a right rectangular pyramid has base dimensions 7yd. by 5yd. and a height of 10yd. determine the surface area of the pyramid to the nearest square yard.
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To determine the surface area of the right rectangular pyramid, we need to calculate the area of each face separately and then add them together.
1. Base:
The base of the pyramid is a rectangle with dimensions 7 yards by 5 yards. The area of the base is calculated as:
Area of base = length × width = 7yd × 5yd = 35 square yards.
2. Side faces:
There are four side faces in a pyramid. Each side face is a triangle with the base equal to the length of the rectangle (7 yards) and the height equal to the height of the pyramid (10 yards). The area of each side face is calculated as:
Area of a side face = (1/2) × base × height = (1/2) × 7yd × 10yd = 35 square yards.
3. Total surface area:
To find the total surface area of the pyramid, we need to add the area of the base and the four side faces:
Total surface area = Area of base + 4 × Area of a side face
Total surface area = 35 square yards + 4 × 35 square yards
Total surface area = 35 square yards + 140 square yards
Total surface area = 175 square yards.
Therefore, the surface area of the right rectangular pyramid is approximately 175 square yards.
To find the surface area of a right rectangular pyramid, we need to calculate the area of each face and then sum them up.
The pyramid has five faces: the base and four triangular faces.
First, let's calculate the area of the base. The base is a rectangle with dimensions 7 yards by 5 yards. So, the area of the base is 7 yards * 5 yards = 35 square yards.
Next, let's calculate the area of each triangular face. Since the pyramid is right-angled, the two triangular faces adjacent to the base are congruent.
To find the area of a triangle, we need the base length and the height. The base of each triangular face is one of the sides of the base rectangle, which is 5 yards. The height of each triangular face can be found using the Pythagorean theorem. The height is the distance between the base and the apex of the pyramid, which is given as 10 yards.
Using the Pythagorean theorem, we can find the height of each triangular face:
height = √(10^2 - (5/2)^2)
= √(100 - 6.25)
= √(93.75)
= 9.68 yards (rounded to the nearest hundredth)
Now, we can calculate the area of each triangular face:
area of each triangular face = (1/2) * base * height
= (1/2) * 5 yards * 9.68 yards
= 24.2 square yards (rounded to the nearest tenth)
Since there are four triangular faces, the total area of the four triangular faces is 4 * 24.2 square yards = 96.8 square yards.
Finally, we can find the surface area by summing the area of the base and the four triangular faces:
surface area = area of the base + area of the four triangular faces
= 35 square yards + 96.8 square yards
≈ 131.8 square yards (rounded to the nearest square yard)
Therefore, the surface area of the right rectangular pyramid is approximately 131.8 square yards.