A right prism has bases that are squares. The area of one base is 81 square feet. The lateral area of the prism is 144 square feet. What is the altitude of the prism.
Right prism has both bases the same.
laterial area= 4*(side)height
where side= sqrt 81=9
144= 4*9*height
solve for height
38
To find the altitude of the prism, we need to first find the side length of the square base.
1. Since the area of one base is given as 81 square feet, we can find the side length of the square base by taking the square root of its area.
√81 = 9 feet
2. We know that the lateral area of the prism is 144 square feet. A right prism has two congruent bases and faces that are all rectangles. The lateral area of a prism is the sum of the areas of all the rectangular faces.
3. Since the bases are squares, the rectangular faces of the prism are also squares. The area of each rectangular face can be found by subtracting the area of one square base from the total lateral area and dividing it by 4 (since there are 4 rectangular faces in total).
(Total lateral area - Area of one base) / 4 = (144 - 81) / 4 = 63 / 4 = 15.75 square feet
4. The area of each rectangular face is 15.75 square feet. Since the height of a rectangular face is the same as the altitude of the prism, the altitude of the prism is 15.75 feet.
To find the altitude of the prism, we can start by understanding the properties of a right prism with square bases.
A right prism is a three-dimensional shape that has two parallel and congruent polygonal bases. The sides connecting the corresponding vertices of the bases are known as lateral faces, and their total area is called the lateral area.
In this case, the bases of the prism are squares, which means they have equal side lengths. Let's call the side length of one base "s". Therefore, the area of one base is given as s^2 = 81 square feet.
To find the altitude, we need to determine the height of the prism, which is the perpendicular distance between the bases. In a right prism, the lateral area is equal to the perimeter of one base multiplied by the height.
Since the lateral area is given as 144 square feet, and the perimeter of one square base is equal to 4 times the side length, we can set up the equation:
perimeter of one base × height = lateral area
4s × height = 144
4s × height = 144
To solve for the height, divide both sides of the equation by 4s:
height = 144 / (4s)
height = 36 / s
Now, we can substitute the value of s from the given information that s^2 = 81:
s^2 = 81
s = √81
s = 9
Substituting this into the equation for height:
height = 36 / 9
height = 4 feet
Therefore, the altitude of the prism is 4 feet.