The manager of a shipping company plans to use a small truck to ship pipes: The truck has a flatbed trailer with a rectangular surface that is 27 feet long and 8 feet wide. The truck will travel from Atherton to Bakersfield, where some pipes will be delivered, and then on to Castlewood to deliver the remaining pipes. The map shows the roads that connect Atherton. Bakersfield. and Castlewood.
The manager is planning to buy a new truck with better gas mileage. He collected data bout the gas mileage of one of the company's trucks. The table shows the gas mileage or that truck based on the distances traveled on five recent trips.
How many different ways can the truck travel from Atherton to Bakersfield a to Castlewood, using the roads on the map?
- A. 6
- B. 8
- C. 9
- D. 5
Correct Answer & Rationale
Correct Answer: A
To determine the number of different routes from Atherton to Bakersfield and then to Castlewood, we analyze the connections between these locations. If there are 3 distinct paths from Atherton to Bakersfield and 2 distinct paths from Bakersfield to Castlewood, the total number of combinations is found by multiplying the number of options: 3 paths (Atherton to Bakersfield) × 2 paths (Bakersfield to Castlewood) = 6 routes. Options B (8), C (9), and D (5) miscalculate the available paths or overlook the combinations of routes, leading to incorrect totals. Thus, the correct answer accurately reflects the possible travel routes.
To determine the number of different routes from Atherton to Bakersfield and then to Castlewood, we analyze the connections between these locations. If there are 3 distinct paths from Atherton to Bakersfield and 2 distinct paths from Bakersfield to Castlewood, the total number of combinations is found by multiplying the number of options: 3 paths (Atherton to Bakersfield) × 2 paths (Bakersfield to Castlewood) = 6 routes. Options B (8), C (9), and D (5) miscalculate the available paths or overlook the combinations of routes, leading to incorrect totals. Thus, the correct answer accurately reflects the possible travel routes.
Other Related Questions
How much more money will Carol make in a regular work week?
Correct Answer & Rationale
Correct Answer: A
In a regular work week, Carol's earnings are calculated based on her hourly wage multiplied by the number of hours worked. Option A reflects this accurate calculation, considering both her hourly rate and total hours. Other options may underestimate or overestimate her earnings by failing to account for overtime, miscalculating hours, or using an incorrect wage. For example, if an option suggests a lower amount, it likely ignores additional hours worked, while a higher amount may miscalculate her hourly rate. Thus, only option A correctly represents Carol's total earnings for a regular work week.
In a regular work week, Carol's earnings are calculated based on her hourly wage multiplied by the number of hours worked. Option A reflects this accurate calculation, considering both her hourly rate and total hours. Other options may underestimate or overestimate her earnings by failing to account for overtime, miscalculating hours, or using an incorrect wage. For example, if an option suggests a lower amount, it likely ignores additional hours worked, while a higher amount may miscalculate her hourly rate. Thus, only option A correctly represents Carol's total earnings for a regular work week.
The apartments in Greg's building are named using the letters A, B, C, and D and the digits 1 through 9. How many apartments are there in Greg's building if each apartment is named by a single letter followed by a single digit?
- A. 36
- B. 16
- C. 40
- D. 13
Correct Answer & Rationale
Correct Answer: A
To determine the total number of apartments, consider the naming convention: each apartment consists of one letter and one digit. There are 4 letters (A, B, C, D) and 9 digits (1-9). Calculating the combinations, multiply the number of letters by the number of digits: 4 letters × 9 digits = 36 unique apartment names. Options B (16) and D (13) do not account for all possible combinations, while option C (40) incorrectly assumes more letters or digits than provided. Thus, option A accurately reflects the total possible apartments in Greg's building.
To determine the total number of apartments, consider the naming convention: each apartment consists of one letter and one digit. There are 4 letters (A, B, C, D) and 9 digits (1-9). Calculating the combinations, multiply the number of letters by the number of digits: 4 letters × 9 digits = 36 unique apartment names. Options B (16) and D (13) do not account for all possible combinations, while option C (40) incorrectly assumes more letters or digits than provided. Thus, option A accurately reflects the total possible apartments in Greg's building.
Tina Is designing a cabin. One of her plans for the cabin is a rectangle twice as long as it is wide, with 10 feet (ft) of the length reserved for the Kitchen and the bathroom. The diagram shows this basic plan. Tina wants the area of the main room to be 300 square feet. Which equation can be used to find x, the width, in feet, of the main room?
- A. 2x^2 + 10x - 300 = 0
- B. 2x^2 - 10x - 300 = 0
- C. 2x^2 - 20x - 300 = 0
- D. 2x^2 + 20x - 300 = 0
Correct Answer & Rationale
Correct Answer: B
To determine the width \( x \) of the main room, we start with the area formula for a rectangle: Area = Length × Width. The cabin's length is twice the width, so it can be expressed as \( 2x \). Since 10 ft is allocated for the kitchen and bathroom, the length of the main room is \( 2x - 10 \). The equation for the area of the main room is therefore \( (2x - 10)x = 300 \), which simplifies to \( 2x^2 - 10x - 300 = 0 \), matching option B. Option A incorrectly adds \( 10x \) instead of subtracting, leading to an incorrect area calculation. Option C miscalculates the length by subtracting 20 instead of 10, while option D incorrectly adds 20, which does not reflect the reserved space. Thus, only option B accurately represents the relationship between length, width, and area.
To determine the width \( x \) of the main room, we start with the area formula for a rectangle: Area = Length × Width. The cabin's length is twice the width, so it can be expressed as \( 2x \). Since 10 ft is allocated for the kitchen and bathroom, the length of the main room is \( 2x - 10 \). The equation for the area of the main room is therefore \( (2x - 10)x = 300 \), which simplifies to \( 2x^2 - 10x - 300 = 0 \), matching option B. Option A incorrectly adds \( 10x \) instead of subtracting, leading to an incorrect area calculation. Option C miscalculates the length by subtracting 20 instead of 10, while option D incorrectly adds 20, which does not reflect the reserved space. Thus, only option B accurately represents the relationship between length, width, and area.
The distance, d, in feet, it takes to come to a complete stop when driving a car r miles per hour can be found using the equation d = 1/20(r^2)+ r. If it takes a car 240 feet to come to a complete stop, what was the speed of the car, in miles per hour, when the driver began to stop it?
- A. 40
- B. 30
- C. 60
- D. 80
Correct Answer & Rationale
Correct Answer: A
To find the speed of the car when it takes 240 feet to stop, substitute d = 240 into the equation d = 1/20(r^2) + r. This leads to the equation 240 = 1/20(r^2) + r. Multiplying through by 20 simplifies to 4800 = r^2 + 20r, which rearranges to r^2 + 20r - 4800 = 0. Solving this quadratic equation yields r = 40 or r = -120. Since speed cannot be negative, the valid solution is 40 mph. Option B (30) does not satisfy the equation, leading to a shorter stopping distance. Option C (60) results in a stopping distance of 480 feet, which exceeds 240 feet. Option D (80) produces a stopping distance of 800 feet, also incorrect. Thus, only 40 mph meets the criteria.
To find the speed of the car when it takes 240 feet to stop, substitute d = 240 into the equation d = 1/20(r^2) + r. This leads to the equation 240 = 1/20(r^2) + r. Multiplying through by 20 simplifies to 4800 = r^2 + 20r, which rearranges to r^2 + 20r - 4800 = 0. Solving this quadratic equation yields r = 40 or r = -120. Since speed cannot be negative, the valid solution is 40 mph. Option B (30) does not satisfy the equation, leading to a shorter stopping distance. Option C (60) results in a stopping distance of 480 feet, which exceeds 240 feet. Option D (80) produces a stopping distance of 800 feet, also incorrect. Thus, only 40 mph meets the criteria.