A circus tent is in the shape of a square pyramid. Each edge of the square base is 20 m and its slant height is 13.5 m. Find the height of the tent. Round the answer to the nearest tenth. (1 point)
O 9.1 m
• 24.1 cm
О 16.8 m
• 14.8 cm
To find the height of the pyramid (the tent), we can use the Pythagorean theorem. The slant height of the pyramid forms the hypotenuse of a right triangle, while the height (which we need to find) and half the base of the pyramid form the two legs of the right triangle.
Given:
- The slant height (l) of the pyramid is 13.5 m.
- The edge of the square base (b) is 20 m.
To find the height (h), we first calculate half the length of the base to get the length of one leg of the right triangle:
Half the length of the base = b/2 = 20 m / 2 = 10 m
Now we have a right triangle with one leg as half the base (10 m) and the hypotenuse as the slant height (13.5 m). We can label the half of the base as 'a', the height we need to find as 'h', and the hypotenuse as 'l'.
According to the Pythagorean theorem:
l² = a² + h²
Let's solve for h²:
h² = l² - a²
h² = (13.5 m)² - (10 m)²
h² = 182.25 m² - 100 m²
h² = 82.25 m²
Now, let's find h:
h = √(82.25 m²)
h ≈ 9.1 m
Rounding our answer to the nearest tenth gives us 9.1 m. Therefore, the height of the tent is approximately 9.1 meters.
The correct answer from your list is: O 9.1 m.
Melissa has a gerbil that lives in a cage in the shape of a rectangular prism. She wants to put a ramp in the cage to give her gerbil exercise. The cage has a length of 70 cm, a width of 35 cm, and a height of 50 cm.
How long does the ramp need to be to fit diagonally in the cage? Round the answer to the nearest tenth.
(1 point)
• 60.2 cm
• 92.9 cm
O 78.6 cm
• 116.3 cm
To fit the ramp diagonally inside the cage, we need to find the length of the longest diagonal of the rectangular prism (the cage). This diagonal will go from one corner of the cage to the opposite corner and can be found using the 3D version of the Pythagorean theorem:
d² = l² + w² + h²
where:
- d is the length of the diagonal
- l is the length of the cage (70 cm)
- w is the width of the cage (35 cm)
- h is the height of the cage (50 cm)
We'll plug in the values and calculate:
d² = (70 cm)² + (35 cm)² + (50 cm)²
d² = 4900 cm² + 1225 cm² + 2500 cm²
d² = 4900 + 1225 + 2500
d² = 8625 cm²
Now we find d by taking the square root of d²:
d = √(8625 cm²)
d ≈ 92.9 cm
Rounding the answer to the nearest tenth gives us 92.9 cm. Therefore, the ramp needs to be approximately 92.9 cm long to fit diagonally in the cage.
The correct answer from your list is: • 92.9 cm.
To find the height of the tent, you can use the Pythagorean theorem. The slant height of the pyramid and the height of the pyramid form a right triangle, with the slant height as the hypotenuse and the height as one of the legs.
In this case, the slant height is 13.5 m. Let's denote the height of the pyramid as h.
Using the Pythagorean theorem, we have:
h^2 + (20/2)^2 = (13.5)^2
Simplifying the equation, we get:
h^2 + 10^2 = 13.5^2
h^2 + 100 = 182.25
h^2 = 82.25
Taking the square root of both sides, we find:
h ≈ √82.25
h ≈ 9.1
Rounding the answer to the nearest tenth, the height of the tent is approximately 9.1 m.
Therefore, the correct answer is O) 9.1 m.