A pyramid has a height of 5 in and a surface area of 90 in find the surface area of a similar period with the height of 10 in
The surface area of the pyramid with a height of 10 in would be 180 in.
To find the surface area of a similar pyramid with a different height, we need to use the concept of ratios.
The surface area of a pyramid is determined by the square of the length of its base and its height. Let's denote the base length of the original pyramid as 'b' and the height as 'h'. We are given that the original pyramid has a height of 5 in and a surface area of 90 in².
Using the formula for the surface area of a pyramid, we can write the equation as:
90 = b² + (b × (sqrt(b²/4 + h²)))
Now, we want to find the surface area of a similar pyramid with a height of 10 in. Let's call the base length of the similar pyramid 'B'. We want to find the surface area of this similar pyramid.
Using the same formula, we can write the equation as:
Surface Area = B² + (B × (sqrt(B²/4 + 10²)))
To solve this equation, we need to find the ratio between the original height (5 in) and the new height (10 in). The ratio of the heights is 10/5 = 2.
Since the base length and height of similar pyramids are proportional, we can conclude that the ratio between the base lengths is also 2. Therefore, the ratio of the surface areas of similar pyramids is the square of the ratio of their corresponding sides.
In this case, since the ratio of the heights is 2, the ratio of the surface areas will be 2² = 4.
Hence, 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 with a height of 10 in can be calculated as:
Surface Area = 4 × 90 = 360 in².
To find the surface area of the similar pyramid with a height of 10 in, we can use the concept of "similar figures."
Similar figures have proportional side lengths, which means the ratio of the corresponding sides is the same. In this case, since we know the height of the original pyramid is 5 in and the surface area is 90 in², we can find the ratio between the height and surface area.
Original height: 5 in
Original surface area: 90 in²
Ratio of heights: 10 in / 5 in = 2
Ratio of surface areas: ( ) / 90 in² = 2
To find the missing value, we can set up a proportion:
x / 90 = 2/1
Simplifying the proportion, we get:
x = 90 * 2
x = 180
Therefore, the surface area of the similar pyramid with a height of 10 in is 180 in².