Fix It Fast is an auto repair shop that employs 10 mechanics. Each day, the shop owner randomly picks 1 mechanic to receive a free lunch. What is the probability the shop owner will pick the same mechanic to receive a free lunch 2 days in a row?
- A. 1\20
- B. 1/100
- C. 1\5
- D. 1\10
Correct Answer & Rationale
Correct Answer: B
To determine the probability of picking the same mechanic two days in a row, we start by recognizing that there are 10 mechanics. On the first day, any mechanic can be chosen, which does not affect the overall probability. On the second day, to pick the same mechanic again, there is only 1 favorable outcome (the chosen mechanic) out of 10 possible mechanics. Thus, the probability of selecting that same mechanic on the second day is 1/10. Since the first day's choice does not influence this, we multiply the probabilities: (1/10) * (1/10) = 1/100. - Option A (1/20) is incorrect as it miscalculates the favorable outcomes. - Option C (1/5) incorrectly assumes a higher likelihood without considering the second day's requirement. - Option D (1/10) only reflects the probability of picking a mechanic on day two, not the two-day scenario.
To determine the probability of picking the same mechanic two days in a row, we start by recognizing that there are 10 mechanics. On the first day, any mechanic can be chosen, which does not affect the overall probability. On the second day, to pick the same mechanic again, there is only 1 favorable outcome (the chosen mechanic) out of 10 possible mechanics. Thus, the probability of selecting that same mechanic on the second day is 1/10. Since the first day's choice does not influence this, we multiply the probabilities: (1/10) * (1/10) = 1/100. - Option A (1/20) is incorrect as it miscalculates the favorable outcomes. - Option C (1/5) incorrectly assumes a higher likelihood without considering the second day's requirement. - Option D (1/10) only reflects the probability of picking a mechanic on day two, not the two-day scenario.
Other Related Questions
((5^3 * 2^4)^2)(5^(-2) * 2^5)
- A. 5^3 * 2^11
- B. 5^(-12) * 2^40
- C. 5^4 * 2^13
- D. (-5)^8 * 2^13
Correct Answer & Rationale
Correct Answer: C
To simplify the expression \(((5^3 * 2^4)^2)(5^{-2} * 2^5)\), first apply the power of a product rule. This gives \(5^{6} * 2^{8}\) from the first part. Next, combine this with the second part, \(5^{-2} * 2^{5}\). Adding the exponents for the base 5: \(6 + (-2) = 4\). For base 2: \(8 + 5 = 13\). Thus, the final expression simplifies to \(5^4 * 2^{13}\). Option A is incorrect as it miscalculates the exponents. Option B has incorrect exponents and signs. Option D introduces an unnecessary negative sign and does not match the simplified expression.
To simplify the expression \(((5^3 * 2^4)^2)(5^{-2} * 2^5)\), first apply the power of a product rule. This gives \(5^{6} * 2^{8}\) from the first part. Next, combine this with the second part, \(5^{-2} * 2^{5}\). Adding the exponents for the base 5: \(6 + (-2) = 4\). For base 2: \(8 + 5 = 13\). Thus, the final expression simplifies to \(5^4 * 2^{13}\). Option A is incorrect as it miscalculates the exponents. Option B has incorrect exponents and signs. Option D introduces an unnecessary negative sign and does not match the simplified expression.
Dominic built a dog pen with a perimeter of 72 feet (ft). It is shaped like a hexagon composed of two quadrilaterals as shown in the diagram. Side g of the dog pen is a gate. What is the length, in feet, of the gate?
- A. 10
- B. 5
- C. 8
- D. 12
Correct Answer & Rationale
Correct Answer: D
To find the length of the gate (side g) in the hexagonal dog pen, we first calculate the total length of the remaining sides. Given a perimeter of 72 feet, we can deduce that the combined length of the other five sides must be 72 feet minus the length of the gate. Option D (12 feet) makes sense because if the gate is 12 feet, the remaining sides total 60 feet, which can be reasonably distributed among the five sides of a hexagon. Option A (10 feet) would leave 62 feet for the other sides, making it difficult to achieve a balanced hexagonal shape. Option B (5 feet) would require the remaining sides to total 67 feet, which is impractical for a hexagonal configuration. Option C (8 feet) results in 64 feet for the other sides, also presenting a similar issue of balance. Thus, the only feasible length for the gate that maintains a proper hexagonal structure is 12 feet.
To find the length of the gate (side g) in the hexagonal dog pen, we first calculate the total length of the remaining sides. Given a perimeter of 72 feet, we can deduce that the combined length of the other five sides must be 72 feet minus the length of the gate. Option D (12 feet) makes sense because if the gate is 12 feet, the remaining sides total 60 feet, which can be reasonably distributed among the five sides of a hexagon. Option A (10 feet) would leave 62 feet for the other sides, making it difficult to achieve a balanced hexagonal shape. Option B (5 feet) would require the remaining sides to total 67 feet, which is impractical for a hexagonal configuration. Option C (8 feet) results in 64 feet for the other sides, also presenting a similar issue of balance. Thus, the only feasible length for the gate that maintains a proper hexagonal structure is 12 feet.
A shipping box for a refrigerator is shaped like a rectangular prism. The box has a depth of 34.25 inches (in.), a height of 69.37 in., and a width of 32.62 in. To the nearest hundredth cubic inch, what is the volume of the shipping bax?
- A. 2,262.85
- B. 77,502.59
- C. 136.24
- D. 25,834.20
Correct Answer & Rationale
Correct Answer: B
To determine the volume of a rectangular prism, the formula \( V = \text{length} \times \text{width} \times \text{height} \) is applied. Given the dimensions—depth (length) of 34.25 in., width of 32.62 in., and height of 69.37 in.—the calculation yields a volume of approximately 77,502.59 cubic inches. Option A (2,262.85) is far too small, indicating a miscalculation. Option C (136.24) is implausibly low, likely resulting from using incorrect units or dimensions. Option D (25,834.20) is also incorrect, as it does not reflect the correct multiplication of the given dimensions. Thus, only option B accurately represents the calculated volume.
To determine the volume of a rectangular prism, the formula \( V = \text{length} \times \text{width} \times \text{height} \) is applied. Given the dimensions—depth (length) of 34.25 in., width of 32.62 in., and height of 69.37 in.—the calculation yields a volume of approximately 77,502.59 cubic inches. Option A (2,262.85) is far too small, indicating a miscalculation. Option C (136.24) is implausibly low, likely resulting from using incorrect units or dimensions. Option D (25,834.20) is also incorrect, as it does not reflect the correct multiplication of the given dimensions. Thus, only option B accurately represents the calculated volume.
Solve the equation for x: ½ x + 9 = -2/3 x
- A. x=-9/7
- B. x=-54/7
- C. x=-6
- D. x=-54
Correct Answer & Rationale
Correct Answer: B
To solve the equation \( \frac{1}{2}x + 9 = -\frac{2}{3}x \), start by eliminating the fractions. Multiply the entire equation by 6 (the least common multiple of 2 and 3) to obtain \( 3x + 54 = -4x \). Next, combine like terms: adding \( 4x \) to both sides gives \( 7x + 54 = 0 \), leading to \( 7x = -54 \) and thus \( x = -\frac{54}{7} \). Option A is incorrect as it simplifies to a different value. Option C, \( x = -6 \), does not satisfy the original equation. Option D, \( x = -54 \), is also incorrect as it does not balance the equation. Therefore, the only viable solution is \( x = -\frac{54}{7} \).
To solve the equation \( \frac{1}{2}x + 9 = -\frac{2}{3}x \), start by eliminating the fractions. Multiply the entire equation by 6 (the least common multiple of 2 and 3) to obtain \( 3x + 54 = -4x \). Next, combine like terms: adding \( 4x \) to both sides gives \( 7x + 54 = 0 \), leading to \( 7x = -54 \) and thus \( x = -\frac{54}{7} \). Option A is incorrect as it simplifies to a different value. Option C, \( x = -6 \), does not satisfy the original equation. Option D, \( x = -54 \), is also incorrect as it does not balance the equation. Therefore, the only viable solution is \( x = -\frac{54}{7} \).