Point C is the center of the regular hexagon shown above. Which of the following expressions represents the area of this hexagon?
- A. 12xy
- B. 6xy
- C. 3xy
- D. xy
Correct Answer & Rationale
Correct Answer: B
The area of a regular hexagon can be calculated using the formula \( \frac{3\sqrt{3}}{2} s^2 \), where \( s \) is the length of a side. The expression \( 6xy \) aligns with this area formula when considering specific dimensions of the hexagon defined by \( x \) and \( y \). Option A, \( 12xy \), overestimates the area, suggesting a larger hexagon than the dimensions allow. Option C, \( 3xy \), and Option D, \( xy \), both underestimate the area, not accounting for the full extent of the hexagon's geometry. Thus, \( 6xy \) accurately represents the area based on the given variables.
The area of a regular hexagon can be calculated using the formula \( \frac{3\sqrt{3}}{2} s^2 \), where \( s \) is the length of a side. The expression \( 6xy \) aligns with this area formula when considering specific dimensions of the hexagon defined by \( x \) and \( y \). Option A, \( 12xy \), overestimates the area, suggesting a larger hexagon than the dimensions allow. Option C, \( 3xy \), and Option D, \( xy \), both underestimate the area, not accounting for the full extent of the hexagon's geometry. Thus, \( 6xy \) accurately represents the area based on the given variables.
Other Related Questions
Which of the following could be an equation of the line graphed in the xy-plane above?
- A. y=-x-3
- B. y=-x+3
- C. y=x-3
- D. y=x+3
Correct Answer & Rationale
Correct Answer: D
To determine the equation of the line, we analyze its slope and y-intercept. The line in the graph has a positive slope, indicating that as \(x\) increases, \(y\) also increases. Option D, \(y = x + 3\), has a positive slope of 1 and a y-intercept of 3, aligning with the graph's characteristics. Option A, \(y = -x - 3\), has a negative slope and would decrease as \(x\) increases, which contradicts the graph. Option B, \(y = -x + 3\), also has a negative slope, leading to a downward trend. Option C, \(y = x - 3\), has a positive slope but a y-intercept of -3, placing it below the graph. Thus, D is the only option that fits the observed line.
To determine the equation of the line, we analyze its slope and y-intercept. The line in the graph has a positive slope, indicating that as \(x\) increases, \(y\) also increases. Option D, \(y = x + 3\), has a positive slope of 1 and a y-intercept of 3, aligning with the graph's characteristics. Option A, \(y = -x - 3\), has a negative slope and would decrease as \(x\) increases, which contradicts the graph. Option B, \(y = -x + 3\), also has a negative slope, leading to a downward trend. Option C, \(y = x - 3\), has a positive slope but a y-intercept of -3, placing it below the graph. Thus, D is the only option that fits the observed line.
For how many values of k is (x, y) = (k, -k) a solution to the equation 2x +2y = 0?
- A. None
- B. One
- C. Two
- D. More than two
Correct Answer & Rationale
Correct Answer: D
To determine how many values of \( k \) make \( (x, y) = (k, -k) \) a solution to the equation \( 2x + 2y = 0 \), substitute \( x \) and \( y \) into the equation. This gives \( 2k + 2(-k) = 0 \), which simplifies to \( 0 = 0 \). This statement is always true, meaning any value of \( k \) satisfies the equation. Option A (None) is incorrect; there are indeed solutions. Option B (One) is also wrong since infinitely many values of \( k \) work. Option C (Two) is insufficient, as there are not just two but infinitely many solutions. Hence, the correct interpretation is that there are more than two values of \( k \) that satisfy the equation.
To determine how many values of \( k \) make \( (x, y) = (k, -k) \) a solution to the equation \( 2x + 2y = 0 \), substitute \( x \) and \( y \) into the equation. This gives \( 2k + 2(-k) = 0 \), which simplifies to \( 0 = 0 \). This statement is always true, meaning any value of \( k \) satisfies the equation. Option A (None) is incorrect; there are indeed solutions. Option B (One) is also wrong since infinitely many values of \( k \) work. Option C (Two) is insufficient, as there are not just two but infinitely many solutions. Hence, the correct interpretation is that there are more than two values of \( k \) that satisfy the equation.
If the length of a rectangle is increased by 30% and the width of the same rectangle is decreased by 30%, what is the effect on the area of the rectangle?
- A. It is increased by 60%.
- B. It is unchanged.
- C. It is decreased by 15%.
- D. It is decreased by 9%.
Correct Answer & Rationale
Correct Answer: D
Increasing the length of a rectangle by 30% results in a new length of 1.3L, while decreasing the width by 30% gives a new width of 0.7W. The new area can be calculated as A' = (1.3L)(0.7W) = 0.91LW, indicating a decrease in area. Option A is incorrect because a 60% increase does not occur; the area actually decreases. Option B is wrong as the area changes due to the modifications in dimensions. Option C suggests a decrease of 15%, which miscalculates the area change. The area decreases by 9%, confirming the effect of the opposing percentage changes in length and width.
Increasing the length of a rectangle by 30% results in a new length of 1.3L, while decreasing the width by 30% gives a new width of 0.7W. The new area can be calculated as A' = (1.3L)(0.7W) = 0.91LW, indicating a decrease in area. Option A is incorrect because a 60% increase does not occur; the area actually decreases. Option B is wrong as the area changes due to the modifications in dimensions. Option C suggests a decrease of 15%, which miscalculates the area change. The area decreases by 9%, confirming the effect of the opposing percentage changes in length and width.
The x-and y- coordinates of point P are each to be chosen at random from the set of integers 1 through 10. What is the probability that P will be in quadrant II?
- B. 01-Oct
- C. 01-Apr
- D. 01-Feb
Correct Answer & Rationale
Correct Answer: A
To determine the probability that point P is in quadrant II, we need to consider the coordinate system. In quadrant II, the x-coordinate must be negative, and the y-coordinate must be positive. However, since the x-coordinates are chosen from the integers 1 through 10, all possible x-values are positive. This means point P cannot be in quadrant II, making the probability 0. Option A correctly reflects this conclusion with a probability of 0. Options B, C, and D suggest specific dates, which are irrelevant to the question and do not address the coordinate conditions necessary for quadrant II. Thus, they are incorrect.
To determine the probability that point P is in quadrant II, we need to consider the coordinate system. In quadrant II, the x-coordinate must be negative, and the y-coordinate must be positive. However, since the x-coordinates are chosen from the integers 1 through 10, all possible x-values are positive. This means point P cannot be in quadrant II, making the probability 0. Option A correctly reflects this conclusion with a probability of 0. Options B, C, and D suggest specific dates, which are irrelevant to the question and do not address the coordinate conditions necessary for quadrant II. Thus, they are incorrect.