# Algebra 2

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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 -

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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