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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?(1 point)

Responses

10,933.33 cm3
10,933.33 cm cubed

32,800 cm3
32,800 cm cubed

98,400 cm3
98,400 cm cubed

32.67 cm3

7 answers

The volume \( V \) of a rectangular pyramid can be calculated using the formula:

\[ V = \frac{1}{3} \times \text{base area} \times \text{height} \]

For a rectangular base, the base area is given by:

\[ \text{base area} = \text{length} \times \text{width} \]

Given:

  • Length = 32 cm
  • Width = 25 cm
  • Height (altitude) = 41 cm

First, calculate the base area:

\[ \text{base area} = 32 , \text{cm} \times 25 , \text{cm} = 800 , \text{cm}^2 \]

Now, use the base area to find the volume:

\[ V = \frac{1}{3} \times 800 , \text{cm}^2 \times 41 , \text{cm} \] \[ V = \frac{1}{3} \times 32,800 , \text{cm}^3 \] \[ V = 10,933.33 , \text{cm}^3 \]

Thus, the volume of the rectangular pyramid is \( 10,933.33 , \text{cm}^3 \).

So, the correct response is:

10,933.33 cm³

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?(1 point)
Responses

95 ft.3
95 feet cubed

25,000 ft.3
25,000 feet cubed

75,000 ft.3
75,000 feet cubed

8,333.33 ft.3

To find the volume of the scuba diving lagoon shaped like an upside-down rectangular pyramid, we can use the formula for the volume \( V \) of a pyramid:

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

In this case:

  • The base is a rectangle of size 50 feet by 20 feet.
  • The depth of the lagoon, which serves as the height of the pyramid, is 25 feet.

First, calculate the base area:

\[ \text{Base Area} = \text{Length} \times \text{Width} = 50 , \text{ft} \times 20 , \text{ft} = 1,000 , \text{ft}^2 \]

Now, use the base area to calculate the volume:

\[ V = \frac{1}{3} \times 1,000 , \text{ft}^2 \times 25 , \text{ft} \]

Calculating this gives:

\[ V = \frac{1}{3} \times 25,000 , \text{ft}^3 \] \[ V = 8,333.33 , \text{ft}^3 \]

Thus, the lagoon holds 8,333.33 ft³ of water.

The correct response is:

8,333.33 ft³

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. Show all your work.(2 points)

To calculate the height of the rectangular pyramid using its volume, we can use the formula for the volume \( V \) of a pyramid:

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

Given:

  • Volume \( V = 231 , \text{cm}^3 \)
  • Length of the base \( = 7 , \text{cm} \)
  • Width of the base \( = 9 , \text{cm} \)

Step 1: Calculate the base area.

The base area \( A \) of a rectangle is given by:

\[ A = \text{Length} \times \text{Width} \]

Substituting in the values:

\[ A = 7 , \text{cm} \times 9 , \text{cm} = 63 , \text{cm}^2 \]

Step 2: Substitute the values into the volume formula and solve for height.

Now we plug the base area and volume into the volume formula:

\[ 231 = \frac{1}{3} \times 63 , \text{cm}^2 \times h \]

Step 3: Multiply both sides by 3 to eliminate the fraction.

\[ 3 \times 231 = 63 , \text{cm}^2 \times h \] \[ 693 = 63 , \text{cm}^2 \times h \]

Step 4: Solve for h (height).

Now, divide both sides by \( 63 , \text{cm}^2 \):

\[ h = \frac{693}{63} \]

Calculating this gives:

\[ h = 11 , \text{cm} \]

Final Result:

The height of the rectangular pyramid is \( \boxed{11 , \text{cm}} \).

put everything in a sentence

To find the height of a rectangular pyramid with a length of 7 cm, a width of 9 cm, and a volume of 231 cubic centimeters, we first calculated the base area by multiplying the length and width, obtaining a base area of 63 cm²; then, using the volume formula \( V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \), we substituted the values into the equation \( 231 = \frac{1}{3} \times 63 \times h \), and by multiplying both sides by 3 to eliminate the fraction, we arrived at \( 693 = 63h \); finally, dividing both sides by 63 yielded a height of 11 cm.