David has a rectangular backyard and built a right triangle sandbox in the backyard. The cost of the sand is $5.50 per square foot and the border of the sandbox is $3.00 per linear foot. What was the cost of the materials of the sandbox?
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To find the cost of the materials for the sandbox, we need to first calculate the area of the sandbox and the length of its border.
Let's assume the length of the rectangular backyard is L and the width of the rectangular backyard is W. Since the sandbox is a right triangle, one of its sides will be the same as the width of the rectangular backyard, and the other side will be the same as the length.
The area of a triangle is given by the formula: A = 1/2 * base * height.
The base of the sandbox is the width of the rectangular backyard (W), and the height is the length of the rectangular backyard (L). So the area of the sandbox is A = 1/2 * W * L.
The length of the border is equal to the perimeter of the sandbox. Since the sandbox is a right triangle, the perimeter can be found using the formula: P = a + b + c, where a and b are the two legs of the triangle and c is the hypotenuse.
Since one leg of the triangle is W and the other leg is L, the perimeter is P = W + L + c. The length of the hypotenuse (c) can be found using the Pythagorean theorem: c^2 = W^2 + L^2.
Now, we have all the necessary information to calculate the area of the sandbox, the length of the border, and the cost of the materials.
1. Calculate the area of the sandbox:
A = 1/2 * W * L
2. Calculate the length of the hypotenuse:
c^2 = W^2 + L^2
c = √(W^2 + L^2)
3. Calculate the length of the border:
P = W + L + c
4. Calculate the cost of the sand:
Cost of sand = Area of the sandbox * Cost per square foot
5. Calculate the cost of the border:
Cost of border = Length of the border * Cost per linear foot
6. Add the cost of the sand and the cost of the border to get the total cost of the materials for the sandbox.