Question

A rectangle is formed with the base on the x-axis and the top corners on the function y= 25 - x^2. Find thedimensions of the rectangle with the largest area.

Answer 1

If we draw a rectangle under the graph, we can find that the width w of the rectangle is 2x and the height h is 25 - x^2.

So, the area of the rectangle is,

`\begin{gathered} A=2x(25-x^2) \\ A=50x-2x^3 \\ \end{gathered}`

Differentiate the function to find maxiumum.

`A^(\prime)=50-6x^2`

Putting A'=0,

`\begin{gathered} x^2=(50)/(6)=(25)/(3) \\ x=+\frac{5}{\sqrt[]{3}}or-\frac{5}{\sqrt[]{3}} \end{gathered}`

x should be positive So,

`x=\frac{5}{\sqrt[]{3}}`

Differentiate A' with respect to x to find if it is a maximum.

`\begin{gathered} A^(\doubleprime)=-12x<0 \\ \text{when x=}\frac{\text{5}}{\sqrt[]{3}} \end{gathered}`

It confirms that area is a maximum for x.

So, the width of the rectangle with largest area is ,

`w=2x=2*\frac{5}{\sqrt[]{3}}=\frac{10}{\sqrt[]{3}}`
`\begin{gathered} h=25-x^2 \\ h=25-(\frac{5}{\sqrt[]{3}})^2 \\ h=25-(25)/(3) \\ h=(50)/(3) \end{gathered}`