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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

31.2 ft.
31.2 ft.

15.0 ft.
15.0 ft.

47.2 ft.
47.2 ft.

32.0 ft.
32.0 ft.
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8 answers

To solve for the height \( h \) of the square pyramid using the Pythagorean theorem, we can identify the right triangle formed by the height, the slant height, and half of the length of the base.

Given:

  • Slant height \( l = 25 \) feet
  • Length of the base edge \( = 40 \) feet
  • Half of the base edge \( = \frac{40}{2} = 20 \) feet

We can set up the Pythagorean theorem as follows:

\[ l^2 = h^2 + \left(\frac{\text{base edge length}}{2}\right)^2 \]

Substituting in the known values:

\[ 25^2 = h^2 + 20^2 \]

Calculating the squares:

\[ 625 = h^2 + 400 \]

Now, subtract 400 from both sides:

\[ 625 - 400 = h^2 \] \[ 225 = h^2 \]

Taking the square root of both sides:

\[ h = \sqrt{225} = 15 \]

Thus, the height \( h \) of the square pyramid is \( 15.0 \) feet.

The final answer is:

15.0 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

54.4 cm
54.4 cm

46.6 cm
46.6 cm

59.5 cm
59.5 cm

51.2 cm
51.2 cm
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To find the length of the diagonal of the rectangular prism, we can use the three-dimensional version of the Pythagorean theorem. The formula for the diagonal \( d \) of a rectangular prism is given by:

\[ d = \sqrt{\text{length}^2 + \text{width}^2 + \text{height}^2} \]

Given:

  • Length \( = 32 \) cm
  • Width \( = 24 \) cm
  • Height \( = 44 \) cm

Substituting the values into the formula:

\[ d = \sqrt{32^2 + 24^2 + 44^2} \]

Calculating each term:

\[ 32^2 = 1024 \] \[ 24^2 = 576 \] \[ 44^2 = 1936 \]

Now, add these values together:

\[ d = \sqrt{1024 + 576 + 1936} \] \[ d = \sqrt{3536} \]

Now, we find the square root:

\[ d \approx 59.5 \text{ cm} \]

Thus, the length of the diagonal \( BE \) of the rectangular prism is approximately 59.5 cm.

Final answer:

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 

 8.8 cm
 8.8 cm

11.5 cm
11.5 cm

14.3 cm
14.3 cm
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To find the slant height of the paper cone using the Pythagorean theorem, we can set up a right triangle where:

  • The height \( h \) of the cone is one leg.
  • The radius \( r \) of the base of the cone is the other leg.
  • The slant height \( l \) is the hypotenuse.

Given:

  • Height \( h = 6 \) cm
  • Diameter \( d = 13 \) cm, so the radius \( r = \frac{d}{2} = \frac{13}{2} = 6.5 \) cm

According to the Pythagorean theorem:

\[ l^2 = h^2 + r^2 \]

Substituting the values:

\[ l^2 = 6^2 + (6.5)^2 \]

Calculating each term:

\[ 6^2 = 36 \] \[ (6.5)^2 = 42.25 \]

Now, add these values together:

\[ l^2 = 36 + 42.25 = 78.25 \]

Next, take the square root to find \( l \):

\[ l = \sqrt{78.25} \approx 8.8 \text{ cm} \]

Thus, the slant height of the cone must be approximately 8.8 cm.

Final answer:

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

16.8 m 
16.8 m 

9.1 m 
9.1 m 

24.1 cm
24.1 cm

14.8 cm

To find the height of the square pyramid (the circus tent), we can use the Pythagorean theorem. In this case, we have a right triangle formed by the height of the pyramid \( h \), half of the base edge length \( r \), and the slant height \( l \) as the hypotenuse.

Given:

  • Edge length of the square base = 20 m, so half of this length (radius to the center of the base) is: \[ r = \frac{20}{2} = 10 \text{ m} \]
  • Slant height \( l = 13.5 \text{ m} \)

According to the Pythagorean theorem:

\[ l^2 = h^2 + r^2 \]

Substituting in the known values:

\[ (13.5)^2 = h^2 + (10)^2 \]

Calculating the squares:

\[ 182.25 = h^2 + 100 \]

Now, subtract 100 from both sides:

\[ 182.25 - 100 = h^2 \] \[ 82.25 = h^2 \]

Now, take the square root to find \( h \):

\[ h = \sqrt{82.25} \approx 9.1 \text{ m} \]

Thus, the height of the tent is approximately 9.1 m.

Final answer:

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

60.2 cm
60.2 cm

78.6 cm
78.6 cm

92.9 cm
92.9 cm

116.3 cm