The relationship between h, a person's height in inches, and f, the length in inches of the person's femur, is modeled by the equation: h = 1.88f + 32. Which statement correctly identifies and describes the slope of the equation?
- A. The slope of the equation is 1.88, and it represents the femur length, in inches, when the height is 32 inches.
- B. The slope of the equation is 1.88, and it represents the number of inches the height increases for each inch the femur length increases
- C. The slope of the equation is 1.88, and it represents the number of inches the femur length increases for each inch the height increases
- D. The slope of the equation is 32, and it represents the number of inches the height increases for each inch the femur length increases.
- E. The slope of the equation is 32, and it represents the height, in inches, when the femur length is 1.88 inches.
Correct Answer & Rationale
Correct Answer: B
The slope of 1.88 in the equation h = 1.88f + 32 indicates that for every additional inch in femur length (f), height (h) increases by 1.88 inches. This relationship highlights the direct impact of femur length on height. Option A misinterprets the slope, incorrectly stating it represents femur length at a specific height. Option C reverses the relationship, suggesting femur length increases with height, which is inaccurate. Option D incorrectly identifies the slope as 32 and misrepresents the relationship. Option E also incorrectly identifies the slope and misinterprets its meaning in the context of the equation.
The slope of 1.88 in the equation h = 1.88f + 32 indicates that for every additional inch in femur length (f), height (h) increases by 1.88 inches. This relationship highlights the direct impact of femur length on height. Option A misinterprets the slope, incorrectly stating it represents femur length at a specific height. Option C reverses the relationship, suggesting femur length increases with height, which is inaccurate. Option D incorrectly identifies the slope as 32 and misrepresents the relationship. Option E also incorrectly identifies the slope and misinterprets its meaning in the context of the equation.
Other Related Questions
The volume of 1 cup of water is 14.4 cubic inches. The diameter of an empty cylindrical can is 3.0 inches. The can holds 2.0 cups of water. What is the height of the can, to the nearest 0.1 inch?
- A. 1
- B. 2
- C. 3.1
- D. 4.1
- E. 6.2
Correct Answer & Rationale
Correct Answer: D
To find the height of the can, first determine the total volume of water it holds. Since 1 cup is 14.4 cubic inches, 2 cups equal 28.8 cubic inches (2 x 14.4). The formula for the volume of a cylinder is V = πr²h. The radius (r) of the can is half the diameter: 1.5 inches. Plugging in the values: 28.8 = π(1.5)²h. Calculating the area of the base gives approximately 7.07. Rearranging the equation for height (h) results in h ≈ 4.1 inches. Options A (1), B (2), C (3.1), and E (6.2) do not satisfy the volume calculation, as they yield heights inconsistent with the required volume based on the diameter provided.
To find the height of the can, first determine the total volume of water it holds. Since 1 cup is 14.4 cubic inches, 2 cups equal 28.8 cubic inches (2 x 14.4). The formula for the volume of a cylinder is V = πr²h. The radius (r) of the can is half the diameter: 1.5 inches. Plugging in the values: 28.8 = π(1.5)²h. Calculating the area of the base gives approximately 7.07. Rearranging the equation for height (h) results in h ≈ 4.1 inches. Options A (1), B (2), C (3.1), and E (6.2) do not satisfy the volume calculation, as they yield heights inconsistent with the required volume based on the diameter provided.
What is the product of the two polynomials: (x - 5)(x² - 3x + 6)?
- A. x³ - 8x² + 21x - 30
- B. x³ - 8x² - 21x - 30
- C. x³ - 8x² - 9x - 30
- D. x³ + 8x² + 21x + 30
- E. x³ + 8x² - 9x + 30
Correct Answer & Rationale
Correct Answer: A
To find the product of the polynomials (x - 5)(x² - 3x + 6), we apply the distributive property (FOIL method). 1. Multiply x by each term in the second polynomial: - x * x² = x³ - x * (-3x) = -3x² - x * 6 = 6x 2. Multiply -5 by each term in the second polynomial: - -5 * x² = -5x² - -5 * (-3x) = 15x - -5 * 6 = -30 Combining these results yields: x³ + (-3x² - 5x²) + (6x + 15x) - 30 = x³ - 8x² + 21x - 30. Option A matches this result. Options B and C have incorrect signs for the x terms. Option D has incorrect signs for all terms, and option E has incorrect signs for the x² and x terms. Thus, only option A accurately represents the product of the polynomials.
To find the product of the polynomials (x - 5)(x² - 3x + 6), we apply the distributive property (FOIL method). 1. Multiply x by each term in the second polynomial: - x * x² = x³ - x * (-3x) = -3x² - x * 6 = 6x 2. Multiply -5 by each term in the second polynomial: - -5 * x² = -5x² - -5 * (-3x) = 15x - -5 * 6 = -30 Combining these results yields: x³ + (-3x² - 5x²) + (6x + 15x) - 30 = x³ - 8x² + 21x - 30. Option A matches this result. Options B and C have incorrect signs for the x terms. Option D has incorrect signs for all terms, and option E has incorrect signs for the x² and x terms. Thus, only option A accurately represents the product of the polynomials.
What are the solutions to the equation: x² - 10?
- A. ±5
- B. ±√10
- C. ±10
- D. ±10²
- E. ±20
Correct Answer & Rationale
Correct Answer: B
To solve the equation \( x^2 - 10 = 0 \), we first isolate \( x^2 \) by adding 10 to both sides, resulting in \( x^2 = 10 \). Taking the square root of both sides gives us \( x = \pm\sqrt{10} \), which corresponds to option B. Option A, \( \pm5 \), is incorrect as \( 5^2 = 25 \), not 10. Option C, \( \pm10 \), is also wrong because \( 10^2 = 100 \). Option D, \( \pm10^2 \), misinterprets the operation, yielding \( \pm100 \), which is not relevant here. Lastly, option E, \( \pm20 \), is incorrect since \( 20^2 = 400 \). Thus, only option B accurately represents the solutions to the equation.
To solve the equation \( x^2 - 10 = 0 \), we first isolate \( x^2 \) by adding 10 to both sides, resulting in \( x^2 = 10 \). Taking the square root of both sides gives us \( x = \pm\sqrt{10} \), which corresponds to option B. Option A, \( \pm5 \), is incorrect as \( 5^2 = 25 \), not 10. Option C, \( \pm10 \), is also wrong because \( 10^2 = 100 \). Option D, \( \pm10^2 \), misinterprets the operation, yielding \( \pm100 \), which is not relevant here. Lastly, option E, \( \pm20 \), is incorrect since \( 20^2 = 400 \). Thus, only option B accurately represents the solutions to the equation.
Each month, the charge for a lawn care service consists of a flat fee of $25, plus $5 each time the lawn is mowed. Which of the following equations represents the total monthly charge, A(m), in dollars, as a function of the number of times the lawn is mowed, m?
- A. A(m) = 5(25)m
- B. A(m) = 5 + 25m
- C. A(m) = 5m + 25
- D. A(m) = 25m + 5
- E. A(m) = m + 5 + 25
Correct Answer & Rationale
Correct Answer: C
The equation A(m) = 5m + 25 accurately represents the total monthly charge for the lawn care service. Here, the term 5m accounts for the $5 charge per mowing, and the flat fee of $25 is added to this total. Option A incorrectly multiplies the flat fee by the number of mowings, which misrepresents the structure of the charges. Option B misplaces the flat fee, summing it with the number of mowings instead of adding it as a fixed cost. Option D incorrectly places the flat fee as a coefficient of m, which distorts the relationship. Option E combines the charges incorrectly, failing to clearly separate the flat fee from the per-mow charge.
The equation A(m) = 5m + 25 accurately represents the total monthly charge for the lawn care service. Here, the term 5m accounts for the $5 charge per mowing, and the flat fee of $25 is added to this total. Option A incorrectly multiplies the flat fee by the number of mowings, which misrepresents the structure of the charges. Option B misplaces the flat fee, summing it with the number of mowings instead of adding it as a fixed cost. Option D incorrectly places the flat fee as a coefficient of m, which distorts the relationship. Option E combines the charges incorrectly, failing to clearly separate the flat fee from the per-mow charge.