Posted by **Brandon** on Friday, April 22, 2011 at 1:22pm.

You want to make a rectangular box that is x cm high, (x+5) cm. long and (10-x) cm. wide. What is the greatest volume possible? What will the dimensions of the box be?

I need all of the steps to get to the answer.

- Algebra 2 -
**MathMate**, Friday, April 22, 2011 at 7:44pm
The volume as a function of x is given by:

V(x)=x(x+5)(10-x)=-x^3+5*x^2+50*x

To find the maximum volume, we differentiate V(x) with respect to x and equate the derivative to zero:

dV/dx = -3*x^2+10*x+50 =0

Solve for x to get:

x=-2.7 or x=6.08

We reject the negative value of x to retain

x=6.08.

To check if the Volume is a maximum, we calculate

V"(6.08)=d²V/dx²=10-6x=-26.48<0

so V(6.08) is a maximum.

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