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.
Other Related Questions
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.
Square S has area 2√2 square units. What is the length of a side of square S?
- A. ∜128
- B. ∜32
- C. ∜8
- D. ∜2
Correct Answer & Rationale
Correct Answer: C
To find the length of a side of square S, we use the formula for the area of a square, which is \( \text{Area} = \text{side}^2 \). Given that the area is \( 2\sqrt{2} \), we set up the equation \( \text{side}^2 = 2\sqrt{2} \). Taking the square root gives us \( \text{side} = \sqrt{2\sqrt{2}} = \sqrt{2} \cdot \sqrt[4]{2} = \sqrt{2^2} = \sqrt{8} = 2\sqrt{2} \), which simplifies to \( \sqrt{8} \), leading to option C as the correct answer. Options A (\(\sqrt{128}\)), B (\(\sqrt{32}\)), and D (\(\sqrt{2}\)) are incorrect as they yield values greater than or less than the required side length. Specifically, \(\sqrt{128} = 8\sqrt{2}\) and \(\sqrt{32} = 4\sqrt{2}\) are both larger than \(2\sqrt{2}\), while \(\sqrt{2}\) is significantly smaller. Thus, option C accurately represents the side length of square S.
To find the length of a side of square S, we use the formula for the area of a square, which is \( \text{Area} = \text{side}^2 \). Given that the area is \( 2\sqrt{2} \), we set up the equation \( \text{side}^2 = 2\sqrt{2} \). Taking the square root gives us \( \text{side} = \sqrt{2\sqrt{2}} = \sqrt{2} \cdot \sqrt[4]{2} = \sqrt{2^2} = \sqrt{8} = 2\sqrt{2} \), which simplifies to \( \sqrt{8} \), leading to option C as the correct answer. Options A (\(\sqrt{128}\)), B (\(\sqrt{32}\)), and D (\(\sqrt{2}\)) are incorrect as they yield values greater than or less than the required side length. Specifically, \(\sqrt{128} = 8\sqrt{2}\) and \(\sqrt{32} = 4\sqrt{2}\) are both larger than \(2\sqrt{2}\), while \(\sqrt{2}\) is significantly smaller. Thus, option C accurately represents the side length of square S.
A salesperson's commission is k percent of the selling price of a car. Which of the following represents the commission, in dollars, on 2 cars that sold for $14,000 each?
- A. 280k
- B. 28,000k
- C. 14,000/(100+2k)
- D. (28,000+k)/100
Correct Answer & Rationale
Correct Answer: A
To determine the commission on 2 cars sold for $14,000 each, first calculate the total selling price: 2 × $14,000 = $28,000. The commission, being k percent of this total, is expressed as (k/100) × $28,000, which simplifies to $280k. Option B, 28,000k, incorrectly suggests the commission is k percent of the total without dividing by 100. Option C, 14,000/(100+2k), misrepresents the calculation entirely by altering the formula. Option D, (28,000+k)/100, incorrectly adds k to the total selling price before calculating the percentage, which is not aligned with commission calculation principles.
To determine the commission on 2 cars sold for $14,000 each, first calculate the total selling price: 2 × $14,000 = $28,000. The commission, being k percent of this total, is expressed as (k/100) × $28,000, which simplifies to $280k. Option B, 28,000k, incorrectly suggests the commission is k percent of the total without dividing by 100. Option C, 14,000/(100+2k), misrepresents the calculation entirely by altering the formula. Option D, (28,000+k)/100, incorrectly adds k to the total selling price before calculating the percentage, which is not aligned with commission calculation principles.
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.