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.
Other Related Questions
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.
sqrt(45) is between what two consecutive whole numbers?
- A. 4 and 5
- B. 5 and 6
- C. 6 and 7
- D. 14 and 15
- E. 22 and 23
Correct Answer & Rationale
Correct Answer: C
To determine between which two consecutive whole numbers \(\sqrt{45}\) lies, we can evaluate the squares of whole numbers around it. Calculating, \(6^2 = 36\) and \(7^2 = 49\). Since \(36 < 45 < 49\), it follows that \(6 < \sqrt{45} < 7\). Therefore, \(\sqrt{45}\) is between 6 and 7. Option A (4 and 5) is incorrect as \(4^2 = 16\) and \(5^2 = 25\), which are both less than 45. Option B (5 and 6) is also wrong since \(5^2 = 25\) and \(6^2 = 36\) are still below 45. Option D (14 and 15) and Option E (22 and 23) are far too high, as \(14^2 = 196\) and \(22^2 = 484\) exceed 45.
To determine between which two consecutive whole numbers \(\sqrt{45}\) lies, we can evaluate the squares of whole numbers around it. Calculating, \(6^2 = 36\) and \(7^2 = 49\). Since \(36 < 45 < 49\), it follows that \(6 < \sqrt{45} < 7\). Therefore, \(\sqrt{45}\) is between 6 and 7. Option A (4 and 5) is incorrect as \(4^2 = 16\) and \(5^2 = 25\), which are both less than 45. Option B (5 and 6) is also wrong since \(5^2 = 25\) and \(6^2 = 36\) are still below 45. Option D (14 and 15) and Option E (22 and 23) are far too high, as \(14^2 = 196\) and \(22^2 = 484\) exceed 45.
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.
Let g(x) = x². What is the average rate of change of the function from x = 4 to x = 8?
- A. 1/12
- B. $2
- C. $4
- D. $12
- E. $48
Correct Answer & Rationale
Correct Answer: C
To determine the average rate of change of the function g(x) = x² from x = 4 to x = 8, we use the formula: (g(b) - g(a)) / (b - a), where a = 4 and b = 8. Calculating g(4) = 4² = 16 and g(8) = 8² = 64. Thus, the average rate of change is (64 - 16) / (8 - 4) = 48 / 4 = 12. Option A (1/12) is incorrect as it underestimates the change. Option B ($2) and Option D ($12) miscalculate the average rate. Option E ($48) represents the total change but does not account for the interval length. The correct average rate of change is $12, reflecting the consistent increase of the function over the specified interval.
To determine the average rate of change of the function g(x) = x² from x = 4 to x = 8, we use the formula: (g(b) - g(a)) / (b - a), where a = 4 and b = 8. Calculating g(4) = 4² = 16 and g(8) = 8² = 64. Thus, the average rate of change is (64 - 16) / (8 - 4) = 48 / 4 = 12. Option A (1/12) is incorrect as it underestimates the change. Option B ($2) and Option D ($12) miscalculate the average rate. Option E ($48) represents the total change but does not account for the interval length. The correct average rate of change is $12, reflecting the consistent increase of the function over the specified interval.