Choose the best answer. If necessary, use the paper you were given.
A playground at a mall is in the shape of a rectangle, and there is a 144 foot long fence around it. If the rectangle is 6 feet longer than it is wide, what is the width, in feet, of the rectangle?
- A. 33
- B. 39
- C. 69
- D. 75
Correct Answer & Rationale
Correct Answer: A
To find the width of the rectangle, let the width be represented as \( w \). The length, being 6 feet longer, can be expressed as \( w + 6 \). The perimeter of a rectangle is given by the formula \( P = 2(l + w) \). Here, the perimeter is 144 feet, leading to the equation \( 2(w + (w + 6)) = 144 \). Simplifying this gives \( 2(2w + 6) = 144 \), which reduces to \( 4w + 12 = 144 \), and further simplifies to \( 4w = 132 \), resulting in \( w = 33 \). Option B (39) is incorrect as it gives a perimeter of 156 feet. Option C (69) would lead to an impossible perimeter of 150 feet. Option D (75) results in a perimeter of 162 feet, which exceeds the given value. Thus, only option A satisfies all conditions, confirming the width as 33 feet.
To find the width of the rectangle, let the width be represented as \( w \). The length, being 6 feet longer, can be expressed as \( w + 6 \). The perimeter of a rectangle is given by the formula \( P = 2(l + w) \). Here, the perimeter is 144 feet, leading to the equation \( 2(w + (w + 6)) = 144 \). Simplifying this gives \( 2(2w + 6) = 144 \), which reduces to \( 4w + 12 = 144 \), and further simplifies to \( 4w = 132 \), resulting in \( w = 33 \). Option B (39) is incorrect as it gives a perimeter of 156 feet. Option C (69) would lead to an impossible perimeter of 150 feet. Option D (75) results in a perimeter of 162 feet, which exceeds the given value. Thus, only option A satisfies all conditions, confirming the width as 33 feet.
Other Related Questions
Which of the following represents the cost, in dollars, of renting a car for d days and driving m miles?
- A. 45+.25+ d+m
- B. 45.25+ dm
- C. 45d +.25m
- D. 45/d +25/m
Correct Answer & Rationale
Correct Answer: C
Option C accurately represents the cost of renting a car, where $45 is a fixed daily rental fee multiplied by the number of days (d) and $0.25 is the cost per mile multiplied by the number of miles driven (m). Option A incorrectly adds the fixed cost and variable costs without proper multiplication, leading to an illogical expression. Option B misrepresents the relationship by multiplying the daily rate by the miles driven, which does not reflect the cost structure. Option D divides the fixed cost by days and the cost per mile by miles, which does not align with standard cost calculations for renting a car.
Option C accurately represents the cost of renting a car, where $45 is a fixed daily rental fee multiplied by the number of days (d) and $0.25 is the cost per mile multiplied by the number of miles driven (m). Option A incorrectly adds the fixed cost and variable costs without proper multiplication, leading to an illogical expression. Option B misrepresents the relationship by multiplying the daily rate by the miles driven, which does not reflect the cost structure. Option D divides the fixed cost by days and the cost per mile by miles, which does not align with standard cost calculations for renting a car.
A bowl contains 6 green grapes, 10 red grapes, and 8 black grapes.Which of the following is the correct calculation for the probability of choosing a red grape and then without putting the red grape back into the bowl, choosing a green grape?
- A. 10/24+6/24
- B. 10/24+6/23
- C. 10/24*6/24
- D. 10/24*6/23
Correct Answer & Rationale
Correct Answer: D
To determine the probability of selecting a red grape followed by a green grape without replacement, the first step involves calculating the probability of the first event (selecting a red grape). There are 10 red grapes out of a total of 24 grapes, giving a probability of 10/24. After choosing a red grape, there are now 23 grapes left in the bowl, including 6 green grapes. Thus, the probability of then selecting a green grape is 6/23. Option A incorrectly adds the probabilities, which is not appropriate for sequential events. Option B uses the correct second probability but fails to multiply the probabilities of the two events. Option C mistakenly adds both probabilities instead of multiplying them. Only option D correctly multiplies the probabilities of the two dependent events.
To determine the probability of selecting a red grape followed by a green grape without replacement, the first step involves calculating the probability of the first event (selecting a red grape). There are 10 red grapes out of a total of 24 grapes, giving a probability of 10/24. After choosing a red grape, there are now 23 grapes left in the bowl, including 6 green grapes. Thus, the probability of then selecting a green grape is 6/23. Option A incorrectly adds the probabilities, which is not appropriate for sequential events. Option B uses the correct second probability but fails to multiply the probabilities of the two events. Option C mistakenly adds both probabilities instead of multiplying them. Only option D correctly multiplies the probabilities of the two dependent events.
If (2w + 7)(3w - 1) = 0 which of the following is a possible value of w?
- A. -3
- B. -0.28571
- C. 01-Mar
- D. 07-Feb
Correct Answer & Rationale
Correct Answer: D
To solve the equation (2w + 7)(3w - 1) = 0, we set each factor to zero. 1. For 2w + 7 = 0, solving gives w = -3. This corresponds to option A, which is a valid solution. 2. For 3w - 1 = 0, solving gives w = 1/3, approximately 0.333. Option B, -0.28571, does not match this value. 3. Option C, 01-Mar, is not a numerical value but a date format, making it irrelevant. 4. Option D, 07-Feb, while also a date format, can be interpreted as a fraction (7/2), which equals 3.5, not a solution to the equation. Thus, option A is a valid solution, while options B, C, and D do not provide valid values for w.
To solve the equation (2w + 7)(3w - 1) = 0, we set each factor to zero. 1. For 2w + 7 = 0, solving gives w = -3. This corresponds to option A, which is a valid solution. 2. For 3w - 1 = 0, solving gives w = 1/3, approximately 0.333. Option B, -0.28571, does not match this value. 3. Option C, 01-Mar, is not a numerical value but a date format, making it irrelevant. 4. Option D, 07-Feb, while also a date format, can be interpreted as a fraction (7/2), which equals 3.5, not a solution to the equation. Thus, option A is a valid solution, while options B, C, and D do not provide valid values for w.
Allison drives her car at an average speed of x miles per hour for y hours and travels 150 miles. Which of the following equations represents this situation?
- A. x + y = 150
- B. xy = 150
- C. y/x = 150
- D. x/y = 150
Correct Answer & Rationale
Correct Answer: B
The relationship between speed, time, and distance is expressed by the formula: distance = speed × time. In this scenario, Allison travels 150 miles at an average speed of x miles per hour for y hours, which translates to the equation xy = 150. Option A (x + y = 150) incorrectly suggests that speed and time add up to distance, which is not accurate. Option C (y/x = 150) misrepresents the relationship by implying that the ratio of time to speed equals distance, which is incorrect. Option D (x/y = 150) also misinterprets the relationship, suggesting that the ratio of speed to time equals distance. Thus, option B correctly captures the relationship among the variables.
The relationship between speed, time, and distance is expressed by the formula: distance = speed × time. In this scenario, Allison travels 150 miles at an average speed of x miles per hour for y hours, which translates to the equation xy = 150. Option A (x + y = 150) incorrectly suggests that speed and time add up to distance, which is not accurate. Option C (y/x = 150) misrepresents the relationship by implying that the ratio of time to speed equals distance, which is incorrect. Option D (x/y = 150) also misinterprets the relationship, suggesting that the ratio of speed to time equals distance. Thus, option B correctly captures the relationship among the variables.