Choose the best answer. If necessary, use the paper you were given.
The expressions x - 2 and x + 3 represent the length and width of a rectangle, respectively. If the area of the rectangle is 24, what is the perimeter of the rectangle?
- A. 20
- B. 22
- C. 24
- D. 28
Correct Answer & Rationale
Correct Answer: B
To find the perimeter of the rectangle, first calculate its dimensions using the area formula. The area is given by multiplying length and width: \[ (x - 2)(x + 3) = 24 \] Expanding this, we get: \[ x^2 + x - 6 = 24 \implies x^2 + x - 30 = 0 \] Factoring yields: \[ (x - 5)(x + 6) = 0 \implies x = 5 \text{ (valid)} \text{ or } x = -6 \text{ (not valid)} \] Using \(x = 5\), the dimensions are \(3\) (length) and \(8\) (width). The perimeter is: \[ 2(3 + 8) = 22 \] Options A (20), C (24), and D (28) do not match the calculated perimeter of 22, confirming they are incorrect.
To find the perimeter of the rectangle, first calculate its dimensions using the area formula. The area is given by multiplying length and width: \[ (x - 2)(x + 3) = 24 \] Expanding this, we get: \[ x^2 + x - 6 = 24 \implies x^2 + x - 30 = 0 \] Factoring yields: \[ (x - 5)(x + 6) = 0 \implies x = 5 \text{ (valid)} \text{ or } x = -6 \text{ (not valid)} \] Using \(x = 5\), the dimensions are \(3\) (length) and \(8\) (width). The perimeter is: \[ 2(3 + 8) = 22 \] Options A (20), C (24), and D (28) do not match the calculated perimeter of 22, confirming they are incorrect.
Other Related Questions
Which equation is a correct way to calculate x?
- A. sin x=5,000 /7,000
- B. sin x= 7,000 /5,000
- C. tan x= 5,000/7,000
- D. tan x=7,000/5,000
Correct Answer & Rationale
Correct Answer: C
To solve for \( x \), the correct relationship involves the tangent function, as \( \tan \) is defined as the ratio of the opposite side to the adjacent side in a right triangle. Option C, \( \tan x = \frac{5,000}{7,000} \), accurately represents this ratio. Option A misapplies the sine function, which should represent the ratio of the opposite side to the hypotenuse, not the adjacent side. Similarly, option B incorrectly uses sine but with the sides reversed, leading to an inaccurate representation. Option D misuses tangent, suggesting the opposite and adjacent sides are swapped, which does not align with the definition of tangent. Thus, only option C correctly applies the tangent function to find \( x \).
To solve for \( x \), the correct relationship involves the tangent function, as \( \tan \) is defined as the ratio of the opposite side to the adjacent side in a right triangle. Option C, \( \tan x = \frac{5,000}{7,000} \), accurately represents this ratio. Option A misapplies the sine function, which should represent the ratio of the opposite side to the hypotenuse, not the adjacent side. Similarly, option B incorrectly uses sine but with the sides reversed, leading to an inaccurate representation. Option D misuses tangent, suggesting the opposite and adjacent sides are swapped, which does not align with the definition of tangent. Thus, only option C correctly applies the tangent function to find \( x \).
What is the range of her scores?
- A. 100
- B. 120
- C. 440
- D. 2,250
Correct Answer & Rationale
Correct Answer: B
To determine the range of her scores, we subtract the lowest score from the highest score. If the highest score is 220 and the lowest is 100, the calculation is 220 - 100 = 120, which represents the range. Option A (100) misrepresents the range as it does not account for the difference between the highest and lowest scores. Option C (440) and Option D (2,250) are excessively high and do not reflect the actual spread of scores based on the provided data. Thus, 120 accurately represents the range of her scores.
To determine the range of her scores, we subtract the lowest score from the highest score. If the highest score is 220 and the lowest is 100, the calculation is 220 - 100 = 120, which represents the range. Option A (100) misrepresents the range as it does not account for the difference between the highest and lowest scores. Option C (440) and Option D (2,250) are excessively high and do not reflect the actual spread of scores based on the provided data. Thus, 120 accurately represents the range of her scores.
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.
(a ^ 9 * b ^ 12)/(a ^ 3 * b) =
- A. a ^ 3 * b ^ 11
- B. a ^ 6 * b ^ 12
- C. a ^ 3 * b ^ 12
- D. a ^ 6 * b ^ 11
Correct Answer & Rationale
Correct Answer: D
To simplify the expression \((a^9 * b^{12})/(a^3 * b)\), apply the laws of exponents. For the \(a\) terms, subtract the exponents: \(9 - 3 = 6\), giving \(a^6\). For the \(b\) terms, also subtract the exponents: \(12 - 1 = 11\), resulting in \(b^{11}\). Thus, the simplified expression is \(a^6 * b^{11}\). Option A is incorrect because it miscalculates the exponent of \(b\). Option B incorrectly maintains the exponent of \(b\) at 12. Option C fails to adjust the exponent of \(a\) correctly. Only option D accurately reflects the simplification.
To simplify the expression \((a^9 * b^{12})/(a^3 * b)\), apply the laws of exponents. For the \(a\) terms, subtract the exponents: \(9 - 3 = 6\), giving \(a^6\). For the \(b\) terms, also subtract the exponents: \(12 - 1 = 11\), resulting in \(b^{11}\). Thus, the simplified expression is \(a^6 * b^{11}\). Option A is incorrect because it miscalculates the exponent of \(b\). Option B incorrectly maintains the exponent of \(b\) at 12. Option C fails to adjust the exponent of \(a\) correctly. Only option D accurately reflects the simplification.