In still water, a speed boat travels 5 times faster than the current of the river. If the speed boat can travel 48 miles upstream and 10 back in 5 hours, find the rate.
Angles and dgrees
u = 5 c
t = time upstream
(5-t) = time downstream
(u-c)t = 48
(u+c)(5-t) = 10
4 c t = 48 so t = 12/c
6 c (5-t) = 10
6 c (5 - 12/c) = 10
30 c - 72 = 10
30 c = 82
c = 2.7333
u = 5 c = 13 2/3 miles/hour
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by the way t = 12/2.73333 = 4.4
hours so 5 - t = .6 hours
check
u+c = 16.4
u-c = 10.9
10.9 *4.4 = 48 close enough
16.4 * .6 = 10 close enough
To solve this problem, we can break it down into a few steps:
Step 1: Assign variables.
Let's assume the speed of the current in still water is represented by "C" (in miles per hour), and the speed of the speedboat in still water is represented by "S" (in miles per hour).
Step 2: Set up the equations.
We know that the speed boat travels 5 times faster than the current, so the speed of the speed boat in still water can be expressed as:
S = 5C
Step 3: Calculate the time taken to travel upstream.
Using the formula: Time = Distance / Speed, we can calculate the time it takes for the speedboat to travel upstream. The speed of the boat relative to the current when traveling upstream is (S - C), so we have:
Time upstream = 48 / (S - C)
Step 4: Calculate the time taken to travel downstream.
Similarly, the speed of the boat relative to the current when traveling downstream is (S + C). Therefore, the time it takes for the speedboat to travel downstream can be calculated as:
Time downstream = 10 / (S + C)
Step 5: Use the given information to create an equation.
We are told that the total time taken for the journey (to travel 48 miles upstream and 10 miles back downstream) is 5 hours. This can be expressed as:
Time upstream + Time downstream = 5
Step 6: Substitute the equations.
Now we can substitute the equations we established earlier into the total time equation:
48 / (S - C) + 10 / (S + C) = 5
Step 7: Solve the equation.
To solve this equation, we can simplify it further by finding a common denominator and combining the fractions. Multiplying both sides by (S - C)(S + C), we get:
48(S + C) + 10(S - C) = 5(S - C)(S + C)
Solve this equation for S, and you will have the value of the speed of the speedboat in still water (S), which represents the rate.