Question

Please help!!!!!

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. Every day, his route takes him 22.5 miles each way against the current and back to his dock, and he needs to make 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 is no current. How fast must the boat go on the lake in order for it to
serve the ferry operator's needs?

Answer 1

Answer: the speed of the boat on the lake is 9 mph

Step-by-step explanation:

Let x represent the speed of the boat on the lake or in still water.

The speed of the current in a river is 6 mph. This means that if the boat goes upstream against the speed of the current, its total speed would be (x - 6)mph. If the boat goes downstream against the speed of the current, its total speed would be (x + 6)mph.

Time = distance/ speed

Every day, his route takes him 22.5 miles each way against the current and back to his dock, and he needs to make this trip in a total of 9 hours. This means that the time taken to travel upstream is

22.5/(x - 6). The time taken to travel downstream is

22.5/(x + 6)

Since the total time is 9 hours, it means that

22.5/(x - 6) = 22.5/(x + 6)

Cross multiplying, it becomes

22.5(x + 6) + 22.5(x - 6) = 9

Multiplying through by (x + 6)(x - 6), it becomes

22.5(x - 6) + 22.5(x + 6) = 9[(x + 6)(x - 6)]

22.5x - 135 + 22.5x + 135 = 9(x² - 6x + 6x - 36)

22.5x + 22.5x = 9x² - 324

9x² - 45x - 324 = 0

Dividing through by 9, it becomes

x² - 5x - 36 = 0

x² + 4x - 9x - 36 = 0

x(x + 4) - 9(x + 4) = 0

x - 9 = 0 or x + 4 = 0

x = 9 or x = - 4

Since the speed cannot be negative, then x = 9

Answer 2

The boat must go 9mph on the lake in order for it to serve the ferry operator's needs

Represent the speed of the boat on the lake or in still water with x

Given that:

The current in a river has a speed of 6 mph.

• When the boat goes upstream, the speed is x - 6;
• When the boat comes downstream, the speed is x + 6;

So, the time taken by the boat each trip are:

$t_1= 22.5 \times (x - 6)$

$t_2= 22.5 \times (x + 6)$

$T = 9 \times (x + 6) \times (x - 6)$

The equation that calculates the required time (x) is:

$t_1 + t_2 =T$

This gives

$22.5 \times (x - 6) + 22.5 \times (x + 6) = 9 \times (x + 6)\times (x - 6)$

Expand

$22.5x - 135 + 22.5x + 135 = 9x\² - 324$

Evaluate the like terms

$45x = 9x\² - 324$

Divide through by 9

$5x = x\² - 36$

Collect like terms

$x\² -5x - 36 = 0$

Factorize

$(x + 4) (x - 9) = 0$

Solve for x

$x = 9\ or\ x = - 4$

x cannot be negative.

So, we have:

$x = 9$

Hence, the boat must go 9mph on the lake in order for it to serve the ferry operator's needs

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