You are adding a three-sided pyramid to your climbing wall. You make the three lateral faces of the pyramid from a 4-foot by 8-foot sheet of plywood. How many square feet of plywood are left over?
To find out how many square feet of plywood are left over after making the three lateral faces of the pyramid, we first need to calculate the total area of the plywood sheet.
The area of a rectangular plywood sheet can be calculated by multiplying its length by its width. In this case, the plywood sheet is 4 feet by 8 feet, so its total area is:
Area = Length × Width = 4 feet × 8 feet = 32 square feet
Now, let's calculate the total area of the three lateral faces of the pyramid.
Since each face of the pyramid is a triangle, we can calculate the area of one triangle and then multiply it by three to account for all three faces.
The formula for the area of a triangle is:
Area = (Base × Height) / 2
In this case, the height of the triangle is given as 4 feet, and the base can be calculated using the Pythagorean theorem. The Pythagorean theorem states that the square of the hypotenuse (side opposite the right angle) is equal to the sum of the squares of the other two sides.
In the case of this triangle, one of the sides is the length of the plywood sheet (8 feet), and the other side is half the width of the plywood sheet (4 feet ÷ 2 = 2 feet). So, the base of the triangle can be calculated as:
Base = √(8^2 - 2^2)
Base = √(64 - 4)
Base = √60
Base ≈ 7.746 feet (rounded to three decimal places)
Now, let's calculate the area of one triangle using the formula:
Area = (Base × Height) / 2
Area = (7.746 feet × 4 feet) / 2
Area ≈ 15.492 square feet (rounded to three decimal places)
Since we have three identical triangles, the total area of the three lateral faces is:
Total Area = 3 × Area of one triangle
Total Area = 3 × 15.492 square feet
Total Area ≈ 46.476 square feet (rounded to three decimal places)
Finally, to find out how many square feet of plywood are left over, subtract the total area of the three lateral faces from the total area of the plywood sheet:
Leftover Area = Total Area of plywood sheet - Total Area of three lateral faces
Leftover Area = 32 square feet - 46.476 square feet
Leftover Area ≈ -14.476 square feet
Since the result is negative, it means that you would require additional plywood to cover all the three lateral faces of the pyramid.