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.

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

15.0 ft
32.0 ft
47.2 ft
32.2 ft

GPT-4o mini · Answer

To solve for the height $ h $ of the square pyramid using the Pythagorean theorem, we can visualize the problem as forming a right triangle with: - The height $ h $ of the pyramid as one leg. - The distance from the center of the base to the midpoint of the front base edge as the other leg. - The slant height (25 feet) as the hypotenuse. 1. **Identify dimensions**: - The length of the base edge is 40 feet. Therefore, the distance from the center of the base to the midpoint of the front base edge is half of 40 feet: $$ \frac{40}{2} = 20 \text{ feet} $$ 2. **Set up the Pythagorean theorem**: According to the Pythagorean theorem: $$ \text{(slant height)}^2 = h^2 + \text{(distance from center to midpoint)}^2 $$ Plugging in the values: $$ 25^2 = h^2 + 20^2 $$ 3. **Calculate the values**: $$ 625 = h^2 + 400 $$ $$ h^2 = 625 - 400 $$ $$ h^2 = 225 $$ 4. **Solve for $ h $**: $$ h = \sqrt{225} = 15 $$ Therefore, the height $ h $ of the square pyramid is **15.0 feet**. So, the answer is: **15.0 ft**.

ok · Answer

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

59.5 cm
54.4 cm
46.6 cm
51.2 cm

GPT-4o mini · Answer

To find the length of the diagonal $ BE $ of the rectangular prism, we can use the 3-dimensional version of the Pythagorean theorem. The diagonal $ d $ of a rectangular prism can be calculated using the formula: $$ d = \sqrt{l^2 + w^2 + h^2} $$ where $ l $ is the length, $ w $ is the width, and $ h $ is the height of the prism. Given: - Length $ l = 32 $ cm - Width $ w = 24 $ cm - Height $ h = 44 $ cm We will now substitute these values into the formula: 1. Calculate $ l^2 $: $$ l^2 = 32^2 = 1024 $$ 2. Calculate $ w^2 $: $$ w^2 = 24^2 = 576 $$ 3. Calculate $ h^2 $: $$ h^2 = 44^2 = 1936 $$ 4. Add these values: $$ l^2 + w^2 + h^2 = 1024 + 576 + 1936 = 3536 $$ 5. Now, find $ d $: $$ d = \sqrt{3536} $$ 6. Calculate the square root: $$ d \approx 59.5 \text{ cm} \quad (\text{rounded to one decimal place}) $$ Thus, the length of the diagonal of the rectangular prism $ BE $ is **59.5 cm**.

ok · Answer

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 want 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.

8.8 cm
2.5 cm
11.5 cm
14.3 cm

GPT-4o mini · Answer

To find the slant height of the cone using the Pythagorean theorem, we need to interpret the given dimensions in the context of a right triangle formed by the height, the radius of the base, and the slant height.

Given:

Height $ h = 6 $ cm
Diameter $ d = 13 $ cm

1. Find the radius: The radius $ r $ is half of the diameter: \[ r = \frac{d}{2} = \frac{13 \text{ cm}}{2} = 6.5 \text{ cm} \]

2. Set up the Pythagorean theorem: In a right triangle formed by the height (perpendicular), the radius (base), and the slant height (hypotenuse), we have: \[ \text{(slant height)}^2 = \text{(height)}^2 + \text{(radius)}^2 \] Writing it down with the numbers: \[ L^2 = h^2 + r^2 \]

3. Substitute the given values: \[ L^2 = 6^2 + 6.5^2 \]

Calculating the squares:

$ 6^2 = 36 $
$ 6.5^2 = 42.25 $

4. Add the squares: \[ L^2 = 36 + 42.25 = 78.25 \]

5. Take the square root to find the slant height: \[ L = \sqrt{78.25} \]

Calculating the square root: \[ L \approx 8.8 \text{ cm} \quad (\text{rounded to one decimal place}) \]

Thus, the required slant height of the cone is 8.8 cm.

ok · Answer

A circus tent is in the shape of a square pyramid. Each edge of the square base in 20 m and its slant height is 13.5 m. Find the height of the tent. Round the answer to the nearest tenth.

9.1 m
24.1 m
16.8 m
14.8 m

GPT-4o mini · Answer

To find the height of the square pyramid (circus tent), we can use the Pythagorean theorem.

Given:

Each edge of the square base ($ a $) = 20 m
Slant height ($ L $) = 13.5 m

1. Find the radius of the base (half the length of the base edge):

The center of the base to the midpoint of one side is half the length of the edge: \[ r = \frac{a}{2} = \frac{20}{2} = 10 \text{ m} \]

2. Set up the relationship using the Pythagorean theorem:

We have a right triangle formed by:

The height of the pyramid ($ h $)
The radius of the base ($ r $)
The slant height ($ L $) as the hypotenuse

According to the Pythagorean theorem: \[ L^2 = h^2 + r^2 \]

3. Substitute the values into the equation:

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

4. Calculate the squares:

$ 13.5^2 = 182.25 $
$ 10^2 = 100 $

5. Replace the values:

\[ 182.25 = h^2 + 100 \]

6. Solve for $ h^2 $:

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

7. Find $ h $ by taking the square root:

\[ h = \sqrt{82.25} \approx 9.1 \text{ m} \quad (\text{rounded to one decimal place}) \]

Thus, the height of the tent is 9.1 m.

ok · Answer

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 the diagonally in the cage? round the answer to the nearest tenth.

60.2 cm
78.6 cm
116.3 cm
92.9

GPT-4o mini · Answer

To find the length of the ramp that fits diagonally in the rectangular prism-shaped cage, we use the 3-dimensional version of the Pythagorean theorem. The formula for the diagonal $ d $ of a rectangular prism is given by:

\[ d = \sqrt{l^2 + w^2 + h^2} \]

where:

$ l $ is the length of the prism,
$ w $ is the width of the prism,
$ h $ is the height of the prism.

Given dimensions:

Length $ l = 70 $ cm
Width $ w = 35 $ cm
Height $ h = 50 $ cm

Step 1: Calculate $ l^2 $, $ w^2 $, and $ h^2 $

$ l^2 = 70^2 = 4900 $
$ w^2 = 35^2 = 1225 $
$ h^2 = 50^2 = 2500 $

Step 2: Sum the squares

\[ l^2 + w^2 + h^2 = 4900 + 1225 + 2500 = 8625 \]

Step 3: Take the square root to find the diagonal

\[ d = \sqrt{8625} \]

Calculating the square root: \[ d \approx 93.0 \text{ cm} \quad (\text{rounded to one decimal place}) \]

Conclusion

The length of the ramp needed to fit diagonally in the cage is approximately 93.0 cm.

Given the provided options, it would round to 92.9 cm as the closest answer.

Given:

1. Find the radius of the base (half the length of the base edge):

2. Set up the relationship using the Pythagorean theorem:

3. Substitute the values into the equation:

4. Calculate the squares:

5. Replace the values:

6. Solve for \( h^2 \):

7. Find \( h \) by taking the square root:

Given dimensions:

Step 1: Calculate \( l^2 \), \( w^2 \), and \( h^2 \)

Step 2: Sum the squares

Step 3: Take the square root to find the diagonal

Conclusion