Find the dimensions of the square-based prism with the greatest volume given each surface area. a) 700 cm2
How do I do this?
The height is variable and the square base area are variable.
Zero volume corresponds to zero height and also to zero base area.
In between these extremes is a maximum volume.
Write the prism volume equation in terms of one variable only using the surface area information.
Differentiate the volume equation,
finding the maximum value of the volume equation.
Hi there kymcd, welcome to MHF!
First you need to consider the formula to surface area for a square based prism.
It is, $\displaystyle \text{SA} = 4xh+2x^2 $
So making this equal 700 and solving for h we will be able to find an expressin for volume to terms of x.
$\displaystyle 700= 4xh+2x^2 $
$\displaystyle 700-2x^2= 4xh $
$\displaystyle \frac{700}{4x}-\frac{2x^2}{4x}= h $
$\displaystyle h = \frac{175}{x}-\frac{x}{2} $
Now volume is
$\displaystyle \text{V} = x^2h$
$\displaystyle \text{V} = x^2\left(\frac{175}{x}-\frac{x}{2}\right)$
$\displaystyle \text{V} = 175x-\frac{x^3}{2}$
Now to finish it, finding the maximum make $\displaystyle \frac{dV}{dx}=0$ and solve for $\displaystyle x$
Can you take it from here?