The speed of the current is 5m/h. Angie goes upstream 25 miles and returns to the doc. The whole trip takes 12 hours. What is the speed of the boat in still water?
To find the speed of the boat in still water, we need to set up an equation based on the information given. Let's start by identifying the relevant variables and values.
Let's say the speed of the boat in still water is "x" mph.
Given:
Speed of the current = 5 mph
Distance traveled upstream = 25 miles
Total time for the round trip = 12 hours
Now, let's break down the problem into two parts: Angie traveling upstream and Angie traveling downstream.
When Angie is traveling upstream:
Speed of the boat relative to the ground = Speed of the boat in still water - Speed of the current = (x - 5) mph
Time taken to travel 25 miles upstream = Distance / Speed = 25 / (x - 5) hours
When Angie is traveling downstream:
Speed of the boat relative to the ground = Speed of the boat in still water + Speed of the current = (x + 5) mph
Time taken to travel 25 miles downstream = Distance / Speed = 25 / (x + 5) hours
Since the total time for the round trip is 12 hours, we can set up the following equation:
Time upstream + Time downstream = 12 hours
25 / (x - 5) + 25 / (x + 5) = 12
To solve this equation, we can multiply through by the least common denominator (x - 5)(x + 5), which gives us:
25(x + 5) + 25(x - 5) = 12(x - 5)(x + 5)
Now, let's solve for x.
25x + 125 + 25x - 125 = 12(x^2 - 25)
50x = 12x^2 - 300
12x^2 - 50x - 300 = 0
Simplifying the equation, we have a quadratic equation:
6x^2 - 25x - 150 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula.
Factoring the equation, we have:
(2x - 15)(3x + 10) = 0
Setting each factor equal to zero, we get:
2x - 15 = 0 or 3x + 10 = 0
Solving for x, we have:
2x = 15 or 3x = -10
x = 15/2 or x = -10/3
Since speed cannot be negative, we discard the negative value. Therefore, the speed of the boat in still water is:
x = 15/2 = 7.5 mph
So, the speed of the boat in still water is 7.5 mph.