Question

the area of a rectangle is 65 square units, if the length is 2 times the width, then find the dimensions of the rectangle

Answer 1

Dimensions of a rectangle

Writing an expression for the dimensions

Let's call each dimension by a letter:

W: width

L: length

We have that the length is 2 times the width.

This means that the length is equal to 2 times the width:

L = 2W

Area of a rectangle: equation for the unkowns

We have that the area of a rectangle is given by the product of the width and the length:

area = W · L

Since L = 2W, then:

area = W · L

↓ replacing L by 2W

area = W · 2W

area = 2W²

Since area = 65, then

65 = 2W²

Now, we have an equation that we can use to find the value of W.

Finding each dimension

Width

Using the previous equation, we can find the value for the width W:

65 = 2W²

↓ taking 2 to the left side

65/2 = W²

↓ square root of both sides

√(65/2) = √W² = W

Then,

`W=\sqrt[]{(65)/(2)}=\frac{\sqrt[]{65}}{\sqrt[]{2}}`

We have an expression for the width now, we are going to find a better expression:

`\begin{gathered} W=\frac{\sqrt[]{65}}{\sqrt[]{2}}=\frac{\sqrt[]{65}\cdot\sqrt[]{2}}{\sqrt[]{2}\cdot\sqrt[]{2}} \\ =\frac{\sqrt[]{65\cdot2}}{\sqrt[]{2}^2}=\frac{\sqrt[]{130}}{2} \end{gathered}`

Then, we can say that:

`W=\frac{\sqrt[]{130}}{2}`

Length

Since the length is 2 times W:

`\begin{gathered} L=2W=2\cdot\frac{\sqrt[]{130}}{2} \\ L=\sqrt[]{130} \end{gathered}`

Answer

We have that the measures for the width, W, and length, L, are:

`W=\frac{\sqrt[]{130}}{2}\text{ and }L=\sqrt[]{130}`