Let's go through each problem one by one:
### 1. Finding the Height of the Square Pyramid
The problem involves a square pyramid with a base edge of 40 feet and a slant height of 25 feet. The height \( h \) of the pyramid can be found using the Pythagorean Theorem in a right triangle formed by the height, half of the base edge, and the slant height.
The half of the base edge is:
\[ \frac{40}{2} = 20 \text{ ft} \]
Using the Pythagorean theorem:
\[ h^2 + 20^2 = 25^2 \]
\[ h^2 + 400 = 625 \]
\[ h^2 = 225 \]
\[ h = \sqrt{225} \]
\[ h \approx 15.0 \text{ ft} \]
So, the correct answer is:
\[ \boxed{15.0 \text{ ft}} \]
### 2. Finding the Length of the Diagonal of the Rectangular Prism
The given dimensions are: length \( l = 32 \) cm, width \( w = 24 \) cm, height \( h = 44 \) cm. The diagonal \( \text{BE} \) in 3D can be found using the Pythagorean theorem in three dimensions:
First, find the diagonal of the base \( \text{BH} \) (which is given as 40 cm, already done for us):
\[ \text{BH} = \sqrt{32^2 + 24^2} = 40 \text{ cm} \]
Now use the value of \( \text{BH} \) to find the space diagonal \( \text{BE} \):
\[ \text{BE}^2 = \text{BH}^2 + h^2 \]
\[ \text{BE}^2 = 40^2 + 44^2 \]
\[ \text{BE}^2 = 1600 + 1936 \]
\[ \text{BE}^2 = 3536 \]
\[ \text{BE} = \sqrt{3536} \]
\[ \text{BE} \approx 59.5 \text{ cm} \]
So, the correct answer is:
\[ \boxed{59.5 \text{ cm}} \]
### 3. Finding the Slant Height of the Paper Cone
The height \( h \) is given as 6 cm and the diameter is 13 cm (hence radius \( r = 6.5 \) cm). Using the Pythagorean theorem to find the slant height \( L \):
\[ L^2 = h^2 + r^2 \]
\[ L^2 = 6^2 + 6.5^2 \]
\[ L^2 = 36 + 42.25 \]
\[ L^2 = 78.25 \]
\[ L = \sqrt{78.25} \]
\[ L \approx 8.8 \text{ cm} \]
So, the correct answer is:
\[ \boxed{8.8 \text{ cm}} \]
### 4. Finding the Height of the Circus Tent
The base edge is 20 m and the slant height is 13.5 m. The height \( h \) of the tent can be calculated similarly to problem 1:
Half of the base edge:
\[ \frac{20}{2} = 10 \text{ m} \]
Using the Pythagorean theorem:
\[ h^2 + 10^2 = 13.5^2 \]
\[ h^2 + 100 = 182.25 \]
\[ h^2 = 82.25 \]
\[ h = \sqrt{82.25} \]
\[ h \approx 9.1 \text{ m} \]
So, the correct answer is:
\[ \boxed{9.1 \text{ m}} \]
### 5. Finding the Length of the Ramp in the Gerbil Cage
The length \( l = 70 \) cm, width \( w = 35 \) cm, and height \( h = 50 \) cm. The length of the ramp (diagonal of the prism) can be found using the Pythagorean theorem in three dimensions:
First, find the diagonal of the base:
\[ \text{Diagonal of base} = \sqrt{70^2 + 35^2} \]
\[ = \sqrt{4900 + 1225} \]
\[ = \sqrt{6125} \]
\[ = 78.2 \text{ cm} \]
Then use this to find the space diagonal:
\[ D^2 = (\text{diagonal of base})^2 + h^2 \]
\[ D^2 = 78.2^2 + 50^2 \]
\[ D^2 = 6125 + 2500 \]
\[ D^2 = 8625 \]
\[ D = \sqrt{8625} \]
\[ D \approx 92.9 \text{ cm} \]
So, the correct answer is:
\[ \boxed{92.9 \text{ cm}} \]
The Pythagorean Theorem in Three Dimensions Quick Check
1. 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
32.0 ft.
32.0 ft.
47.2 ft.
47.2 ft.
15.0 ft.
15.0 ft.
31.2 ft.
2. 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
51.2 cm
51.2 cm
59.5 cm
59.5 cm
54.4 cm
3. 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
14.3 cm
14.3 cm
8.8 cm
8.8 cm
2.5 cm
2.5 cm
11.5 cm
4. 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
16.8 m
16.8 m
24.1 cm
5.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
116.3 cm
116.3 cm
60.2 cm
3 answers
u sure?
Yes, I'm confident in the calculations I provided. Let's review each step again to ensure the accuracy.
### 1. Height of the Square Pyramid
Given:
- Base edge: 40 ft (therefore half of the base edge is 20 ft)
- Slant height: 25 ft
Using the Pythagorean theorem for the right triangle formed by the height \( h \), half of the base, and the slant height:
\[ h^2 + 20^2 = 25^2 \]
\[ h^2 + 400 = 625 \]
\[ h^2 = 225 \]
\[ h = \sqrt{225} \]
\[ h = 15 \text{ ft} \]
So, the correct answer is:
\[ \boxed{15.0 \text{ ft}} \]
### 2. Diagonal of the Rectangular Prism
Given:
- Length \( l \) = 32 cm
- Width \( w \) = 24 cm
- Height \( h \) = 44 cm
- Diagonal of the base \( \text{BH} \) = 40 cm (provided, and correctly calculated from \( l \) and \( w \))
Finding the diagonal \( \text{BE} \) in 3D space:
\[ \text{BE}^2 = \text{BH}^2 + h^2 \]
\[ \text{BE}^2 = 40^2 + 44^2 \]
\[ \text{BE}^2 = 1600 + 1936 \]
\[ \text{BE}^2 = 3536 \]
\[ \text{BE} = \sqrt{3536} \]
\[ \text{BE} \approx 59.5 \text{ cm} \]
So, the correct answer is:
\[ \boxed{59.5 \text{ cm}} \]
### 3. Slant Height of the Paper Cone
Given:
- Height \( h \) = 6 cm
- Diameter = 13 cm (therefore radius \( r = 6.5 \) cm)
Using the Pythagorean theorem:
\[ L^2 = h^2 + r^2 \]
\[ L^2 = 6^2 + 6.5^2 \]
\[ L^2 = 36 + 42.25 \]
\[ L^2 = 78.25 \]
\[ L = \sqrt{78.25} \]
\[ L \approx 8.8 \text{ cm} \]
So, the correct answer is:
\[ \boxed{8.8 \text{ cm}} \]
### 4. Height of the Circus Tent
Given:
- Edge of the square base = 20 m (therefore half of the base edge is 10 m)
- Slant height: 13.5 m
Using the Pythagorean theorem:
\[ h^2 + 10^2 = 13.5^2 \]
\[ h^2 + 100 = 182.25 \]
\[ h^2 = 82.25 \]
\[ h = \sqrt{82.25} \]
\[ h \approx 9.1 \text{ m} \]
So, the correct answer is:
\[ \boxed{9.1 \text{ m}} \]
### 5. Length of the Ramp in the Gerbil Cage
Given:
- Length \( l = 70 \) cm
- Width \( w = 35 \) cm
- Height \( h = 50 \) cm
First, find the diagonal of the base:
\[ \text{Diagonal of base} = \sqrt{70^2 + 35^2} \]
\[ = \sqrt{4900 + 1225} \]
\[ = \sqrt{6125} \]
\[ \approx 78.2 \text{ cm} \]
Then use this to find the diagonal of the rectangular prism:
\[ \text{Diagonal}^2 = (\text{diagonal of base})^2 + h^2 \]
\[ \text{Diagonal}^2 = 78.2^2 + 50^2 \]
\[ \text{Diagonal}^2 = 6125 + 2500 \]
\[ \text{Diagonal}^2 = 8625 \]
\[ \text{Diagonal} = \sqrt{8625} \]
\[ \text{Diagonal} \approx 92.9 \text{ cm} \]
So, the correct answer is:
\[ \boxed{92.9 \text{ cm}} \]
These calculations are precise, and the answers are correct as provided. If you have any uncertainties or questions about any specific part of the calculations, please let me know!
### 1. Height of the Square Pyramid
Given:
- Base edge: 40 ft (therefore half of the base edge is 20 ft)
- Slant height: 25 ft
Using the Pythagorean theorem for the right triangle formed by the height \( h \), half of the base, and the slant height:
\[ h^2 + 20^2 = 25^2 \]
\[ h^2 + 400 = 625 \]
\[ h^2 = 225 \]
\[ h = \sqrt{225} \]
\[ h = 15 \text{ ft} \]
So, the correct answer is:
\[ \boxed{15.0 \text{ ft}} \]
### 2. Diagonal of the Rectangular Prism
Given:
- Length \( l \) = 32 cm
- Width \( w \) = 24 cm
- Height \( h \) = 44 cm
- Diagonal of the base \( \text{BH} \) = 40 cm (provided, and correctly calculated from \( l \) and \( w \))
Finding the diagonal \( \text{BE} \) in 3D space:
\[ \text{BE}^2 = \text{BH}^2 + h^2 \]
\[ \text{BE}^2 = 40^2 + 44^2 \]
\[ \text{BE}^2 = 1600 + 1936 \]
\[ \text{BE}^2 = 3536 \]
\[ \text{BE} = \sqrt{3536} \]
\[ \text{BE} \approx 59.5 \text{ cm} \]
So, the correct answer is:
\[ \boxed{59.5 \text{ cm}} \]
### 3. Slant Height of the Paper Cone
Given:
- Height \( h \) = 6 cm
- Diameter = 13 cm (therefore radius \( r = 6.5 \) cm)
Using the Pythagorean theorem:
\[ L^2 = h^2 + r^2 \]
\[ L^2 = 6^2 + 6.5^2 \]
\[ L^2 = 36 + 42.25 \]
\[ L^2 = 78.25 \]
\[ L = \sqrt{78.25} \]
\[ L \approx 8.8 \text{ cm} \]
So, the correct answer is:
\[ \boxed{8.8 \text{ cm}} \]
### 4. Height of the Circus Tent
Given:
- Edge of the square base = 20 m (therefore half of the base edge is 10 m)
- Slant height: 13.5 m
Using the Pythagorean theorem:
\[ h^2 + 10^2 = 13.5^2 \]
\[ h^2 + 100 = 182.25 \]
\[ h^2 = 82.25 \]
\[ h = \sqrt{82.25} \]
\[ h \approx 9.1 \text{ m} \]
So, the correct answer is:
\[ \boxed{9.1 \text{ m}} \]
### 5. Length of the Ramp in the Gerbil Cage
Given:
- Length \( l = 70 \) cm
- Width \( w = 35 \) cm
- Height \( h = 50 \) cm
First, find the diagonal of the base:
\[ \text{Diagonal of base} = \sqrt{70^2 + 35^2} \]
\[ = \sqrt{4900 + 1225} \]
\[ = \sqrt{6125} \]
\[ \approx 78.2 \text{ cm} \]
Then use this to find the diagonal of the rectangular prism:
\[ \text{Diagonal}^2 = (\text{diagonal of base})^2 + h^2 \]
\[ \text{Diagonal}^2 = 78.2^2 + 50^2 \]
\[ \text{Diagonal}^2 = 6125 + 2500 \]
\[ \text{Diagonal}^2 = 8625 \]
\[ \text{Diagonal} = \sqrt{8625} \]
\[ \text{Diagonal} \approx 92.9 \text{ cm} \]
So, the correct answer is:
\[ \boxed{92.9 \text{ cm}} \]
These calculations are precise, and the answers are correct as provided. If you have any uncertainties or questions about any specific part of the calculations, please let me know!
5 answers