Question

1 what is one first step in calculating the volume of a cone?
a. divide the radius by the height
b. multiply the radius by the height
c. calculate the area of the base
d. multiply the height by π

2 what is the formula for the volume of a cone?
a. ( v = hr )
b. ( v = \frac{1}{2}pi r^2 h )
c. ( v = \frac{1}{3}pi r^2 h )
d. ( v = \frac{1}{3}pi r h )

3 if the volume of a cone is ( 30pi , \text{cm}^3 ) and its height is 10 cm, what is its radius? choose the exact answer.
a. 3 cm
b. ( sqrt{8} , \text{cm} )
c. ( 3sqrt{8} , \text{cm} )
d. ( 9sqrt{8} , \text{cm} )

4 which mathematical constant is used in the formula for the volume of a cylinder?
a. π
b. 2
c. 8
d. ( pi \times 100 )

5 if the radius of a cone is 5 cm and its volume is ( 250pi , \text{cm}^3 ), what is its height?
a. 4 cm
b. 9 cm
c. 3 cm
d. 6 cm

6 a cone and a pyramid have the same height of 18 cm. the cone’s circular base has a radius of 3 cm. if they have the same volume, what is the side length of the square base of the pyramid?
a. 6.2 cm
b. 4.2 cm
c. 5.3 cm
d. 8.0 cm

7 if the diameter of a sphere is 12 cm, what is its radius?
a. 6 cm
b. 30 cm
c. 12 cm
d. 24 cm

Explanation:

Step1: Answer Q1 (Cone volume first step)

The volume of a cone relies on the base area (a circle, $\pi r^2$) first.

Step2: Answer Q2 (Cone volume formula)

The standard formula for the volume of a cone is $V=\frac{1}{3}\pi r^2 h$.

Step3: Solve Q3 (Find cone radius)

Given $V=180\pi$, $h=10$. Substitute into $V=\frac{1}{3}\pi r^2 h$:
$$180\pi = \frac{1}{3}\pi r^2 (10)$$
Cancel $\pi$, multiply both sides by 3: $540=10r^2$ → $r^2=54$ → $r=3\sqrt{6}$.

Step4: Answer Q4 (Cylinder constant)

Cylinder volume formula is $V=\pi r^2 h$, uses $\pi$.

Step5: Solve Q5 (Find cone height)

Given $V=240\pi$, $r=6$. Substitute into $V=\frac{1}{3}\pi r^2 h$:
$$240\pi = \frac{1}{3}\pi (6)^2 h$$
Cancel $\pi$, calculate $6^2=36$: $240=12h$ → $h=20$.

Step6: Solve Q6 (Pyramid base edge)

Cone volume: $V_{cone}=\frac{1}{3}\pi r^2 h = \frac{1}{3}\pi (3)^2(10)=30\pi$.
Pyramid volume $V_{pyramid}=\frac{1}{3}s^2 h$, set equal to $30\pi$, $h=10$:
$$30\pi = \frac{1}{3}s^2(10)$$
Multiply by 3: $90\pi=10s^2$ → $s^2=9\pi$ → $s=3\sqrt{\pi}\approx5.3$.

Step7: Answer Q7 (Sphere radius)

Radius = $\frac{\text{diameter}}{2} = \frac{12}{2}=6$.

Answer:

1. c. Calculate the area of the base
2. b. $V = \frac{1}{3}\pi r^2 h$
3. c. $3\sqrt{6}$ cm
4. a. $\pi$
5. c. 20 cm
6. c. 5.3 cm
7. a. 6 cm