Question

each unit on the coordinate grid represents 1 yard. the rectangular pool and triangular hot tub shown are both in need of covers. how much total material is needed to cover both? square yards

Explanation:

Step1: Find the area of the rectangle (pool)

First, we can use the distance formula or count the grid units to find the length and width. Alternatively, we can use the formula for the area of a parallelogram (since the rectangle is slanted, it's a parallelogram) which is base times height. By counting the grid, we can see that the base (horizontal distance) and height (vertical distance) or we can use the formula for the area of a parallelogram. Let's find the length of the sides. Using the coordinates, let's assume the rectangle has length and width. Alternatively, we can use the formula for the area of a parallelogram: \( A = base \times height \). By counting the grid, the base can be calculated as the distance between two points, but maybe easier to use the formula for the area of a parallelogram by finding the length of the base and the height. Alternatively, we can use the shoelace formula for the coordinates. Let's list the coordinates of the rectangle: Let's say the vertices are (0,4), (3,10), (7,8), (4,2). Using the shoelace formula:

Shoelace formula: For vertices \((x_1,y_1), (x_2,y_2), (x_3,y_3), (x_4,y_4)\), area is \( \frac{1}{2} |x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1)| \)

Plugging in (0,4), (3,10), (7,8), (4,2):

\( x_1=0, y_1=4; x_2=3, y_2=10; x_3=7, y_3=8; x_4=4, y_4=2 \)

\( x_1y_2 = 0*10 = 0 \)

\( x_2y_3 = 3*8 = 24 \)

\( x_3y_4 = 7*2 = 14 \)

\( x_4y_1 = 4*4 = 16 \)

Sum: \( 0 + 24 + 14 + 16 = 54 \)

\( y_1x_2 = 4*3 = 12 \)

\( y_2x_3 = 10*7 = 70 \)

\( y_3x_4 = 8*4 = 32 \)

\( y_4x_1 = 2*0 = 0 \)

Sum: \( 12 + 70 + 32 + 0 = 114 \)

Area = \( \frac{1}{2} |54 - 114| = \frac{1}{2} | -60 | = 30 \) square yards.

Step2: Find the area of the triangle (hot tub)

The triangle has a base and height. From the grid, the base is 2 units (from x=5 to x=7, y=2) and height is 3 units (from y=2 to y=5). So area of triangle is \( \frac{1}{2} \times base \times height = \frac{1}{2} \times 2 \times 3 = 3 \) square yards.

Step3: Total material needed

Add the area of the rectangle and the area of the triangle: \( 30 + 3 = 33 \) square yards.

Answer:

33