An airplane is 5,000 ft above ground and has to land on a runway that is 7,000 ft away as shown above. Let x be the angle the pilot takes to land the airplane at the beginning of the runway. Which equation is a correct way to calculate x?
- A. sin x = 5000/7000
- B. sin x = 7000/5000
- C. tan x = 5000/7000
- D. tan x = 7/5000
Correct Answer & Rationale
Correct Answer: C
To determine the angle \( x \) for landing, we need to consider the relationship between the height of the airplane and the distance to the runway. The height (5000 ft) is the opposite side of the right triangle formed, while the distance to the runway (7000 ft) is the adjacent side. The tangent function relates these two sides, hence \( \tan x = \frac{\text{opposite}}{\text{adjacent}} \) leads to \( \tan x = \frac{5000}{7000} \). Option A incorrectly uses the sine function, which relates the opposite side to the hypotenuse. Option B also misapplies sine but swaps the sides, leading to an incorrect ratio. Option D incorrectly uses tangent but misrepresents the sides, making it invalid. Thus, option C accurately represents the relationship needed to calculate angle \( x \).
To determine the angle \( x \) for landing, we need to consider the relationship between the height of the airplane and the distance to the runway. The height (5000 ft) is the opposite side of the right triangle formed, while the distance to the runway (7000 ft) is the adjacent side. The tangent function relates these two sides, hence \( \tan x = \frac{\text{opposite}}{\text{adjacent}} \) leads to \( \tan x = \frac{5000}{7000} \). Option A incorrectly uses the sine function, which relates the opposite side to the hypotenuse. Option B also misapplies sine but swaps the sides, leading to an incorrect ratio. Option D incorrectly uses tangent but misrepresents the sides, making it invalid. Thus, option C accurately represents the relationship needed to calculate angle \( x \).
Other Related Questions
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.
The price P, in dollars, that a store sets for an item is given by the equation P = C + 1/10 * C where C dollars is the store's cost for the item. If the store sets a price of $55.00 for an item, what is the store's cost for the item?
- A. $50.00
- B. $54.90
- C. $55.10
- D. $60.50
Correct Answer & Rationale
Correct Answer: A
To find the store's cost \( C \), we start with the equation \( P = C + \frac{1}{10}C \). This can be simplified to \( P = 1.1C \). Given that \( P = 55 \), we can set up the equation \( 55 = 1.1C \). Solving for \( C \) gives \( C = \frac{55}{1.1} = 50 \). Option A ($50.00) is correct, as it satisfies the equation. Option B ($54.90) incorrectly suggests a cost that would lead to a higher price than $55 when applying the markup. Option C ($55.10) implies a cost greater than the set price, which is illogical. Option D ($60.50) is also incorrect as it would result in a price far exceeding $55, making it unfeasible.
To find the store's cost \( C \), we start with the equation \( P = C + \frac{1}{10}C \). This can be simplified to \( P = 1.1C \). Given that \( P = 55 \), we can set up the equation \( 55 = 1.1C \). Solving for \( C \) gives \( C = \frac{55}{1.1} = 50 \). Option A ($50.00) is correct, as it satisfies the equation. Option B ($54.90) incorrectly suggests a cost that would lead to a higher price than $55 when applying the markup. Option C ($55.10) implies a cost greater than the set price, which is illogical. Option D ($60.50) is also incorrect as it would result in a price far exceeding $55, making it unfeasible.
If an item regularly costs d dollars and is discounted 12 percent, which of the following represents the discounted price in dollars?
- A. 0.12d
- B. 0.88d
- C. 1.12d
- D. d-0.12
Correct Answer & Rationale
Correct Answer: B
To find the discounted price after a 12 percent discount on an item that costs d dollars, we first calculate the amount of the discount, which is 12% of d, or 0.12d. To determine the final price, we subtract this discount from the original price: d - 0.12d = 0.88d. Option A (0.12d) represents only the discount amount, not the final price. Option C (1.12d) incorrectly suggests an increase in price. Option D (d - 0.12) does not account for the percentage; it inaccurately represents the discount as a flat dollar amount rather than a percentage of the original price. Thus, 0.88d correctly reflects the discounted price.
To find the discounted price after a 12 percent discount on an item that costs d dollars, we first calculate the amount of the discount, which is 12% of d, or 0.12d. To determine the final price, we subtract this discount from the original price: d - 0.12d = 0.88d. Option A (0.12d) represents only the discount amount, not the final price. Option C (1.12d) incorrectly suggests an increase in price. Option D (d - 0.12) does not account for the percentage; it inaccurately represents the discount as a flat dollar amount rather than a percentage of the original price. Thus, 0.88d correctly reflects the discounted price.
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.