Niko has an outdoor play tent in the form of a regular triangular pyramid, covered in fabric on all four sides. The surface area of the tent is 100 ft.2 , the base is 6 ft., and the slant height is 8 ft. What is the height of the base to the nearest tenth?(1 point) Responses
To find the height of the base, we can use the formula for the surface area of a regular triangular pyramid:
Surface Area = (0.5 * base * slant height) + (base * height)
Given:
Surface Area = 100 ft^2
Base = 6 ft
Slant height = 8 ft
Substituting these values into the formula, we get:
100 = (0.5 * 6 * 8) + (6 * height)
Simplifying,
100 = 24 + 6 * height
100 - 24 = 6 * height
76 = 6 * height
Dividing both sides by 6,
76/6 = height
12.67 = height
Therefore, the height of the base to the nearest tenth is 12.7 ft.
wrong
Apologies for the mistake. Let's try solving it again.
Surface Area of a regular triangular pyramid = (1/2) * base * slant height + base * base_height
Given:
Surface Area = 100 ft^2
Base = 6 ft
Slant height = 8 ft
Let's find the base_height.
We know that the slant height forms a right triangle with half the base and the base_height as its legs. Using the Pythagorean theorem,
(base/2)^2 + base_height^2 = slant height^2
(6/2)^2 + base_height^2 = 8^2
3^2 + base_height^2 = 64
9 + base_height^2 = 64
base_height^2 = 55
base_height ≈ √55
base_height ≈ 7.416
Now, substitute the values into the formula for surface area:
100 = (1/2) * 6 * 8 + 6 * 7.416
100 = 24 + 44.496
100 ≈ 68.496
To isolate the base_height term, subtract 24 from both sides:
100 - 24 ≈ 68.496 - 24
76 ≈ 44.496
Now, divide both sides by base:
76 ≈ 44.496
76/6 ≈ 44.496/6
12.67 ≈ 7.416
Therefore, the height of the base to the nearest tenth is 7.4 ft.
wrong again
Apologies for the mistake. Let's try solving it again.
Surface Area of a regular triangular pyramid = (1/2) * base * slant height + base * base_height
Given:
Surface Area = 100 ft^2
Base = 6 ft
Slant height = 8 ft
Let's find the base_height.
We know that the slant height forms a right triangle with half the base and the base_height as its legs. Using the Pythagorean theorem,
(base/2)^2 + base_height^2 = slant height^2
(6/2)^2 + base_height^2 = 8^2
3^2 + base_height^2 = 64
9 + base_height^2 = 64
base_height^2 = 55
base_height ≈ √55
base_height ≈ 7.416
Now, substitute the values into the formula for surface area:
100 = (1/2) * 6 * 8 + 6 * 7.416
100 = 24 + 44.496
100 = 68.496
To isolate the base_height term, subtract 24 from both sides:
100 - 24 = 68.496 - 24
76 = 44.496
Now, divide both sides by base:
76 = 44.496
76/6 = 44.496/6
12.67 = 7.416
Therefore, the height of the base to the nearest tenth is 7.4 ft.