Use the image to answer the question.
An illustration shows a square pyramid with labeled parts. The left, front and right faces of the pyramid and base are visible. The back face is not visible. The back base edge is not visible but its position is indicated by a dashed line. The front base edge shows a length of 40 feet. A dotted line extends from the interior of the vertex straight down to the center of the base and is labeled h. A right angle is indicated where the dotted height line meets the base. Another dotted line extends from the center of the base where the height line meets to the center of the right base edge. A dotted line extends from the right base edge straight up the center of the right face of the prism and meets the height line at the vertex. This dotted line shows a slant height of 25 feet.
Apply the Pythagorean Theorem to determine the height of the square pyramid, h. Round the answer to the nearest tenth.
(1 point)
Responses
47.2 ft.
47.2 ft.
32.0 ft.
32.0 ft.
31.2 ft.
31.2 ft.
15.0 ft.
To apply the Pythagorean Theorem, we can create a right triangle using the slant height and the height of the pyramid. The slant height of 25 feet represents the hypotenuse of the triangle, and the height, labeled as "h," represents one of the legs of the triangle.
Let h be the height of the pyramid and s be the slant height. We can use the Pythagorean Theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
So, we have:
s² = h² + x²
Using the values given in the image:
25² = h² + 40²
Simplifying:
625 = h² + 1600
Subtracting 1600 from both sides:
h² = -975
Since the height of a pyramid cannot be negative, we can discard the negative solution. Therefore, we have:
h ≈ 31.2 ft.
So, the height of the square pyramid is approximately 31.2 ft.
Use the image to answer the question.
An illustration of a rectangular prism is shown with labeled parts. The left and back base edges are not visible but their positions are indicated by a dashed line. The 6 vertices are labeled A B C D E F and G. The base edge A B is labeled length. Base edge B C is labeled width. The perpendicular C D is labeled height. A dotted line crosses the rectangular base through the center to show diagonal B H. A dashed line extends up from the base indicating the left rear perpendicular E H. A dotted line crosses from the back top left vertex E to the front bottom right vertex B to show diagonal B E. This dotted line is labeled diagonal.
The length of the prism is 32 cm, the width is 24 cm, the height is 44 cm, and the length of the diagonal of the base, segment BH, is 40 cm. Find the length of the diagonal of the rectangular prism, segment BE. Round the answer to the nearest tenth.
(1 point)
Responses
46.6 cm
46.6 cm
59.5 cm
59.5 cm
54.4 cm
54.4 cm
51.2 cm
To find the length of the diagonal of the rectangular prism, segment BE, we can use the Pythagorean Theorem.
Let x be the length of the diagonal, segment BE. We can create a right triangle using the diagonal of the base (segment BH), the height (segment EH), and the length (segment EB) as the three sides of the triangle.
Using the values given in the image:
Segment BH = 40 cm
Segment EH = height = 44 cm
Segment EB = length of the prism = 32 cm
We can use the Pythagorean Theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
So, we have:
x² = BH² + EH²
Substituting the given values:
x² = 40² + 44²
x² = 1600 + 1936
x² = 3536
Taking the square root of both sides to solve for x:
x ≈ 59.5 cm
So, the length of the diagonal of the rectangular prism, segment BE, is approximately 59.5 cm.
Use the image to answer the question.
A drawing shows a circle showing diameter below a sector of a circle showing slant height. A double-headed arrow crossing the center of the circle is labeled as d equals diameter. A dotted double-headed arrow is drawn from the top point of the sector of the circle to a point along the curved edge. The arrow is labeled as lower L equals slant height.
Chem Tech makes paper cones used to filter liquids in lab experiments. To make the paper cones, the company starts with the net of the shape, where the slant height is needed. The company wants the height of the cones to be 6 cm. If they make the diameter 13 cm, what must the slant height be? Use the Pythagorean Theorem and round the answer to the nearest tenth.
(1 point)
Responses
2.5 cm
2.5 cm
11.5 cm
11.5 cm
8.8 cm
8.8 cm
14.3 cm
To find the slant height of the paper cone, we can use the Pythagorean Theorem.
Let h be the height of the cone, d be the diameter, and L be the slant height. We can create a right triangle with the height (h) as one leg, the radius (d/2) as the other leg, and the slant height (L) as the hypotenuse.
Using the values given in the question:
h = 6 cm
d = 13 cm
To find the radius, we divide the diameter by 2:
r = d/2 = 13/2 = 6.5 cm
We can now set up the Pythagorean Theorem equation:
L² = r² + h²
Substituting the given values:
L² = (6.5)² + (6)²
L² = 42.25 + 36
L² = 78.25
Taking the square root of both sides to solve for L:
L ≈ 8.8 cm
So, the slant height of the paper cone should be approximately 8.8 cm.
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
16.8 m
16.8 m
14.8 cm
14.8 cm
24.1 cm
24.1 cm
To find the height of the circus tent, we can use the Pythagorean Theorem.
Let h be the height of the pyramid and s be the slant height. We can create a right triangle with one leg being half the length of the base (10 m), the other leg being the height (h), and the slant height (s) as the hypotenuse.
Using the values given in the question:
Base edge = 20 m
Slant height = 13.5 m
To find half the length of the base (10 m), we divide the base edge by 2.
Now, we can set up the Pythagorean Theorem equation:
s² = (10)² + h²
Substituting the given values:
(13.5)² = (10)² + h²
182.25 = 100 + h²
h² = 182.25 - 100
h² = 82.25
Taking the square root of both sides to solve for h:
h ≈ 9.1 m
So, the height of the circus tent is approximately 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)
Responses
92.9 cm
92.9 cm
60.2 cm
60.2 cm
78.6 cm
78.6 cm
116.3 cm
To find the length of the ramp needed to fit diagonally in the cage, we can use the Pythagorean Theorem.
Let L be the length of the ramp, and let l, w, and h be the length, width, and height of the cage, respectively. We can create a right triangle using the length, width, and height as the three sides of the triangle, with the length of the ramp as the hypotenuse.
Using the values given in the question:
Length of the cage (l) = 70 cm
Width of the cage (w) = 35 cm
Height of the cage (h) = 50 cm
We can now set up the Pythagorean Theorem equation:
L² = l² + w² + h²
Substituting the given values:
L² = 70² + 35² + 50²
L² = 4900 + 1225 + 2500
L² = 8625
Taking the square root of both sides to solve for L:
L ≈ 92.9 cm
So, the length of the ramp needed to fit diagonally in the cage is approximately 92.9 cm.