What is the volume of a rectangular pyramid with a length of 32 cm, a width of 25 cm, and an altitude of 41 cm

To calculate the volume of a rectangular pyramid, you can use the following formula:

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

The base area of a rectangular pyramid can be calculated by multiplying its length and width:

\[ \text{Base Area} = \text{Length} \times \text{Width} = 32 , \text{cm} \times 25 , \text{cm} = 800 , \text{cm}^2 \]

Next, substitute the base area and the height (altitude) into the volume formula:

\[ V = \frac{1}{3} \times 800 , \text{cm}^2 \times 41 , \text{cm} \]

Calculating this gives:

\[ V = \frac{1}{3} \times 800 \times 41 = \frac{32800}{3} , \text{cm}^3 \approx 10933.33 , \text{cm}^3 \]

Therefore, the volume of the rectangular pyramid is approximately \( 10933.33 , \text{cm}^3 \).

A resort is building a scuba diving lagoon in the shape of an upside down pyramid. The top of the lagoon is a rectangle that is 50 feet by 20 feet, and the deepest point of the lagoon is in the middle at 25 feet deep. How much water does the lagoon hold

To calculate the volume of the scuba diving lagoon shaped like an upside-down rectangular pyramid, we can use the formula for the volume of a pyramid:

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

In this case, the base area is the area of the rectangle at the top of the lagoon, and the height is the depth of the lagoon.

1. Calculate the base area:

   • The dimensions of the rectangle are 50 feet and 20 feet.
   • Base Area = Length × Width = \(50 , \text{ft} \times 20 , \text{ft} = 1000 , \text{ft}^2\)

2. The height of the pyramid:

   • The height of the lagoon (which is the depth) is 25 feet.

3. Substitute into the volume formula: \[ V = \frac{1}{3} \times 1000 , \text{ft}^2 \times 25 , \text{ft} \]

4. Calculate the volume: \[ V = \frac{1}{3} \times 1000 \times 25 = \frac{25000}{3} \approx 8333.33 , \text{ft}^3 \]

Thus, the lagoon can hold approximately \(8333.33 , \text{ft}^3\) of water.

A rectangular pyramid has a length of 7 cm and a width of 9 cm. Its volume is 231 cubic centimeters. Use the formula for the volume of a pyramid to calculate the height of this pyramid

To find the height of the rectangular pyramid, we can use the formula for the volume of a pyramid:

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

Given that the volume \(V\) is 231 cubic centimeters, the length \(L\) is 7 cm, and the width \(W\) is 9 cm, we can first calculate the base area:

\[ \text{Base Area} = \text{Length} \times \text{Width} = 7 , \text{cm} \times 9 , \text{cm} = 63 , \text{cm}^2 \]

Now we substitute the base area and the volume into the volume formula to solve for the height \(H\):

\[ 231 = \frac{1}{3} \times 63 \times H \]

To isolate \(H\), first multiply both sides by 3:

\[ 3 \times 231 = 63 \times H \]

\[ 693 = 63 \times H \]

Now, divide both sides by 63 to find \(H\):

\[ H = \frac{693}{63} = 11 , \text{cm} \]

Therefore, the height of the pyramid is \(11 , \text{cm}\).