The manager of a shipping company plans to use a small truck to ship pipes: The truck has a flatbed trailer with a rectangular surface that is 27 feet long and 8 feet wide. The truck will travel from Atherton to Bakersfield, where some pipes will be delivered, and then on to Castlewood to deliver the remaining pipes. The map shows the roads that connect Atherton. Bakersfield. and Castlewood.
The manager is planning to buy a new truck with better gas mileage. He collected data bout the gas mileage of one of the company's trucks. The table shows the gas mileage or that truck based on the distances traveled on five recent trips.
How many different ways can the truck travel from Atherton to Bakersfield a to Castlewood, using the roads on the map?
- A. 6
- B. 8
- C. 9
- D. 5
Correct Answer & Rationale
Correct Answer: A
To determine the number of different routes from Atherton to Bakersfield and then to Castlewood, we analyze the connections between these locations. If there are 3 distinct paths from Atherton to Bakersfield and 2 distinct paths from Bakersfield to Castlewood, the total number of combinations is found by multiplying the number of options: 3 paths (Atherton to Bakersfield) × 2 paths (Bakersfield to Castlewood) = 6 routes. Options B (8), C (9), and D (5) miscalculate the available paths or overlook the combinations of routes, leading to incorrect totals. Thus, the correct answer accurately reflects the possible travel routes.
To determine the number of different routes from Atherton to Bakersfield and then to Castlewood, we analyze the connections between these locations. If there are 3 distinct paths from Atherton to Bakersfield and 2 distinct paths from Bakersfield to Castlewood, the total number of combinations is found by multiplying the number of options: 3 paths (Atherton to Bakersfield) × 2 paths (Bakersfield to Castlewood) = 6 routes. Options B (8), C (9), and D (5) miscalculate the available paths or overlook the combinations of routes, leading to incorrect totals. Thus, the correct answer accurately reflects the possible travel routes.
Other Related Questions
Multiply: (x^2 - 3)(x^5 + 2x^3)
- A. x^7,-3x^5,-6x^3
- B. x^10,2x^5,-6x^3
- C. 5x^5,2x^6,-6x^3
- D. x^7,2x^5,-6
Correct Answer & Rationale
Correct Answer: A
To find the product of (x^2 - 3)(x^5 + 2x^3), we apply the distributive property (FOIL method). 1. **First Terms**: x^2 * x^5 = x^7. 2. **Outer Terms**: x^2 * 2x^3 = 2x^5. 3. **Inner Terms**: -3 * x^5 = -3x^5. 4. **Last Terms**: -3 * 2x^3 = -6x^3. Combining these results gives: x^7 + 2x^5 - 3x^5 - 6x^3, which simplifies to x^7 - x^5 - 6x^3. Option A correctly lists the terms as x^7, -3x^5, -6x^3. Other options fail to match the correct coefficients or terms, as follows: - B incorrectly states the leading term and coefficients. - C miscalculates the powers of x and coefficients. - D omits the x terms entirely, providing an incomplete expression.
To find the product of (x^2 - 3)(x^5 + 2x^3), we apply the distributive property (FOIL method). 1. **First Terms**: x^2 * x^5 = x^7. 2. **Outer Terms**: x^2 * 2x^3 = 2x^5. 3. **Inner Terms**: -3 * x^5 = -3x^5. 4. **Last Terms**: -3 * 2x^3 = -6x^3. Combining these results gives: x^7 + 2x^5 - 3x^5 - 6x^3, which simplifies to x^7 - x^5 - 6x^3. Option A correctly lists the terms as x^7, -3x^5, -6x^3. Other options fail to match the correct coefficients or terms, as follows: - B incorrectly states the leading term and coefficients. - C miscalculates the powers of x and coefficients. - D omits the x terms entirely, providing an incomplete expression.
((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.
At a local bank, certificates of deposit (CDs) mature every 9 months. At another bank, CDs mature every 12 months. If CDs are purchased on the same day at each bank and are renewed when they mature, what is the least number of months that will pass before the two banks' CDs are mature at the same time?
- A. 72
- B. 36
- C. 108
- D. 3
Correct Answer & Rationale
Correct Answer: B
To find when the CDs from both banks mature simultaneously, we need to determine the least common multiple (LCM) of their maturity periods: 9 months and 12 months. Calculating the LCM, we see that the multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, and 81. The multiples of 12 are 12, 24, 36, 48, 60, 72, and 84. The smallest common multiple is 36 months. Option A (72) is incorrect as it’s not the smallest shared maturity. Option C (108) is also incorrect; it exceeds the LCM. Option D (3) is far too short, as it does not accommodate either maturity period. Thus, 36 months is the earliest point both CDs will mature together.
To find when the CDs from both banks mature simultaneously, we need to determine the least common multiple (LCM) of their maturity periods: 9 months and 12 months. Calculating the LCM, we see that the multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, and 81. The multiples of 12 are 12, 24, 36, 48, 60, 72, and 84. The smallest common multiple is 36 months. Option A (72) is incorrect as it’s not the smallest shared maturity. Option C (108) is also incorrect; it exceeds the LCM. Option D (3) is far too short, as it does not accommodate either maturity period. Thus, 36 months is the earliest point both CDs will mature together.
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 \( x = -\frac{54}{7} \).