# At a glassware factory, molten cobalt glass is poured into molds to make paperweights. Each mold...

## Question:

At a glassware factory, molten cobalt glass is poured into molds to make paperweights. Each mold is a rectangular prism whose height is 3 inches greater than the length of each side of the square base. A machine pours 20 cubic inches of liquid glass into each mold. What are the dimensions of the mold?

## Volume:

It is the quantity that represents the amount of liquid that any solid can store. Mathematically it is the product of the area of the base to the height of any regular slid like cylinder or cube or even prism in the given case.

Given Data:

• Height is 3 inches greater than the length of each side of the square base
• Volume of prism is {eq}V = 20 {/eq}

Assume the length of side of square is {eq}l {/eq}

The expression of volume of prism is,

{eq}\begin{align*} l \times l \times \left( {l + 3} \right)& = 20\\ {l^3} + 3{l^2} - 20 &= 0 \end{align*} {/eq}

Above equation can be written as,

{eq}\left( {l - 2} \right)\left( {{l^2} + 5l + 10} \right) = 0 {/eq}

Roots of above equations are,

{eq}\begin{align*} l &= 2,\dfrac{{ - 5 \pm \sqrt {{5^2} - 4\left( 1 \right)\left( {10} \right)} }}{{2\left( 1 \right)}}\\ &= 2,\dfrac{{ - 5 \pm \sqrt {15} i}}{{2\left( 1 \right)}} \end{align*} {/eq}

As length cannot be negative thus length is {eq}l = 2 {/eq}

Height of prism is {eq},h = l + 3 {/eq}

Substitute the above values,

{eq}\begin{align*} h& = 2 + 3\\ & = 5 \end{align*} {/eq}

Thus height of prism is, {eq}h = 5 {/eq}