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
3√2- 2/(√2) =
- A. 2√2
- B. √2
- C. 3
- D. 4
Correct Answer & Rationale
Correct Answer: A
To solve the expression \( 3\sqrt{2} - \frac{2}{\sqrt{2}} \), we first simplify \( \frac{2}{\sqrt{2}} \). This can be rewritten as \( \frac{2\sqrt{2}}{2} = \sqrt{2} \). Thus, the expression becomes \( 3\sqrt{2} - \sqrt{2} \), which simplifies to \( 2\sqrt{2} \). Option B (\( \sqrt{2} \)) is incorrect as it does not account for the subtraction from \( 3\sqrt{2} \). Option C (3) is incorrect because it misrepresents the value obtained after simplification. Option D (4) is also incorrect, as it does not relate to the expression at all.
To solve the expression \( 3\sqrt{2} - \frac{2}{\sqrt{2}} \), we first simplify \( \frac{2}{\sqrt{2}} \). This can be rewritten as \( \frac{2\sqrt{2}}{2} = \sqrt{2} \). Thus, the expression becomes \( 3\sqrt{2} - \sqrt{2} \), which simplifies to \( 2\sqrt{2} \). Option B (\( \sqrt{2} \)) is incorrect as it does not account for the subtraction from \( 3\sqrt{2} \). Option C (3) is incorrect because it misrepresents the value obtained after simplification. Option D (4) is also incorrect, as it does not relate to the expression at all.
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.
Lanelle traveled 9.7 miles of her delivery route in 1.2 hours. At this same rate, which of the following is closest to the time it will take for Janelle to travel 20 miles?
- A. 2 hours
- B. 2.5 hours
- C. 5 hours
- D. 5.5 hours
Correct Answer & Rationale
Correct Answer: B
To determine the time it will take for Janelle to travel 20 miles, we first calculate Lanelle's speed. She traveled 9.7 miles in 1.2 hours, giving a speed of approximately 8.08 miles per hour (9.7 miles ÷ 1.2 hours). Using this speed, we can find the time for 20 miles by dividing the distance by the speed: 20 miles ÷ 8.08 mph ≈ 2.48 hours, which rounds to about 2.5 hours. Option A (2 hours) underestimates the time based on Lanelle's speed. Options C (5 hours) and D (5.5 hours) greatly overestimate the time needed. Thus, 2.5 hours is the most accurate estimate for Janelle's travel time.
To determine the time it will take for Janelle to travel 20 miles, we first calculate Lanelle's speed. She traveled 9.7 miles in 1.2 hours, giving a speed of approximately 8.08 miles per hour (9.7 miles ÷ 1.2 hours). Using this speed, we can find the time for 20 miles by dividing the distance by the speed: 20 miles ÷ 8.08 mph ≈ 2.48 hours, which rounds to about 2.5 hours. Option A (2 hours) underestimates the time based on Lanelle's speed. Options C (5 hours) and D (5.5 hours) greatly overestimate the time needed. Thus, 2.5 hours is the most accurate estimate for Janelle's travel time.
Each of the following is a solution to the equation x- 2y = 4 EXCEPT
- A. (-2,-3)
- B. (0,2)
- C. (4,0)
- D. (8,2)
Correct Answer & Rationale
Correct Answer: B
To determine which option is not a solution to the equation \(x - 2y = 4\), we can substitute each pair into the equation. - For A: \((-2, -3)\), substituting gives \(-2 - 2(-3) = -2 + 6 = 4\), which is correct. - For B: \((0, 2)\), substituting gives \(0 - 2(2) = 0 - 4 = -4\), which does not equal 4, making this option incorrect. - For C: \((4, 0)\), substituting gives \(4 - 2(0) = 4\), which is correct. - For D: \((8, 2)\), substituting gives \(8 - 2(2) = 8 - 4 = 4\), which is correct. Thus, option B is the only pair that does not satisfy the equation.
To determine which option is not a solution to the equation \(x - 2y = 4\), we can substitute each pair into the equation. - For A: \((-2, -3)\), substituting gives \(-2 - 2(-3) = -2 + 6 = 4\), which is correct. - For B: \((0, 2)\), substituting gives \(0 - 2(2) = 0 - 4 = -4\), which does not equal 4, making this option incorrect. - For C: \((4, 0)\), substituting gives \(4 - 2(0) = 4\), which is correct. - For D: \((8, 2)\), substituting gives \(8 - 2(2) = 8 - 4 = 4\), which is correct. Thus, option B is the only pair that does not satisfy the equation.