Question

1. find the area and perimeter of the rectangle.

$\sqrt{8}+2\sqrt{18}$
$2\sqrt{2}+5\sqrt{3}$
$\sqrt{8}+2\sqrt{18}$

Explanation:

Step1: Simplify the length and width

Simplify $\sqrt{8}+2\sqrt{18}$:

$$\begin{align*} \sqrt{8}+2\sqrt{18}& = 2\sqrt{2}+2\times3\sqrt{2}\\ &=2\sqrt{2}+6\sqrt{2}\\ & = 8\sqrt{2} \end{align*}$$

The width is $2\sqrt{2}+5\sqrt{3}$ (already in simplest form).

Step2: Calculate the area

Use the formula $A = lw$ (where $l$ is length and $w$ is width).

$$\begin{align*} A&=(8\sqrt{2})(2\sqrt{2}+5\sqrt{3})\\ &=8\sqrt{2}\times2\sqrt{2}+8\sqrt{2}\times5\sqrt{3}\\ &=16\times2 + 40\sqrt{6}\\ &=32+40\sqrt{6} \end{align*}$$

Step3: Calculate the perimeter

Use the formula $P = 2(l + w)$.

$$\begin{align*} P&=2(8\sqrt{2}+2\sqrt{2}+5\sqrt{3})\\ &=2(10\sqrt{2}+5\sqrt{3})\\ &=20\sqrt{2}+10\sqrt{3} \end{align*}$$

Answer:

The area is $32 + 40\sqrt{6}$ and the perimeter is $20\sqrt{2}+10\sqrt{3}$.