please explain the step for this I have no idea.
the speed of the current in a river is 6 mph a ferry operator who works that part of the river is looking to buy a new boat for his business everyday his route takes him 22.5 miles against the current and back to his deck and he needs this trip in a total of 9 hours he has a boat in mind but he can only test it on a lake where there in no current how fast must the boat go on the lake in order for it to serve the ferry operators needs.
This is what I got;
22.5/(x+6)+22.5/(x-6)=9
x=9mph
To solve this problem, we can follow these steps:
Step 1: Write down the given information and assign variables:
- Let's say the speed of the boat on the lake is 'x' mph.
- The speed of the current in the river is given as 6 mph.
- The distance of the route is 22.5 miles.
- The total time taken for the trip is 9 hours.
Step 2: Determine the time taken to travel against the current and with the current:
- When the boat is traveling against the current, its effective speed is reduced by the current's speed. So, the time taken for this part of the trip can be calculated as distance/speed. In this case, it is 22.5/(x-6) (distance/speed).
- Similarly, when the boat is traveling with the current, its effective speed is increased by the current's speed. So, the time taken for this part of the trip can be calculated as distance/speed. In this case, it is 22.5/(x+6) (distance/speed).
Step 3: Write the equation based on the given information:
- The total time taken for the trip (against the current and with the current) is given as 9 hours. So, the equation can be formed as follows: 22.5/(x-6) + 22.5/(x+6) = 9.
Step 4: Solve the equation:
- To solve this equation, we can multiply both sides by (x-6)(x+6) to eliminate the denominators:
22.5(x+6) + 22.5(x-6) = 9(x-6)(x+6).
Step 5: Simplify the equation and solve for x:
- Expanding the equation:
22.5x + 135 + 22.5x - 135 = 9(x^2 - 36)
45x = 9x^2 - 324
9x^2 - 45x - 324 = 0
- To solve this quadratic equation, you can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).
- In this case, a = 9, b = -45, and c = -324.
- Then, plug in these values into the quadratic formula and solve for x:
x = (-(-45) ± √((-45)^2 - 4(9)(-324))) / (2(9))
x = (45 ± √(2025 + 11664)) / 18
x = (45 ± √(13689)) / 18
- Simplifying further, we get:
x = (45 ± 117) / 18
- Now, we have two possible solutions:
x = (45 + 117) / 18 = 162 / 18 = 9 mph (speed cannot be negative, so we only consider the positive value)
x = (45 - 117) / 18 = -72 / 18 = -4 mph (disregard this solution)
Step 6: Determine the answer:
- Therefore, the boat must go at a speed of 9 mph on the lake in order to serve the ferry operator's needs.
So, the boat must go at a speed of 9 mph on the lake in order to serve the ferry operator's needs.
To find the required speed of the boat on the lake, we need to solve the equation 22.5/(x+6) + 22.5/(x-6) = 9, where x represents the speed of the boat on the lake.
Here are the step-by-step instructions to solve the equation:
Step 1: Start with the equation: 22.5/(x+6) + 22.5/(x-6) = 9.
Step 2: To clear the denominators, multiply the entire equation by (x+6)(x-6), which is the least common denominator.
(x+6)(x-6) * [22.5/(x+6) + 22.5/(x-6)] = 9 * (x+6)(x-6).
This simplifies to: 22.5(x-6) + 22.5(x+6) = 9(x+6)(x-6).
Step 3: Distribute and simplify the equation:
22.5x - 135 + 22.5x + 135 = 9(x² - 36).
Combine like terms:
45x = 9x² - 324.
Step 4: Rearrange the equation to set it equal to zero:
9x² - 45x - 324 = 0.
Step 5: Divide the entire equation by 9 to simplify it:
x² - 5x - 36 = 0.
Step 6: Factor the quadratic equation:
(x-9)(x+4) = 0.
Step 7: Set each factor equal to zero and solve for x:
x-9 = 0 or x+4 = 0.
x = 9 or x = -4.
Step 8: Since the speed of the boat cannot be negative, we discard x = -4.
Step 9: Therefore, the required speed of the boat on the lake is x = 9 mph.
So, the boat must go at least 9 mph on the lake in order to serve the ferry operator's needs.