Algebra 2
posted by Brandon on .
You want to make a rectangular box that is x cm high, (x+5) cm. long and (10x) 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.

The volume as a function of x is given by:
V(x)=x(x+5)(10x)=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=106x=26.48<0
so V(6.08) is a maximum.