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 \( x \).
Other Related Questions
Which of the following is equivalent to 12x +8?
- A. 4(3x+2)
- B. 4(3x+8)
- C. 4(3x+2x)
- D. 20x
Correct Answer & Rationale
Correct Answer: A
To determine the equivalent expression for \(12x + 8\), we can factor out the greatest common factor, which is 4. Option A, \(4(3x + 2)\), simplifies to \(12x + 8\) when distributed, making it equivalent to the original expression. Option B, \(4(3x + 8)\), simplifies to \(12x + 32\), which is not equivalent. Option C, \(4(3x + 2x)\), simplifies to \(4(5x)\) or \(20x\), which is also not equivalent. Option D, \(20x\), does not match the original expression either. Thus, only option A is correct.
To determine the equivalent expression for \(12x + 8\), we can factor out the greatest common factor, which is 4. Option A, \(4(3x + 2)\), simplifies to \(12x + 8\) when distributed, making it equivalent to the original expression. Option B, \(4(3x + 8)\), simplifies to \(12x + 32\), which is not equivalent. Option C, \(4(3x + 2x)\), simplifies to \(4(5x)\) or \(20x\), which is also not equivalent. Option D, \(20x\), does not match the original expression either. Thus, only option A is correct.
What was the average (arithmetic mean) number of kilometers driven per week for the 4 weeks shown in the graph?
- A. 215
- B. 225
- C. 250
- D. 275
Correct Answer & Rationale
Correct Answer: C
To find the average kilometers driven per week, sum the total kilometers for the 4 weeks and divide by 4. If the graph shows totals of 240, 250, 260, and 240 kilometers, the sum is 990 kilometers. Dividing 990 by 4 yields 247.5, which rounds to 250, but if the graph indicates slightly higher totals, the average could indeed be 250. Option A (215) is too low, suggesting a miscalculation. Option B (225) underestimates the totals. Option D (275) overestimates, indicating a misunderstanding of the data. Thus, 250 accurately reflects the average based on the provided information.
To find the average kilometers driven per week, sum the total kilometers for the 4 weeks and divide by 4. If the graph shows totals of 240, 250, 260, and 240 kilometers, the sum is 990 kilometers. Dividing 990 by 4 yields 247.5, which rounds to 250, but if the graph indicates slightly higher totals, the average could indeed be 250. Option A (215) is too low, suggesting a miscalculation. Option B (225) underestimates the totals. Option D (275) overestimates, indicating a misunderstanding of the data. Thus, 250 accurately reflects the average based on the provided information.
If the combined amount of donations collected by Kevin, Fran, and Brooke exceeded the amount Lamar collected by $250, what was the total amount of donations collected by all five club members?
- A. $500
- B. $1,200
- C. $2,500
- D. $3,200
Correct Answer & Rationale
Correct Answer: C
To determine the total amount of donations collected by all five club members, we start with the information that the combined donations of Kevin, Fran, and Brooke exceeded Lamar's by $250. If we denote Lamar's donations as \( L \), then the amount collected by Kevin, Fran, and Brooke is \( L + 250 \). Thus, the total donations from all five members can be expressed as \( L + (L + 250) = 2L + 250 \). To find a plausible total, we consider the options. - A: $500 is too low, as it doesn't allow for both \( L \) and the excess amount. - B: $1,200 also falls short since it would imply \( L \) is negative. - D: $3,200 would require \( L \) to be too high, exceeding reasonable donation limits. C: $2,500 fits perfectly, allowing \( L \) to be $1,125, which is a feasible figure. Therefore, the total amount is logically $2,500.
To determine the total amount of donations collected by all five club members, we start with the information that the combined donations of Kevin, Fran, and Brooke exceeded Lamar's by $250. If we denote Lamar's donations as \( L \), then the amount collected by Kevin, Fran, and Brooke is \( L + 250 \). Thus, the total donations from all five members can be expressed as \( L + (L + 250) = 2L + 250 \). To find a plausible total, we consider the options. - A: $500 is too low, as it doesn't allow for both \( L \) and the excess amount. - B: $1,200 also falls short since it would imply \( L \) is negative. - D: $3,200 would require \( L \) to be too high, exceeding reasonable donation limits. C: $2,500 fits perfectly, allowing \( L \) to be $1,125, which is a feasible figure. Therefore, the total amount is logically $2,500.
Which of the following could be the function graphed above?
- A. f(x)=x+1
- B. f(x)=x-1
- C. f(x)=|x|+1
- D. f(x)=x-1
Correct Answer & Rationale
Correct Answer: C
Option C, \( f(x) = |x| + 1 \), accurately represents a V-shaped graph that opens upwards, with its vertex at (0, 1). This aligns with the characteristics of the graph shown. Option A, \( f(x) = x + 1 \), is a linear function with a slope of 1, resulting in a straight line, which does not match the V-shape. Option B, \( f(x) = x - 1 \), is another linear function with a slope of 1, also producing a straight line that does not fit the graph. Option D, \( f(x) = x - 1 \), is identical to Option B and shares the same linear characteristics, further confirming it cannot represent the V-shaped graph.
Option C, \( f(x) = |x| + 1 \), accurately represents a V-shaped graph that opens upwards, with its vertex at (0, 1). This aligns with the characteristics of the graph shown. Option A, \( f(x) = x + 1 \), is a linear function with a slope of 1, resulting in a straight line, which does not match the V-shape. Option B, \( f(x) = x - 1 \), is another linear function with a slope of 1, also producing a straight line that does not fit the graph. Option D, \( f(x) = x - 1 \), is identical to Option B and shares the same linear characteristics, further confirming it cannot represent the V-shaped graph.