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
Looks good to me.
You could have checked your answer:
22.5mi at 3mph = 7.5 hrs
22.5mi @ 15mph = 1.5 hrs
total: 9 hrs
Thanks for checking my work
To solve this problem, follow these steps:
Step 1: Identify the given information.
- The speed of the current in the river is 6 mph.
- The ferry operator needs to complete a round trip of 22.5 miles against the current.
- The ferry operator needs to complete the entire trip in a total of 9 hours.
Step 2: Understand the problem and define the unknown.
- The unknown in this problem is the speed of the boat required on the lake, which we'll call 'x' mph.
Step 3: Set up a formula or equation to represent the problem.
- The time taken to travel against the current and back is equal to 9 hours. We can set up the equation:
22.5 / (x + 6) + 22.5 / (x - 6) = 9
Step 4: Solve the equation.
To solve the equation, we'll follow these steps:
- Multiply both sides of the equation by the common denominators, (x + 6)(x - 6), to eliminate the denominators.
- Simplify the equation and bring all terms to one side to obtain:
22.5(x - 6) + 22.5(x + 6) = 9(x + 6)(x - 6)
- Expand and simplify both sides of the equation:
22.5x - 135 + 22.5x + 135 = 9(x^2 - 36)
45x = 9x^2 - 324
- Rearrange the equation to solve for x:
9x^2 - 45x - 324 = 0
- Factor the equation to obtain:
(3x - 36)(3x + 9) = 0
Step 5: Solve for x.
From the factored equation, set each factor equal to zero:
3x - 36 = 0 or 3x + 9 = 0
Solving each equation, we get:
3x = 36 or 3x = -9
x = 12 or x = -3
Step 6: Evaluate the solution.
Since the speed of a boat cannot be negative, we discard x = -3.
Therefore, the boat must go at a speed of 12 mph on the lake in order to meet the ferry operator's needs.
So, the boat must go 12 mph on the lake in order to serve the ferry operator's needs.
To solve this problem, we can follow these steps:
Step 1: Understand the problem
We are given the speed of the current in a river (6 mph) and the distance the ferry operator needs to travel (22.5 miles). We need to find the speed at which the boat must go on a lake (let's call it x mph) so that the total trip time is 9 hours.
Step 2: Set up the equation
The time taken to travel against the current is given by the formula distance / (speed - current speed). Therefore, the time taken to travel against the current is 22.5 / (x + 6). The time taken to travel with the current is 22.5 / (x - 6). The total trip time is 9 hours, so we can set up the equation as follows:
22.5 / (x + 6) + 22.5 / (x - 6) = 9
Step 3: Solve the equation
To solve the equation, we need to find the value of x that satisfies it. To do this, we can simplify the equation and then solve for x.
Multiplying through by (x + 6)(x - 6) to get rid of the denominators, we get:
22.5(x - 6) + 22.5(x + 6) = 9(x + 6)(x - 6)
Simplifying, we have:
22.5x - 135 + 22.5x + 135 = 9(x^2 - 36)
45x = 9x^2 - 324
Rearranging the equation, we get:
9x^2 - 45x - 324 = 0
Using factoring, the equation can be rewritten as:
9(x^2 - 5x - 36) = 0
(x^2 - 5x - 36) = 0
Factoring further, we have:
(x - 9)(x + 4) = 0
So, x - 9 = 0 or x + 4 = 0
We discard the negative solution, leaving us with x = 9 mph.
Therefore, the boat must go at a speed of 9 mph on the lake in order to serve the ferry operator's needs.