The formula for volume of a cone is,Originally Posted byLilDragonfly

.

Since, (Pythagorean Theorem),

we can express volume in terms of height,

Taking its derivative,

and finding critical points,

thus,

thus,

reject negative, thus, .

To prove that this is a maximum we use the second derivative test, first we need the second derivative,

We can see that at it is negative thus we have a maximum.

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Next, you should tell the truck operator the maximum number because then he would definetly remove all the debris. Which is the Volume evaluated at this height.

Since,

at

Thus,

Thus,

Thus,

Thus,

Of course, you cannot tell the truck operator this number he would have no idea what you mean.