Question

1. the area of the circular base is calculated using which formula?

a. $a = \pi r^2 h$
b. $a = \pi r^2$
c. $a = \pi d h$
d. $a = 2\pi r$

1. if the slant height of a cone is 13 cm and the radius is 5 cm, what is the height of the cone?

a. 14 cm
b. 10 cm
c. 15 cm
d. 12 cm

1. a triangular base pyramid has a base of 6 cm, a height of the base triangle of 4 cm, and a pyramid height of 9 cm. what is its volume?

a. $72\\,\text{cm}^3$
b. $24\\,\text{cm}^3$
c. $18\\,\text{cm}^3$
d. $36\\,\text{cm}^3$

1. if the volume of a cylinder is $54\pi\\,\text{cm}^3$ and its height is 6 cm, what is its radius?

a. 4 cm
b. 6 cm
c. 2 cm
d. 3 cm

1. if the radius of a cone is 5 cm and its height is 10 cm, what is the volume of the cone?

a. $50\pi\\,\text{cm}^3$
b. $100\pi\\,\text{cm}^3$
c. $\frac{250}{3}\pi\\,\text{cm}^3$
d. $\frac{125}{3}\pi\\,\text{cm}^3$

1. if the radius of a cone is 8 cm and its volume is $192\pi\\,\text{cm}^3$, what is its height?

a. 6 cm
b. 4 cm
c. 2 cm
d. 9 cm

1. if the radius of a cylinder is doubled, how does the volume change?

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

Explanation:

Question 8

Step1: Identify circle area formula

The area of a circle (circular base) is given by $A = \pi r^2$.

Question 9

Step1: Use Pythagorean theorem

For a cone, slant height $l$, radius $r$, height $h$ satisfy $l^2 = r^2 + h^2$.
$h = \sqrt{l^2 - r^2} = \sqrt{13^2 - 5^2}$

Step2: Calculate the height

$h = \sqrt{169 - 25} = \sqrt{144} = 12$

Question 10

Step1: Find base triangle area

Area of triangular base: $A = \frac{1}{2} \times b \times h_b = \frac{1}{2} \times 6 \times 4$

Step2: Calculate pyramid volume

Volume of pyramid: $V = \frac{1}{3} \times A \times H = \frac{1}{3} \times 12 \times 9$

Question 11

Step1: Recall cylinder volume formula

Cylinder volume: $V = \pi r^2 H$, solve for $r$: $r = \sqrt{\frac{V}{\pi H}}$

Step2: Substitute values

$r = \sqrt{\frac{54\pi}{6\pi}} = \sqrt{9} = 3$

Question 12

Step1: Recall cone volume formula

Cone volume: $V = \frac{1}{3} \pi r^2 h$

Step2: Substitute values

$V = \frac{1}{3} \pi \times 5^2 \times 10 = \frac{250}{3}\pi$

Question 13

Step1: Rearrange cone volume formula

Solve $V = \frac{1}{3} \pi r^2 h$ for $h$: $h = \frac{3V}{\pi r^2}$

Step2: Substitute values

$h = \frac{3 \times 192\pi}{\pi \times 8^2} = \frac{576}{64} = 9$

Question 14

Step1: Analyze cylinder volume change

Cylinder volume: $V = \pi r^2 H$. If $r$ becomes $2r$, new volume $V' = \pi (2r)^2 H = 4\pi r^2 H = 4V$.

Answer:

1. b. $A = \pi r^2$
2. d. 12 cm
3. d. $36\ \text{cm}^3$
4. d. 3 cm
5. d. $\frac{250}{3}\pi\ \text{cm}^3$
6. d. 9 cm
7. b. Quadruples