Question

julias family has a triangular back yard. they plan to install a rectangular swimming pool and leave the rest of the yard as grass. when planning the placement of their pool, they created a scale representation of the yard and pool on a coordinate grid pictured below. if one grid line represents a length of 5 feet, how many square feet of grass will there be in julias back yard after installing the pool? a. 2,700 square feet b. 3,060 square feet c. 1,650 square feet d. 1,500 square feet

Explanation:

Step1: Calculate the area of the triangular yard

The base of the triangle spans from \(x = - 2\) to \(x = 6\), so the number of grid - lines for the base \(n_{b}=(6 - (-2)) = 8\). The length of the base \(b = 8\times5=40\) feet. The height of the triangle spans from \(y=-9\) to \(y = 9\), so the number of grid - lines for the height \(n_{h}=(9-(-9)) = 18\). The height \(h = 18\times5 = 90\) feet.
Using the formula for the area of a triangle \(A_{\triangle}=\frac{1}{2}\times b\times h\), we have \(A_{\triangle}=\frac{1}{2}\times40\times90=1800\) square feet.

Step2: Calculate the area of the rectangular pool

The length of the rectangle (horizontal side): the number of grid - lines \(n_{l}=4\), so \(l = 4\times5 = 20\) feet. The width of the rectangle (vertical side): the number of grid - lines \(n_{w}=3\), so \(w = 3\times5=15\) feet.
Using the formula for the area of a rectangle \(A_{rect}=l\times w\), we have \(A_{rect}=20\times15 = 300\) square feet.

Step3: Calculate the area of the grass

The area of the grass \(A = A_{\triangle}-A_{rect}\). Substitute the values: \(A=1800 - 300=1500\) square feet.

Answer:

D. 1,500 square feet