Question

what is the formula used to calculate the volume of a pyramid?
a. $v = \frac{1}{2} \times b \times h$
b. $v = b + h$
c. $v = b \times h$
d. $v = \frac{1}{3} \times b \times h$

how is the volume of a pyramid affected if the height is doubled while the base area remains the same?
a. volume is halved
b. volume is doubled
c. volume is quadrupled
d. volume remains the same

which of the following elements is necessary to calculate the volume of a pyramid?
a. slant height
b. base area
c. diagonal length
d. lateral area

the volume of a rectangular pyramid is 216 cubic meters, and the height is 9 meters. the length of the base is 6 meters. what is the width of the base?
a. 12 m
b. 6 m
c. 8 m
d. 10 m

a pyramid has a base area of 64 square meters and a volume of 128 cubic meters. what is the height of the pyramid?
a. 4 m
b. 8 m
c. 5 m
d. 6 m

in a right pyramid, what angle does the height make with the base?
a. 60 degrees
b. 45 degrees
c. 90 degrees
d. 75 degrees

Explanation:

Step1: Identify pyramid volume formula

The standard formula for the volume of a pyramid is $V = \frac{1}{3} \times B \times h$, where $B$ is base area and $h$ is height.

Step2: Analyze height doubling effect

Original volume: $V_1 = \frac{1}{3}Bh$. New height $h_2=2h$, so new volume: $V_2 = \frac{1}{3}B(2h) = 2\times\frac{1}{3}Bh = 2V_1$.

Step3: Identify required volume element

From $V = \frac{1}{3}Bh$, only base area $B$ and height $h$ are needed.

Step4: Solve for base width

Given $V=216\ \text{m}^3$, $h=9\ \text{m}$, base length $l=6\ \text{m}$. First find base area:
$B = \frac{3V}{h} = \frac{3\times216}{9} = 72\ \text{m}^2$.
Since $B = l \times w$, $w = \frac{B}{l} = \frac{72}{6} = 12\ \text{m}$.

Step5: Solve for pyramid height

Given $V=128\ \text{m}^3$, $B=64\ \text{m}^2$. Rearrange volume formula:
$h = \frac{3V}{B} = \frac{3\times128}{64} = 6\ \text{m}$.

Step6: Determine right pyramid height angle

In a right pyramid, height is perpendicular to the base, so the angle is 90 degrees.

Answer:

1. d. $V = \frac{1}{3} \times B \times h$
2. b. Volume is doubled
3. b. Base area
4. a. 12 m
5. d. 6 m
6. c. 90 degrees