A river flows at the rate of 2 mph. If a motorboat can travel 12 miles upstream and return in a total of 2.5 hours, what is the speed of the motorboat in still water?
To find the speed of the motorboat in still water, we need to set up a system of equations and solve them.
Let's assume that the speed of the motorboat in still water is "b" mph, and the speed of the river is "r" mph.
When the motorboat is traveling upstream, the effective speed is decreased by the speed of the river. So, the speed of the motorboat relative to the ground will be (b - r) mph.
When the motorboat is traveling downstream, the effective speed is increased by the speed of the river. So, the speed of the motorboat relative to the ground will be (b + r) mph.
Given that the river flows at a constant rate of 2 mph, we can write the following equations based on the distance, rate, and time concepts:
Upstream journey: Distance = Rate * Time
12 = (b - 2) * T1
Downstream journey: Distance = Rate * Time
12 = (b + 2) * T2
We also have the total time of the journey, which is 2.5 hours:
T1 + T2 = 2.5
Now, we can solve this system of equations to find the value of "b" (the speed of the motorboat in still water).
First, let's solve the equations related to the distance using a substitution method. From the first equation, we can express T1 in terms of 12, b, and r:
T1 = 12 / (b - 2)
Next, substitute this value of T1 into the third equation:
(12 / (b - 2)) + T2 = 2.5
Multiply both sides by (b - 2) to eliminate the denominator:
12 + (b - 2)T2 = 2.5(b - 2)
Expand and simplify the equation:
12 + bT2 - 2T2 = 2.5b - 5
Rearrange the equation:
bT2 - 2.5b = -5 - 12 + 2T2
Combine similar terms:
T2(b - 2.5) = -17 + 2T2
Now, let's solve the equation T2(b - 2.5) = -17 + 2T2 for T2. Since we know T2 cannot be zero, we can divide both sides by it:
b - 2.5 = (-17 + 2T2) / T2
Simplify the right side of the equation:
b - 2.5 = -17 / T2 + 2
Multiply both sides by T2 to eliminate the denominator:
bT2 - 2.5T2 = -17 + 2T2
Simplify the equation:
bT2 - 4.5T2 = -17
Combine similar terms:
T2(b - 4.5) = -17
Since T2 cannot be zero, we can divide both sides by T2:
b - 4.5 = -17 / T2
Add 4.5 to both sides:
b = -17 / T2 + 4.5
We have expressed the speed of the motorboat in still water "b" in terms of the unknown variable T2. To find the speed of the motorboat in still water, we need to find the value of T2.
Now, we can substitute b back into the equation we obtained from the second equation related to the total time:
T1 + T2 = 2.5
Using the derived expression for T1:
(12 / (b - 2)) + T2 = 2.5
Substitute the expression for b:
(12 / ((-17 / T2 + 4.5) - 2)) + T2 = 2.5
Simplify the equation by finding a common denominator:
(12 / ((-17 / T2 + 4.5) - 2)) + T2 = 2.5
Simplify further and eliminate denominators:
12T2 / (-17 + 4.5T2 - 2T2) + T2 = 2.5
Simplify the equation:
12T2 / (-17 + 2.5T2) + T2 = 2.5
Now, we can solve this equation to find the value of T2. We can do this by multiplying through by the common denominator, (-17 + 2.5T2):
12T2 + T2(-17 + 2.5T2) = 2.5(-17 + 2.5T2)
Expand the equation:
12T2 - 17T2 + 2.5T2^2 = -42.5 + 5.625T2
Combine similar terms and arrange the equation in standard form:
2.5T2^2 - 4T2 - 42.5 = 0
Now we have a quadratic equation in terms of T2. We can solve this quadratic equation using factoring, completing the square, or the quadratic formula to find the values of T2.
Once we find the values of T2, we can substitute them back into the equation b = -17 / T2 + 4.5 to get the values of b (the speed of the motorboat in still water).
After obtaining the value(s) of b, we can determine the speed of the motorboat in still water.