Question

Find expressions for the area and perimeter of the shaded region of the figure

Answer 1

The following steps would guide in providing answers to the area and perimeter of the shaded region of the figure.

Step 1: State the shape of the figure

The shape of the figure is a rectangle with length and width

Step 2: Write the formula for finding the area and the perimeter of the figure

The area of a rectangle is calculated by the formula

`A_{\text{rectangle}}=l* w`

The perimeter of a rectangle is calculated by the formula

`P_{\text{rectangle}}=2(l+w)`

Step 3: Calculate the area of the figure

The area of the figure is the area of a rectangle. Input the length and the width of the given figure into the formula for finding the area of a rectangle

`\begin{gathered} A_{\text{rectangle}}=l* w \\ l=x+5,w=x+3 \\ A_{\text{rectangle}}=(x+5)(x+3) \\ A_{\text{rectangle}}=x(x+3)+5(x+3) \\ A_{\text{rectangle}}=x^2+3x+5x+15 \\ A_{\text{rectangle}}=x^2+8x+15 \end{gathered}`

Step 4: Calculate the area of the unshaded region

Input the length and width of the unshaded region into the formula for finding the area of a rectangle

`\begin{gathered} A_{\text{rectangle}}=l* w \\ l=x,w=x-3 \\ A_{\text{rectangle}}=x(x-3) \\ A_{\text{rectangle}}=x^2-3x \end{gathered}`

Step 5: Calculate the area of the shaded portion

From the figure shown, it can be observed that the area of the figure is the addition of the area of the unshaded region and the shaded region. Hence, the area of the shaded region would be the area of the figure minus the area of the unshaded region. This is calculated below

`\begin{gathered} A_{\text{shaded region}}=A_(figure)-A_{unshaded\text{ region}} \\ A_{\text{shaded region}}=(x^2+8x+15)-(x^2-3x) \\ A_{\text{shaded region}}=x^2+8x+15-x^2-3x \\ Collect\text{ like terms} \\ A_{\text{shaded region}}=x^2-x^2+8x-3x+15 \\ A_{\text{shaded region}}=5x+15 \end{gathered}`

Step 6: Calculate the perimeter of the figure

The perimeter of the figure is the perimeter of a rectangle. Input the length and the width of the given figure into the formula for finding the perimeter of a rectangle

`\begin{gathered} P_{\text{rectangle}}=2(l+w) \\ l=x+5,w=x+3 \\ P_{\text{rectangle}}=2(x+5+x+3) \\ P_{\text{rectangle}}=2(x+x+5+3) \\ P_{\text{rectangle}}=2(2x+8) \\ P_{\text{rectangle}}=4x+16 \end{gathered}`

Step 7: Calculate the perimeter of the unshaded portion

The perimeter of the unshaded portion is the perimeter of a rectangle. Input the length and the width of the unshaded portion into the formula for finding the perimeter of a rectangle

`\begin{gathered} P_{\text{rectangle}}=2(l+w) \\ l=x,w=x-3 \\ P_{\text{rectangle}}=2(x+x-3) \\ P_{\text{rectangle}}=2(2x-3) \\ P_{\text{rectangle}}=4x-6 \end{gathered}`

Step 8: Calculate the perimeter of the shaded region

From the figure shown, it can be observed that the perimeter of the figure is the addition of the perimeter of the unshaded region and the shaded portion. Hence, the perimeter of the shaded region would be the

perimeter of the figure minus the perimeter of the unshaded region. This is calculated below

`\begin{gathered} p_{\text{shaded region}}=A_(figure)-A_{unshaded\text{ region}} \\ p_{\text{shaded region}}=(4x+16)-(4x-6) \\ p_{\text{shaded region}}=4x+16-4x+6 \\ Collect\text{ like terms} \\ p_{\text{shaded region}}=4x-4x+16+6 \\ p_{\text{shaded region}}=0+22 \\ p_{\text{shaded region}}=22\text{units} \end{gathered}`

Answer Summary

The expression for the area of the shaded region is 5x + 15 square unit

The perimeter of the shaded region is 22 units