The Pythagorean Theorem in Three Dimensions Quick Check
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Question
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)
Responses
9.1 m
9.1 m
14.8 cm
14.8 cm
24.1 cm
24.1 cm
16.8 m
16.8 m
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To find the height of the tent, we can use the Pythagorean theorem in three dimensions.
Let's label the height of the tent as h.
The slant height of the tent forms a right triangle with the height and one of the edges of the square base.
The length of the edge of the square base is given as 20 m, and the slant height is given as 13.5 m.
Using the Pythagorean theorem, we have:
h^2 + 20^2 = 13.5^2
Simplifying and solving for h, we get:
h^2 + 400 = 182.25
h^2 = 182.25 - 400
h^2 = 217.75
h ≈ √217.75
h ≈ 14.8
Therefore, the height of the tent is approximately 14.8 m.
The correct answer is:
14.8 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)
Responses
92.9 cm
92.9 cm
78.6 cm
78.6 cm
60.2 cm
60.2 cm
116.3 cm
To find the length of the ramp needed to fit diagonally in the cage, we can use the Pythagorean theorem in three dimensions.
Let's label the length of the ramp as r.
The length of the cage is given as 70 cm, the width is given as 35 cm, and the height is given as 50 cm.
To find the diagonal length within the cage, we need to find the diagonal of the rectangular prism. This can be done by finding the diagonal of the base and using it as one side of a right triangle, with the height as the other side.
Using the Pythagorean theorem:
r^2 = 70^2 + 35^2 + 50^2
r^2 = 4900 + 1225 + 2500
r^2 = 8625
r ≈ √8625
r ≈ 92.9
Therefore, the length of the ramp needed to fit diagonally in the cage is approximately 92.9 cm.
The correct answer is:
92.9 cm
Why did Thomas Paine write the pamphlet, Common Sense?
(1 point)
Responses
to support conquering the Native Americans
to support conquering the Native Americans
to argue that more land should go to France and Spain
to argue that more land should go to France and Spain
to show that loyalty to the monarchy was wrong
to show that loyalty to the monarchy was wrong
to pledge his loyalty to the British government
The correct answer is:
to show that loyalty to the monarchy was wrong
To find the height of the circus tent, we can use the Pythagorean theorem in three dimensions. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In three dimensions, we can extend this concept to relate the height, base, and slant height of a pyramid. Here's how to solve the problem:
1. Start by drawing a diagram of the triangular face of the pyramid. Label the height as "h", the base as "b", and the slant height as "s".
2. Use the Pythagorean theorem to relate the height, base, and slant height. The equation is: s^2 = h^2 + (b/2)^2
3. Substitute the given values into the equation. The slant height is given as 13.5 m, and the base length is given as 20 m. The equation becomes: 13.5^2 = h^2 + (20/2)^2
4. Simplify the equation. The equation becomes: 182.25 = h^2 + 100
5. Subtract 100 from both sides of the equation to isolate h^2. The equation becomes: h^2 = 82.25
6. Take the square root of both sides of the equation to solve for h. The height (h) is equal to the square root of 82.25.
7. Round the answer to the nearest tenth, as requested. The height of the tent is approximately 9.1 m.
So, the correct answer to the question is 9.1 m.