# Thread: Calculus Story Problem (optimization/minimization)

1. ## Calculus Story Problem (optimization/minimization)

So, this should be really easy. But it's driving me crazy. I have done like 10 problems pretty similar to this today, but can not figure this one out. We have to do all the steps and explain why we chose to do the steps that we did.

"The cost per hour for fuel to run a train is (v^2)/4 dollars, where v=speed of train in miles per hour. Other costs, including labor, are $300 per hour. How fast should the train travel on a 360 mile trip to minimize the total cost for the trip?" I know that the answer is 34.64 miles per hour, but I have no idea how to get to that answer. If someone could tell me the steps to get there, that would be amazing. Thanks so much. -Hannah 2. Originally Posted by flutters33 So, this should be really easy. But it's driving me crazy. I have done like 10 problems pretty similar to this today, but can not figure this one out. We have to do all the steps and explain why we chose to do the steps that we did. "The cost per hour for fuel to run a train is (v^2)/4 dollars, where v=speed of train in miles per hour. Other costs, including labor, are$300 per hour. How fast should the train travel on a 360 mile trip to minimize the total cost for the trip?"

I know that the answer is 34.64 miles per hour, but I have no idea how to get to that answer. If someone could tell me the steps to get there, that would be amazing. Thanks so much.

-Hannah
time in hours, $t = \frac{360}{v}$

total cost = (fixed cost rate)(time) + (variable cost rate)(time)

$C = 300 \cdot \frac{360}{v} + \frac{v^2}{4} \cdot \frac{360}{v}$

simplify the expression for C and minimize.

3. Originally Posted by skeeter
time in hours, $t = \frac{360}{v}$

total cost = (fixed cost rate)(time) + (variable cost rate)(time)

$C = 300 \cdot \frac{360}{v} + \frac{v^2}{4} \cdot \frac{360}{v}$

simplify the expression for C and minimize.
Thank you! However, I think that I am doing something wrong, as I keep getting

C= 10800V^-1 + 90v as my answer before minimization, and that just isn't working. I don't know what I am doing wrong.

4. Originally Posted by flutters33
Thank you! However, I think that I am doing something wrong, as I keep getting

C= 10800V^-1 + 90v as my answer before minimization, and that just isn't working. I don't know what I am doing wrong.
$C = 300 \cdot 360v^{-1} + 90v$

$\frac{dC}{dv} = -300 \cdot 360v^{-2} + 90
$

$-300 \cdot \frac{360}{v^2} + 90 = 0$

$\frac{120}{v^2} = \frac{1}{10}$

$v = \sqrt{1200} = 36.641...$

confirm it's a minimum.

5. Originally Posted by skeeter
$C = 300 \cdot 360v^{-1} + 90v$

$\frac{dC}{dv} = -300 \cdot 360v^{-2} + 90
$

$-300 \cdot \frac{360}{v^2} + 90 = 0$

$\frac{120}{v^2} = \frac{1}{10}$

$v = \sqrt{1200} = 36.641...$

confirm it's a minimum.
thank you! i did the second derivative, its concave up, and a minimum. thank you!