in simple interest over a period of 15 months to help pay for a vacation. A local credit union offers a short-term investment account with an annual simple interest rate of $4\%$ . Assuming no other deposits or withdrawals are made, how much money must the retiree deposit initially to reach this interest goal?

$3,000

$3,600

$36,000

$45,000

Show answer & explanation

Answer: $36,000

Answer

$36,000

To find the initial deposit (principal), the simple interest formula

I = P \times R \times T

must be rearranged to solve for

P

, yielding

P = \frac{I}{R \times T}

. Because the rate is an annual percentage, the 15-month period must first be converted into years (

15 / 12 = 1.25

years). Substituting the given values gives

P = \frac{1800}{0.04 \times 1.25}

, which simplifies to

1800 / 0.05 = 36000

. Therefore, an initial deposit of

36,000

is required.

Step-by-Step Solution

1

Convert the time duration from months to years.

T = \frac{15}{12} = 1.25

years

The interest rate is provided as an annual rate, so the time factor in the simple interest formula must also be expressed in years.

2

Set up the simple interest formula with the known values.

I = P \times R \times T \rightarrow 1800 = P \times 0.04 \times 1.25

The formula connects the interest earned (

I

) to the principal (

P

), rate (

R

), and time (

T

).

3

Solve for the unknown principal (

P

).

P = \frac{1800}{0.04 \times 1.25} = \frac{1800}{0.05} = 36000

Isolating

P

determines the initial deposit required to reach the interest goal.

Key Concept

Calculating the initial principal using the simple interest formula by isolating the principal variable and properly converting time into years.

Question 449Question →

A base facility manager is purchasing fuel for a backup generator. He needs $42.5$ gallons of diesel fuel. The supplier charges $\$0.90$ per quart of diesel. He also purchases a maintenance logbook for $\$14.25$ . Knowing that $1$ gallon equals $4$ quarts, what is the total cost of the purchase?

\$52.50

\$29.55

\$167.25

\$23.81

Show answer & explanation

Answer: $\$167.25$

Answer

The total cost of the purchase is

\$167.25

.

The correct answer is found by first unifying the units of measurement. Since the price is given per quart, either convert the total volume to quarts (

42.5 \times 4 = 170

quarts) or the price to gallons (

0.90 \times 4 = \$3.60

per gallon). Multiplying the volume by the corresponding rate yields a fuel cost of

\$153.00

. Adding the cost of the logbook (

\$14.25

) results in a final total of

\$167.25

.

Step-by-Step Solution

1

Find the equivalent amount of fuel in quarts.

42.5 \text{ gallons} \times 4 \text{ quarts/gallon} = 170 \text{ quarts}

To align the quantity of fuel with the given unit price per quart.

2

Calculate the total cost of the diesel fuel.

170 \text{ quarts} \times \$0.90/\text{quart} = \$153.00

To determine the primary expense before adding secondary items.

3

Add the cost of the maintenance logbook to the fuel cost.

\$153.00 + \$14.25 = \$167.25

To determine the final total cost of the entire purchase.

Key Concept

Unit conversions and decimal operations in currency problems

Question 450Question →

Two cars leave a parking lot at the same time and travel in opposite directions on a straight highway. One car travels at a constant speed of $40$ mph, and the other car travels at a constant speed of $50$ mph. How many hours will it take for the two cars to be exactly $270$ miles apart?

3

6

27

30

Show answer & explanation

Answer: 3

Answer

It will take 3 hours for the cars to be 270 miles apart.

The correct answer is 3. Since the cars are traveling in opposite directions, their speeds are added to find the total rate at which they are separating:

40 \text{ mph} + 50 \text{ mph} = 90 \text{ mph}

. To find the time it takes to separate by

270

miles, divide the total distance by the combined relative speed:

270 \div 90 = 3

hours.

Step-by-Step Solution

1

Determine the relative speed of the two cars.

40 + 50 = 90

mph

When two objects move in opposite directions away from each other, their relative speed is the sum of their individual speeds.

2

Calculate the time required to reach the target distance.

270 \div 90 = 3

hours

Time equals total distance divided by the relative speed (

T = D \div R

).

Key Concept

Relative Motion (Opposite Directions)

Question 451Question →

A competitive swimmer is walking around the outer edge of a rectangular pool to warm up. If the pool has a length of $50$ meters and a width of $25$ meters, what is the total perimeter of the pool, in meters?

Show answer & explanation

Answer: 150

Answer

The total perimeter of the pool is 150 meters.

The perimeter of a rectangle is found by adding the lengths of all four sides. Using the formula

P = 2l + 2w

, we calculate

2(50) + 2(25) = 150

meters.

Step-by-Step Solution

1

Identify the shape and the goal.

The shape is a rectangle and we need to find its perimeter.

Understanding what needs to be calculated determines the correct formula.

2

Recall the perimeter formula for a rectangle.

P = 2l + 2w

or

P = 2(l + w)

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

3

Substitute the given dimensions into the formula.

P = 2(50) + 2(25)

The length (

l

) is

50

and the width (

w

) is

25

.

4

Perform the arithmetic operations.

100 + 50 = 150

Multiplying the terms and adding them together yields the final perimeter.

Key Concept

Perimeter of Rectangles

Question 452Question →

A cafe sells muffins and scones. A customer purchases $2$ muffins and $1$ scone for a total of $11$ dollars. If the price of one scone is $4$ dollars less than the price of $3$ muffins, what is the price of one scone in dollars?

Show answer & explanation

Answer: 5

Answer

The price of one scone is

5

dollars.

By translating the word problem into a system of two equations (

2m + s = 11

and

s = 3m - 4

) and using substitution, we find that

m = 3

and

s = 5

. Thus, the price of one scone is

5

.

Step-by-Step Solution

1

Define variables and set up the equations based on the word problem.

Let

m

= price of a muffin and

s

= price of a scone. The equations are

2m + s = 11

and

s = 3m - 4

.

Translating the text into algebraic equations allows us to solve for the unknown values.

2

Substitute the expression for

s

from the second equation into the first equation.

2m + (3m - 4) = 11

This eliminates the variable

s

, leaving an equation with only one variable,

m

.

3

Solve the resulting equation for

m

.

5m - 4 = 11 \Rightarrow 5m = 15 \Rightarrow m = 3

Isolating

m

gives us the price of one muffin.

4

Substitute the value of

m

back into the equation for

s

.

s = 3(3) - 4 = 9 - 4 = 5

This calculates the price of the scone, which is the final requested value.

Key Concept

Solving systems of linear equations using the substitution method.

Question 453Question →

A field kitchen consumes $15$ pounds of coffee over a $4$ -hour breakfast service. What is the unit rate of coffee consumption, in ounces per hour?

Show answer & explanation

Answer: 60

Answer

60

The unit rate is found by first converting the total weight into the requested units. Since 1 pound equals 16 ounces, 15 pounds is equal to 240 ounces. Dividing this 240 ounces by the 4 hours of service gives the correct unit rate of 60 ounces per hour.

Step-by-Step Solution

1

Convert the total coffee weight from pounds to ounces.

15 \times 16 = 240

ounces

The question specifically asks for the rate in ounces per hour, and there are 16 ounces in 1 pound.

2

Calculate the unit rate by dividing the total ounces by the total hours.

240 \div 4 = 60

ounces per hour

A unit rate represents the quantity of one variable per single unit of another variable, found here by dividing total consumption by total time.

Key Concept

Calculating Unit Rates with Unit Conversions

Question 454Question →

A riverine patrol boat is tasked with delivering supplies to a forward operating base. The boat needs to cover a distance of $3.5$ miles upstream. If it maintains a constant speed over ground of $14$ miles per hour and departs at $0945$ hours, at what time will the boat arrive at its destination?

1215

hours

1000

hours

1345

hours

1010

hours

Show answer & explanation

Answer: $1000$ hours

Answer

The boat will arrive at

1000

hours.

To find the arrival time, first calculate the travel time. Using the formula Time = Distance / Rate, the time is

3.5 / 14 = 0.25

hours. Next, convert

0.25

hours to minutes by multiplying by

60

, which equals

15

minutes. Finally, add the

15

minutes of travel time to the departure time of

0945

hours. Adding

15

minutes to

0945

brings the clock time exactly to

1000

hours.

Step-by-Step Solution

1

Calculate the travel time in hours by dividing the distance by the rate.

t = \frac{3.5 \text{ miles}}{14 \text{ mph}} = 0.25 \text{ hours}

The formula for time is Time = Distance / Rate.

2

Convert the calculated time from hours to minutes.

0.25 \text{ hours} \times 60 \text{ minutes/hour} = 15 \text{ minutes}

Clock time additions are easier and more accurate when working with minutes rather than fractional hours.

3

Add the travel time in minutes to the departure time to find the arrival time.

0945 \text{ hours} + 15 \text{ minutes} = 1000 \text{ hours}

The arrival time is the departure time plus the elapsed travel time. Adding 15 minutes to 45 minutes completes the 60-minute hour, rolling the time over to 1000.

Key Concept

Calculating elapsed time using the distance formula (

t = d/r

) and converting fractional hours to minutes to determine clock arrival time.

Question 455Question →

A college student invests $\$2,500$ of their scholarship money into a bank savings certificate that pays an annual simple interest rate of $3.6\%$ . How many months will it take for the investment to earn exactly $\$135$ in interest?

Show answer & explanation

Answer: 18

Answer

18

The simple interest formula is

I = P \times r \times t

. Substituting the given values gives

135 = 2500 \times 0.036 \times t

. Solving for

t

yields

1.5

years. Multiplying by 12 converts this to exactly 18 months.

Step-by-Step Solution

1

Identify the simple interest formula and the given values.

I = P \times r \times t

, where

I = \$135

,

P = \$2,500

, and

r = 0.036

.

To set up the equation to solve for the unknown time variable.

2

Substitute the values into the formula and simplify.

135 = 2500 \times 0.036 \times t

, which simplifies to

135 = 90t

.

To isolate the term with the unknown variable.

3

Solve for the time

t

in years.

t = 135 / 90 = 1.5

years.

The standard simple interest formula calculates time in years.

4

Convert the time from years to months.

1.5 \times 12 = 18

months.

The question specifically asks for the duration in months.

Key Concept

Simple Interest

Question 456Question →

During an algebra calculation, an engineer needs to rewrite the expression $2x^2 + 7x - 15$ in its completely factored form. Which of the following is equivalent to this expression?

(2x - 3)(x + 5)

(2x + 3)(x - 5)

(2x - 5)(x + 3)

(2x + 15)(x - 1)

Show answer & explanation

Answer: $(2x - 3)(x + 5)$

Answer

The correct factorization is

(2x - 3)(x + 5)

.

The correct answer is mathematically equivalent to the original polynomial. Expanding it using the FOIL method confirms this:

(2x)(x) + (2x)(5) + (-3)(x) + (-3)(5) = 2x^2 + 10x - 3x - 15 = 2x^2 + 7x - 15

.

Step-by-Step Solution

1

Identify the coefficients of the quadratic expression

ax^2 + bx + c

.

a = 2

,

b = 7

, and

c = -15

.

Identifying these values is required to determine the target product and sum for factoring by grouping.

2

Find two numbers that multiply to

a \times c

and add up to

b

.

The numbers are

10

and

-3

because

10 \times (-3) = -30

and

10 + (-3) = 7

.

These specific numbers allow us to logically split the middle term of the polynomial.

3

Rewrite the middle term

7x

using these two numbers and factor the polynomial by grouping.

2x^2 + 10x - 3x - 15 = 2x(x + 5) - 3(x + 5) = (2x - 3)(x + 5)

.

Extracting the greatest common factor from each pair reveals the common binomial factor, completing the factorization.

Key Concept

Factoring non-monic quadratic trinomials by grouping

Question 457Question →

A corporate travel agent earns a fixed base salary of $\$2,400$ per month, plus a $6\%$ commission on the total value of all bookings she makes. If her goal is to earn exactly $\$4,200$ in gross pay this month, what must be the total value of her bookings?

\$70,000

\$30,000

\$3,000

\$1,794

Show answer & explanation

Answer: $\$30,000$

Answer

\$30,000

The total gross pay consists of the base salary plus the commission. By subtracting the base salary of

\$2,400

from the target gross pay of

\$4,200

, the required commission is determined to be

\$1,800

. Since the commission rate is

6\%

, dividing

\$1,800

by

0.06

yields the total required bookings of

\$30,000

.

Step-by-Step Solution

1

Determine the amount of pay that must come from commission.

\$4,200 - \$2,400 = \$1,800

The base salary is a fixed amount, so the remainder of the target gross pay must be earned through commissions.

2

Convert the commission percentage to a decimal.

6\% = 0.06

Percentages must be converted to decimals before they can be used in division calculations.

3

Calculate the total bookings required to earn the target commission.

\$1,800 \div 0.06 = \$30,000

Since commission is calculated as (Bookings × Commission Rate), dividing the required commission by the rate gives the required bookings.

Key Concept

Calculating required sales to meet a target income when given a base salary and a commission rate.

Question 458Question →

Two inspection drones take off from the same cellular tower at the same time and fly in opposite directions along a straight power line. Drone A flies at $15$ mph and Drone B flies at $25$ mph. How many miles apart are the two drones after $2$ hours?

Show answer & explanation

Answer: 80; 80 miles; 80 mi; eighty

Answer

The drones are 80 miles apart.

Since the drones travel in opposite directions, their relative speed is the sum of their individual speeds (

15 + 25 = 40

mph). Over the course of

2

hours, the distance between them becomes

40 \times 2 = 80

miles.

Step-by-Step Solution

1

Calculate the combined relative speed of both drones.

15 + 25 = 40

mph

Because the drones are flying in opposite directions, the distance between them increases at a rate equal to the sum of their individual speeds.

2

Multiply the relative speed by the time traveled.

40 \times 2 = 80

miles

Total distance is found by multiplying the relative speed by the total time (

D = r \times t

).

Key Concept

Calculating separation distance for objects moving in opposite directions using relative speed.

Estimated Time:45s

Question 459Question →

A rectangular metallic sheet has a length of $28$ inches and a width of $15$ inches. A triangular metallic sheet has the exact same area. If the base of the triangular sheet is $24$ inches, what is its height, in inches?

Show answer & explanation

Answer: 35

Answer

35

The correct height is found by first determining the area of the rectangle (

28 \times 15 = 420

). Since the triangle shares this exact area and has a given base of

24

, solving the triangle area formula

420 = \frac{1}{2}(24)h

yields

12h = 420

, which simplifies to

h = 35

.

Step-by-Step Solution

1

Calculate the area of the rectangular sheet.

Area = 420 square inches

The area of the rectangle is required to find the area of the triangle, as the problem explicitly states their areas are equal.

2

Set up the area equation for the triangular sheet.

420 = 0.5 * 24 * h

Substituting the known area and base into the triangle area formula allows us to solve for the unknown height.

3

Solve the algebraic equation for the height (h).

h = 35

Simplifying 420 = 12h by dividing both sides by 12 isolates the height variable.

Key Concept

Calculating and equating the area of different 2D figures using standard area formulas.

Estimated Time:1m 0s

Question 460Question →

A convex pentagon has three interior angles that are equal in measure. If the remaining two interior angles are supplementary to each other, what is the measure of one of the three equal angles?

60^\circ

120^\circ

150^\circ

180^\circ

Show answer & explanation

Answer: $120^\circ$

Answer

The measure of one of the three equal angles is

120^\circ

.

A pentagon has 5 sides, so the total sum of its interior angles is

(5 - 2) \times 180^\circ = 540^\circ

. The problem states that three angles are equal (we can call each

x

) and the other two are supplementary. Supplementary angles add up to

180^\circ

. Setting up the algebraic equation gives

3x + 180^\circ = 540^\circ

. Subtracting

180^\circ

from both sides yields

3x = 360^\circ

. Dividing by 3, we find that

x = 120^\circ

.

Step-by-Step Solution

1

Calculate the total sum of the interior angles of a pentagon.

The sum is

(5 - 2) \times 180^\circ = 540^\circ

.

The formula for the sum of interior angles of any convex polygon is

(n - 2) \times 180^\circ

, where

n

is the number of sides.

2

Set up an algebraic equation representing the sum of the five interior angles.

3x + 180^\circ = 540^\circ

Three angles are equal (let them be

x

), and the remaining two angles are supplementary, meaning their combined sum is exactly

180^\circ

.

3

Solve the equation for

x

.

3x = 360^\circ \Rightarrow x = 120^\circ

Subtracting

180^\circ

from both sides isolates the

3x

term. Dividing both sides by 3 gives the value of a single equal angle.

Key Concept

The sum of interior angles of a polygon and the geometric property of supplementary angles.

Answer

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