Question

A rectangular prism has a volume of x^3y + xy^3 - 2x^2y^2 with a square base. If it has a lateral area of 30 in ^2, what is the volume in cubic inches?

Answer 1

We will have the following:

First, we know that the volume of a rectangular prism has the form:

`V=l\cdot h\cdot w`

So, we will factor the expression, that is:

`V=x^3y+xy^3-2x^2y^2\Rightarrow V=xy(y^2-2x+x^2)`
`\Rightarrow V=xy((x-y)(x-y))\Rightarrow V=xy(x-y)^2`

Since the base is a square we will have that the only candidate for it to be is "x*(x - y)^2 " or "y*(x - y)^2". If x*y where the base, then the expression would equal "0". This is since the base is an square, then x = y, so (x - y)^2 = (x - x)^2 = 0; or (y - y)^2 = 0.

Then: We will have that the lateral area will be given by "xy", that is:

`4xy=30`

Now, we solve for either "x" or "y" in this last expression, that is:

`y=(15)/(2x)`

From the problem we will also have that:

`V=w\cdot h\cdot l=w^2\cdot h=l^2\cdot h`

So: