Question 1

Question

A)A square-based pyramid has height h and base area a. A cube has the same base area and the same height. If the cube has a volume of 195 cubic centimeters, what is the volume of the pyramid?(1 point)
The pyramid has a volume of $$ cubic centimeters. 
Question 2
A)Latoria has a cube and a square-based pyramid. The two figures have the same height and their bases have the same area. Latoria knows that the volume of the cube is 330 cubic inches. What is the volume of the pyramid?(1 point)
The volume of the pyramid is $$ in.3.
Question 3
A)A square-based pyramid has a height of 3 inches and a base area of 9 square inches. Can you find the volume of a cube with the same dimensions as the pyramid? Enter 1 for yes or 2 for no.(1 point)
$$
Question 4
A)The area of a square-based pyramid is 50 square yards. The volume of the pyramid is 250 cubic yards. What is the height of the pyramid?(1 point)
The height of the pyramid is $$ yards.
Question 5
A)
Use the image to answer the question.

A triangular pyramid with a square base has a dotted line segment labeled h equals 8 meters extending from the top vertex to the base. The base of the pyramid is labeled a equals 60 meters squared.

Joao constructs a square-based pyramid. What is the volume of the pyramid?

(1 point)
The volume of the pyramid is $$ cubic meters.

Answer 1

Let's solve each question step by step.

Question 1: To find the volume of a square-based pyramid, we use the formula: \[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \] Given that the base area \(a\) is the same as the cube, and the cube's volume is \(V_{cube} = \text{Base Area} \times h\). We can also relate the cube's volume to base area as: \[ V_{cube} = a \times h = 195 \text{ cubic cm} \]

Since \(V_{cube} = a \times h = 195\), we can express the height \(h\) in terms of base area \(a\): \[ h = \frac{195}{a} \]

Plugging \(h\) back into the pyramid volume formula: \[ V_{pyramid} = \frac{1}{3} \times a \times h = \frac{1}{3} \times a \times \frac{195}{a} = \frac{195}{3} = 65 \text{ cubic centimeters} \]

Thus, the pyramid has a volume of 65 cubic centimeters.

Question 2: Using the same methods as question 1, since Latoria's cube has the same base area and height as the pyramid, we have: \[ V_{cube} = a \times h = 330 \text{ cubic inches} \] And similarly: \[ V_{pyramid} = \frac{1}{3} \times a \times h = \frac{1}{3} \times a \times \frac{330}{a} = \frac{330}{3} = 110 \text{ cubic inches} \]

Thus, the volume of the pyramid is 110 in³.

Question 3: The volume of a pyramid is computed with the formula: \[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \] Given the pyramid has a height of 3 inches and a base area of 9 square inches: \[ V_{pyramid} = \frac{1}{3} \times 9 \times 3 = \frac{27}{3} = 9 \text{ cubic inches} \]

Now, a cube with the same base area and height would have: \[ V_{cube} = \text{Base Area} \times \text{Height} = 9 \times 3 = 27 \text{ cubic inches} \]

Hence the answer here would be 1 (yes), you can find the volume of the cube.

Question 4: Using the volume formula for a pyramid: \[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \] Given that the volume is 250 cubic yards, and the base area is 50 square yards: \[ 250 = \frac{1}{3} \times 50 \times \text{Height} \] To find the height, rearranging gives: \[ \text{Height} = \frac{250 \times 3}{50} = 15 \text{ yards} \]

Thus, the height of the pyramid is 15 yards.

Question 5: For the triangular pyramid with a square base: Using the formula for the volume of a pyramid: \[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \] Given that the height \(h = 8\) meters and base area \(a = 60\) square meters: \[ V = \frac{1}{3} \times 60 \times 8 = \frac{480}{3} = 160 \text{ cubic meters} \]

The volume of the pyramid is 160 cubic meters.

To summarize:

1. The volume of the pyramid is 65 cubic centimeters.
2. The volume of the pyramid is 110 in³.
3. The answer is 1 (yes).
4. The height of the pyramid is 15 yards.
5. The volume of the pyramid is 160 cubic meters.