A pyramids has a height of 5 in and surface area of 90 in2 find the surface area of 90 in2 find the surface area of a similar pyramid with a height of 10 in Round to the nearest tenth if necessary
1. 360 in2
2. 180 in2
3. 22.5 in2
4. 3.6 in2
1. 360 in2
Explanation:
The surface area of a pyramid is given by the formula:
Surface area = 1/2 * perimeter of base * slant height + base area
Since the two pyramids are similar, their corresponding sides are in proportion. Let x be the length of one side of the base of the smaller pyramid, and y be the length of one side of the base of the larger pyramid. Then, we have:
y/x = 2 (since the height of the larger pyramid is twice that of the smaller pyramid)
y = 2x
The ratio of the surface areas of the two pyramids is:
(surface area of larger pyramid) / (surface area of smaller pyramid) = (y^2 / x^2)
= (2x)^2 / x^2
= 4
Therefore, the surface area of the larger pyramid is 4 times that of the smaller pyramid.
Substituting the given values, we have:
Surface area of smaller pyramid = 90 in2
Height of smaller pyramid = 5 in
Using the formula for surface area, we can solve for the length of one side of the base:
90 = 1/2 * (4x) * (sqrt(x^2 + 5^2)) + x^2
Simplifying, we get a quadratic equation:
2x^2 + 10sqrt(x^2 + 25) - 90 = 0
Solving this equation (by using the quadratic formula or a graphing calculator), we get:
x ≈ 4.67 in
Therefore, the length of one side of the base of the larger pyramid is:
y = 2x ≈ 9.33 in
Now, we can use the formula for surface area to find the surface area of the larger pyramid:
Surface area = 1/2 * (4y) * (sqrt(y^2 + 10^2)) + y^2
= 360 in2 (rounded to the nearest tenth)
To find the surface area of a similar pyramid with a height of 10 in, we can use the concept of similar triangles.
The ratio of the heights of the two pyramids is 10/5 = 2.
Since the two pyramids are similar, the ratio of their surface areas would be equal to the square of the ratio of their corresponding sides. In this case, it would be equal to (2^2) = 4.
So, the surface area of the similar pyramid with a height of 10 in would be 4 times the surface area of the original pyramid.
Therefore, the surface area of the similar pyramid would be 4 * 90 in^2 = 360 in^2.
Hence, the correct answer is 1. 360 in^2.