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
The average of 4 numbers is 9. If one of the numbers is 7, what is the sum of the other 3 numbers?
- A. 2
- B. 12
- C. 29
- D. 36
Correct Answer & Rationale
Correct Answer: C
To find the sum of the other three numbers, start by calculating the total sum of all four numbers. Since the average is 9, multiply this by 4, yielding a total of 36. Given that one of the numbers is 7, subtract this from the total: 36 - 7 = 29. Therefore, the sum of the other three numbers is 29. Option A (2) is too low, as it does not account for the total sum needed. Option B (12) underestimates the remaining numbers. Option D (36) mistakenly includes the known number, rather than calculating the sum of the others.
To find the sum of the other three numbers, start by calculating the total sum of all four numbers. Since the average is 9, multiply this by 4, yielding a total of 36. Given that one of the numbers is 7, subtract this from the total: 36 - 7 = 29. Therefore, the sum of the other three numbers is 29. Option A (2) is too low, as it does not account for the total sum needed. Option B (12) underestimates the remaining numbers. Option D (36) mistakenly includes the known number, rather than calculating the sum of the others.
In the figure above, what is the average (arithmetic mean) of w, x, y, and z?
- A. 90
- B. 100
- C. 120
- D. It cannot be determined from the information given.
Correct Answer & Rationale
Correct Answer: D
To find the average of w, x, y, and z, all values must be known. Option D is valid since the problem does not provide specific values or relationships between these variables, making it impossible to calculate their average. Option A (90), Option B (100), and Option C (120) suggest definitive averages, but without concrete data on w, x, y, and z, these answers cannot be substantiated. Each of these options assumes values that may not exist or be accurate, highlighting the necessity of complete information for such calculations.
To find the average of w, x, y, and z, all values must be known. Option D is valid since the problem does not provide specific values or relationships between these variables, making it impossible to calculate their average. Option A (90), Option B (100), and Option C (120) suggest definitive averages, but without concrete data on w, x, y, and z, these answers cannot be substantiated. Each of these options assumes values that may not exist or be accurate, highlighting the necessity of complete information for such calculations.
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.
Which of the following is a factor of x ^ 3 * y ^ 3 + x * y ^ 5 ?
- A. x ^ 3 - y ^ 3
- B. x ^ 3 + y ^ 3
- C. x ^ 2 + y ^ 2
- D. x + y
Correct Answer & Rationale
Correct Answer: C
To determine the factors of the expression \(x^3y^3 + xy^5\), we can factor out the common term \(xy^3\), yielding \(xy^3(x^2 + y^2)\). Option A, \(x^3 - y^3\), represents a difference of cubes and does not apply here. Option B, \(x^3 + y^3\), is a sum of cubes, which is not a factor of the given expression. Option D, \(x + y\), does not appear in the factorization derived from the original expression. Thus, \(x^2 + y^2\) is the only viable factor, confirming its role in the factorization of the expression.
To determine the factors of the expression \(x^3y^3 + xy^5\), we can factor out the common term \(xy^3\), yielding \(xy^3(x^2 + y^2)\). Option A, \(x^3 - y^3\), represents a difference of cubes and does not apply here. Option B, \(x^3 + y^3\), is a sum of cubes, which is not a factor of the given expression. Option D, \(x + y\), does not appear in the factorization derived from the original expression. Thus, \(x^2 + y^2\) is the only viable factor, confirming its role in the factorization of the expression.