Question

Hi I need help with this geometry calculus 1 problem. I’m in high school and this is a homework. thank you!

Answer 1

Analysis on Perimeter

The shape is made up of a semicircle and a rectangle.

The perimeter of a semicircle is calculated using the formula:

`P_C=\pi r`

From the image, the diameter of the rectangle is given to be x. Since we know that the diameter is twice the length of the radius, this means that:

`r=(d)/(2)=(x)/(2)`

Hence, the perimeter of the semicircle in terms of x becomes:

`P_C=(\pi)/(2)x`

The perimeter of a rectangle is given to be:

`P_R=2l+2w`

Already, we have the base length to be x, therefore:

`w=x`

Note, though, that the connecting sides between the semicircle and the rectangle are not included in the perimeter length given. Therefore, the perimeter of the rectangle will be:

`\begin{gathered} P_R=2l+w \\ P_R=2l+x \end{gathered}`

Since the perimeter is given to be 40 feet, then we have:

`\begin{gathered} P_R+P_C=40 \\ \therefore \\ 2l+x+(\pi)/(2)x=40 \end{gathered}`

We can get the length of the rectangle by making l the subject of the formula:

`\begin{gathered} 2l=40-x-(\pi)/(2)x \\ \text{Dividing through by 2, we have:} \\ l=(40)/(2)-(x)/(2)-(\pi)/(2*2)x \\ l=20-(x)/(2)-(\pi)/(4)x \end{gathered}`

Combining to a single fraction, we have:

`l=(80-2x-\pi x)/(4)`

AREA OF THE WINDOW

The area of the window can be gotten by using the formula:

`A=A_C+A_R`

The area of a semicircle is gotten using the formula:

`A_C=(\pi r^2)/(2)`

Therefore, the area will be:

`\begin{gathered} A_C=(\pi((x)/(2))^2)/(2)=(\pi x^2)/(2*4) \\ A_C=(\pi x^2)/(8) \end{gathered}`

The area of the rectangle will be:

`A_R=l* w`

Inputting our known values, we have:

`\begin{gathered} A_R=(80-2x-\pi x)/(4)* x \\ A_R=x((80-2x-\pi x)/(4)) \end{gathered}`

Therefore, the combined area will be:

`A=(\pi x^2)/(8)+x((80-2x-\pi x)/(4))`

Combining the fraction, we have:

`A=(\pi x^2+2x(80-2x-\pi x))/(8)`

Expanding the bracket, we have:

`A=(\pi x^2+160x-4x^2-2\pi x^2)/(8)=(160x-4x^2-\pi x^2)/(8)`

Therefore, the function of the area is:

`A=(160x-4x^2-\pi x^2)/(8)`