A right rectangular pyramid has an altitude of 35 m, a length of 26 m, and a width of 31 m

Answer 1

To find various properties of the right rectangular pyramid with given dimensions (altitude, length, and width), we can calculate the following:

1. Volume of the pyramid: The formula for the volume \( V \) of a pyramid is: \[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \] The base area \( A \) of a rectangular base is calculated as: \[ A = \text{Length} \times \text{Width} = 26 , \text{m} \times 31 , \text{m} = 806 , \text{m}^2 \] Then, substituting the values into the volume formula: \[ V = \frac{1}{3} \times 806 , \text{m}^2 \times 35 , \text{m} = \frac{1}{3} \times 28210 , \text{m}^3 \approx 9403.33 , \text{m}^3 \]

2. Surface Area of the pyramid: To find the surface area, we calculate the area of the base and the areas of the four triangular faces.

   • Base Area = \( 806 , \text{m}^2 \) (as previously calculated).
   • Area of the triangular faces: Each triangular face has a base equal to the lengths or widths of the rectangle and a slant height calculated using the Pythagorean theorem.

   The dimensions of the triangular faces are as follows:

   Front/Back triangular faces (each has a base 31 m):

   • The height \( h \) of the triangular face can be calculated using the altitude and half of the base as follows: \( h = \sqrt{ (35)^2 + \left(\frac{26}{2}\right)^2 } = \sqrt{ 35^2 + 13^2 } = \sqrt{ 1225 + 169 } = \sqrt{ 1394} \approx 37.34 , \text{m} \) Each triangular face area = \( \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 31 , \text{m} \times 37.34 , \text{m} \approx 578.77 , \text{m}^2 \) Since there are 2 such triangular faces, their combined area is: \[ 2 \times 578.77 \approx 1157.54 , \text{m}^2 \]

   Left/Right triangular faces (each has a base 26 m): For the triangular faces on the shorter side: Each triangular face area = \( \frac{1}{2} \times 26 , \text{m} \times 37.34 , \text{m} \approx 485.84 , \text{m}^2 \) Combined area for two such faces: \[ 2 \times 485.84 \approx 971.68 , \text{m}^2 \]

   Total Surface Area: Combining these, the total surface area \( SA \) is: \[ SA = \text{Base Area} + \text{Front/Back faces Area} + \text{Left/Right faces Area} \] \[ SA = 806 + 1157.54 + 971.68 \approx 2935.22 , \text{m}^2 \]

In summary:

• Volume of the pyramid is approximately \( 9403.33 , \text{m}^3 \).
• Surface area of the pyramid is approximately \( 2935.22 , \text{m}^2 \).

Answer 2

Use the volume formula to determine the volume of the pyramid at the Louvre Museum if it has a base that is 112 feet long on each side and has an altitude of 71 feet

Answer 3

To determine the volume of the pyramid at the Louvre Museum using the volume formula for a pyramid, you can use the following formula:

\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]

1. Calculate the Base Area: For a square base, the area \( A \) is given by the side length squared. Since the base is 112 feet long on each side:

\[ A = \text{Base Side}^2 = 112 , \text{ft} \times 112 , \text{ft} = 12544 , \text{ft}^2 \]

2. Use the Altitude (Height): We have the height (altitude) of the pyramid as 71 feet.

3. Calculate the Volume: Now we can substitute the base area and height into the volume formula:

\[ V = \frac{1}{3} \times 12544 , \text{ft}^2 \times 71 , \text{ft} \]

Calculating the volume step-by-step:

\[ V = \frac{1}{3} \times 12544 \times 71 \]

First, calculate \( 12544 \times 71 \):

\[ 12544 \times 71 = 891904 , \text{ft}^3 \]

Now divide by 3:

\[ V = \frac{891904}{3} \approx 297301.33 , \text{ft}^3 \]

Thus, the volume of the pyramid at the Louvre Museum is approximately:

\[ \boxed{297301.33 , \text{ft}^3} \]

Answer 4

J Juice brand wants to create a new bottle for their products. Their marketing department wants to design a pyramid–shaped bottle with a base that is 8 cm by 15 cm, and has an altitude of 22 cm. How much juice will this new bottle hold

Answer 5

To determine how much juice the pyramid-shaped bottle will hold, we can calculate the volume of the pyramid using the formula:

\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]

Step 1: Calculate the Base Area

Since the base of the pyramid is rectangular, we can calculate its area as follows:

\[ \text{Base Area} = \text{Length} \times \text{Width} = 8 , \text{cm} \times 15 , \text{cm} = 120 , \text{cm}^2 \]

Step 2: Use the Altitude (Height)

The altitude (height) of the pyramid is given as 22 cm.

Step 3: Calculate the Volume

Now we can substitute the base area and height into the volume formula:

\[ V = \frac{1}{3} \times 120 , \text{cm}^2 \times 22 , \text{cm} \]

Calculating it step-by-step:

1. First, compute \( 120 \times 22 \): \[ 120 \times 22 = 2640 , \text{cm}^3 \]

2. Now, divide by 3: \[ V = \frac{2640}{3} \approx 880 , \text{cm}^3 \]

Therefore, the pyramid-shaped bottle will hold approximately:

\[ \boxed{880 , \text{cm}^3} \]

Answer 6

A construction company is digging a drainage trench. It makes the shape of an upside-down rectangular pyramid. The top opening will be 30 ft. wide by 250 ft. long, and the deepest part in the middle will be 15 ft. What is the capacity for this trench

Answer 7

To determine the capacity of the trench shaped like an upside-down rectangular pyramid, we can calculate the volume using the formula for the volume of a pyramid:

\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]

Step 1: Calculate the Base Area

The base area of the trench will be the area of the opening at the top, which is a rectangle:

\[ \text{Base Area} = \text{Width} \times \text{Length} = 30 , \text{ft} \times 250 , \text{ft} = 7500 , \text{ft}^2 \]

Step 2: Use the Height

The height of the pyramid in this case is the depth of the trench, which is given as 15 ft.

Step 3: Calculate the Volume

Now we can substitute the base area and height into the volume formula:

\[ V = \frac{1}{3} \times 7500 , \text{ft}^2 \times 15 , \text{ft} \]

Calculating it step-by-step:

1. First, compute \( 7500 \times 15 \): \[ 7500 \times 15 = 112500 , \text{ft}^3 \]

2. Now, divide by 3: \[ V = \frac{112500}{3} = 37500 , \text{ft}^3 \]

Therefore, the capacity of the drainage trench is:

\[ \boxed{37500 , \text{ft}^3} \]