Question

1. which formula applies to a triangular base pyramid with a base of 6 cm and height of the base triangle of 3 cm?

a. ( a = 6 \times 3 )
b. ( a = 6/3 )
c. ( a = \frac{1}{2} \times 6 \times 3 )
d. ( a = 6 + 3 )

1. if the height of the pyramid is tripled while keeping the base area constant, how does the volume change?

a. it quadruples
b. it triples
c. it doubles
d. it stays the same

1. if the volume of a triangular base pyramid is ( 36 , \text{cm}^3 ), with a base area of ( 12 , \text{cm}^2 ) and height ( 9 , \text{cm} ), what formula was used to calculate it?

a. ( v = \frac{1}{2} bh )
b. ( v = \frac{1}{3} bh )
c. ( v = \frac{1}{4} bh )
d. ( v = bh )

1. which of the following is necessary to calculate the volume of a pyramid?

a. the perimeter of the base and the slant height
b. the surface area and the perimeter of the base
c. the area of the base and the height
d. the side length of the base and the slant height

1. a pyramid has a square base with a side length of 6 units and a height of 9 units. what is the first step in finding its volume?

a. calculate the area of the square base
b. divide the side length by three
c. add the side length to the height
d. multiply the side length by the height

1. what happens to the volume of a pyramid if its height is doubled while the base area remains the same?

a. the volume triples
b. the volume remains the same
c. the volume doubles
d. the volume is halved

Explanation:

Step1: Identify base area formula

The question asks for the formula to find the area of the triangular base (base length 6 cm, height 3 cm). The area of a triangle is $\frac{1}{2} \times \text{base length} \times \text{height}$.

Step2: Match to given options

Substitute the values: $\frac{1}{2} \times 6 \times 3$. This matches option c.

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Step1: Recall pyramid volume formula

Volume of a pyramid: $V = \frac{1}{3}Bh$, where $B$ = base area, $h$ = height.

Step2: Substitute new height

New height $h' = 3h$. New volume $V' = \frac{1}{3}B(3h) = 3 \times \frac{1}{3}Bh = 3V$. Volume triples, matching option b.

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Step1: Verify volume formula

Given $V=36\ \text{cm}^3$, $B=12\ \text{cm}^2$, $h=9\ \text{cm}$. Test $V=\frac{1}{3}Bh$: $\frac{1}{3} \times 12 \times 9 = 36$, which matches. This is option a.

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Step1: Reference volume formula

Volume of a pyramid is $V = \frac{1}{3}Bh$, which requires base area $B$ and height $h$. This matches option c.

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Step1: Analyze volume calculation steps

For a square-based pyramid, first find base area $B = \text{side length}^2$. This is option a.

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Step1: Use volume formula for change

Volume $V = \frac{1}{3}Bh$. New height $h' = 2h$. New volume $V' = \frac{1}{3}B(2h) = 2 \times \frac{1}{3}Bh = 2V$. Volume doubles, matching option c.

Answer:

1. c. $A=\frac{1}{2} \times 6 \times 3$
2. b. It triples
3. a. $V=\frac{1}{3}Bh$
4. c. The area of the base and the height
5. a. Calculate the area of the square base
6. c. The volume doubles