Question

1. (15 points) The roof of a shed is shown below.You need to cover this roof with shingles that measure 2 ft by 2 ft. Notice that the roof has fourdifferent sides that need to be covered.a. What is the area of the roof that needs to be covered?b. How many shingles do you need to buy?

Answer 1

Given the information on the picture, we have to find the area of these two figures:

first, we can find the height of the triangle using the pythagorean theorem:

`\begin{gathered} h=\sqrt[]{(6)^2-(4.5)^2}=\sqrt[]{36-20.25}=\sqrt[]{15.75}=3.96\approx4 \\ \Rightarrow h=4ft \end{gathered}`

now that we have the height, we can calculate both areas to get the following:

`\begin{gathered} \text{ Area of the triangle:} \\ A_1=(9\cdot4)/(2)=(36)/(2)=18ft^2 \\ \text{ Area of the rectangle:} \\ A_2=12\cdot6=72ft^2 \end{gathered}`

since the roof consists of two triangles and two rectangles, the total area of the roof isL:

`\begin{gathered} A=2A_1+2A_2=2(18)+2(72)=36+144=180ft^2 \\ \Rightarrow A=180ft^2 \end{gathered}`

therefore, the area of the roof is 180ft^2

Now, we have that the shingles have measures 2ft by 2ft. This means that they have a square form, and their area is:

`A_3=2\cdot2=4ft^2`

so, dividing the total area of the roof by the area of one roof shingle, we have:

`(180)/(4)=45\text{ roof shingles}`

therefore, you need to buy 45 roof shingles