What system of equations do I use to solve the following question? A boat traveled 336 miles downstream and back. THe trip downstream took 12 hours. THe trio back took 13 hours. What is the speed of the boat in still water What is speed of the current?
(B+W)*12=336
(B-W)*13=336
B is boat speed, W is water speed.
To solve this question, we can use the system of equations:
1. Let's assume the speed of the boat in still water is represented by "b" (in miles per hour), and the speed of the current is represented by "c" (in miles per hour).
2. The downstream trip took 12 hours, during which the boat traveled 336 miles. Since the boat's speed is increased by the speed of the current while going downstream, we can write the equation:
336 = (b + c) * 12
This equation indicates that the distance traveled downstream is equal to the speed of the boat in still water, plus the speed of the current, multiplied by the time taken for the trip downstream.
3. The trip back took 13 hours, during which the boat traveled the same 336 miles. Since the boat's speed is decreased by the speed of the current while going upstream, we can write the equation:
336 = (b - c) * 13
This equation indicates that the distance traveled upstream is equal to the speed of the boat in still water, minus the speed of the current, multiplied by the time taken for the trip back.
4. Now we have a system of two equations:
336 = (b + c) * 12
336 = (b - c) * 13
We can solve this system of equations simultaneously to find the values of "b" and "c", representing the speed of the boat in still water and the speed of the current, respectively.
5. To solve this system of equations, we can use various methods such as substitution, elimination, or graphical representation. Let's use the method of substitution:
Start by rearranging both equations to solve for "b" in terms of "c":
Equation 1: b + c = 336/12
Equation 2: b - c = 336/13
Simplifying both equations:
Equation 1: b + c = 28
Equation 2: b - c = 336/13
Now, we can isolate either "b" or "c" in one of the equations and substitute it into the other equation.
Solving Equation 1 for "b", we get:
b = 28 - c
Substituting this value of "b" into Equation 2:
28 - c - c = 336/13
Simplifying the equation further:
28 - 2c = 336/13
Cross-multiplying and solving for "c", we find:
364 - 26c = 336
-26c = -28
Solving for "c", we get:
c = 1.08
Now that we have the value of "c" (the speed of the current), we can substitute it back into one of the equations to find the value of "b" (the speed of the boat in still water).
Substituting "c = 1.08" into Equation 1:
b + 1.08 = 28
b = 28 - 1.08
b ≈ 26.92
6. Therefore, the speed of the boat in still water is approximately 26.92 miles per hour, and the speed of the current is approximately 1.08 miles per hour.