A boat travels down the river in 2.8 hours. The return trip up river takes 3.8 hours. A one way trip down or up river is a distance of 24 miles. If the boat is traveling at maximum speed, what is the boats speed measured in waters with no current? You need to use systems to solve this problem. Round your speed to the tenth of a mile per hour. You do not need to include units as the problem requires a value in miles per hour
To solve this problem using systems, let's set up two equations based on the given information.
Let's assume the boat's speed in still water is represented by "b" (in miles per hour) and the speed of the river current is represented by "c" (in miles per hour).
For the downstream trip:
The boat's speed relative to the river current is increased by the current's speed, so the effective speed of the boat is (b + c) miles per hour. The distance covered is 24 miles, and it took the boat 2.8 hours. Therefore, the equation is:
Distance = Speed × Time
24 = (b + c) × 2.8
For the upstream trip:
The boat's speed relative to the river current is decreased by the current's speed, so the effective speed of the boat is (b - c) miles per hour. The distance covered is also 24 miles, and it took the boat 3.8 hours. Therefore, the equation is:
24 = (b - c) × 3.8
Now we have a system of two equations:
Equation 1: 24 = (b + c) × 2.8
Equation 2: 24 = (b - c) × 3.8
To solve this system, we can use substitution or elimination. Let's use substitution to eliminate one variable.
Rearrange Equation 1 to solve for b + c:
(b + c) = 24 / 2.8
Simplify:
(b + c) = 8.571428571428571
Now, substitute this value into Equation 2:
24 = (8.571428571428571 - c) × 3.8
Simplify further:
24 = 32.57142857142857 - 3.8c
Rearrange the equation:
3.8c = 32.57142857142857 - 24
Simplify:
3.8c = 8.57142857142857
Divide both sides by 3.8 to solve for c:
c = 8.57142857142857 / 3.8
Simplify to find the speed of the river current:
c ≈ 2.258 miles per hour
Now that we have the speed of the river current, we can substitute it back into Equation 1 and solve for b:
24 = (b + 2.258) × 2.8
Rearrange the equation:
(b + 2.258) = 24 / 2.8
Simplify:
(b + 2.258) ≈ 8.571428571428571
Subtract 2.258 from both sides to solve for b:
b ≈ 8.571428571428571 - 2.258
Simplify to find the speed of the boat in still water:
b ≈ 6.313 miles per hour
Therefore, the boat's speed in still water, rounded to the tenth of a mile per hour, is approximately 6.3 miles per hour.