Your friend is purchasing an umbrella with a slant height of 4 feet. A red umbrella is shaped like a regular pentagonal pyramid with a side length of 3 feet. find the lateral surface area of the red umbrella.
each of the 5 lateral faces is just a triangle with
base = 3
height = 4
I assume you can find the area of such a triangle...
To find the lateral surface area of the red umbrella, we need to calculate the area of each of its five triangular faces and then sum them up.
Step 1: Calculate the perimeter of the base of the pyramid.
Since the red umbrella is a regular pentagon, all of its sides have equal lengths. So, the perimeter of the base is 5 times the length of one side.
Perimeter of the base = 5 * 3 feet = 15 feet
Step 2: Calculate the slant height of one of the triangular faces.
The slant height is the distance from the apex of the pyramid to the midpoint of one of the base edges. Since we know the slant height of the umbrella is 4 feet, we can use this information to calculate the height of the triangular face.
By drawing a right triangle with the slant height as the hypotenuse and the height as one of the legs, we can use the Pythagorean theorem to find the height.
Let h be the height of the triangular face.
Using the Pythagorean theorem:
h^2 + (3/2)^2 = 4^2
h^2 + 9/4 = 16
h^2 = 16 - 9/4
h^2 = 64/4 - 9/4
h^2 = 55/4
h = sqrt(55/4) = sqrt(55)/2 feet
Step 3: Calculate the area of one triangular face.
The area of a triangle can be calculated using the formula A = 1/2 * base * height, where the base is the length of one side of the pentagon and the height is the calculated height from step 2.
A = 1/2 * 3 * (sqrt(55)/2) = 3/2 * sqrt(55) square feet
Step 4: Calculate the total lateral surface area.
Since the umbrella has five triangular faces, we need to multiply the area of one triangular face by 5.
Lateral surface area = 5 * (3/2 * sqrt(55)) = 15/2 * sqrt(55) square feet
Therefore, the lateral surface area of the red umbrella is (15/2) * sqrt(55) square feet.