CE Board Exam Randomizer

Question 1

Spheres and Composite Solids

A sphere is a perfectly round solid — every point on its surface is the same distance (the radius) from the center. Cutting a sphere exactly in half gives a hemisphere, with volume $\frac{1}{2}\cdot\frac{4}{3}\pi r^3=\frac{2}{3}\pi r^3$. Composite solids are real-world shapes built from simpler pieces: a grain silo might be a cylinder topped by a hemisphere, or a bullet is a cylinder with a cone tip. The key strategy is always to identify the component shapes, compute each volume separately, then add or subtract. A sphere's surface area is exactly 4 times the area of a great circle — a useful fact to remember.

$$V_{\text{sphere}}=\frac{4}{3}\pi r^3, \qquad A_{\text{sphere}}=4\pi r^2$$

$$V_{\text{hemisphere}}=\frac{2}{3}\pi r^3, \qquad A_{\text{hemi curved}}=2\pi r^2$$

Answer

The radius is $r=3$ m.

$$V=\frac{4}{3}\pi(3)^3=36\pi=113.10\text{ m}^3$$

$$A=4\pi(3)^2=36\pi=113.10\text{ m}^2$$

Final answer: $V=113.10\text{ m}^3$, $A=113.10\text{ m}^2$.

Answer

Rearrange the volume formula to solve for $r$.

$$V=\frac{4}{3}\pi r^3 \quad\Rightarrow\quad r=\left(\frac{3V}{4\pi}\right)^{1/3}$$

$$r=\left(\frac{3(500)}{4\pi}\right)^{1/3}=\left(119.37\right)^{1/3}=4.924\text{ m}$$

Final answer: $r\approx4.92$ m.

Answer

A hemisphere is half a sphere.

$$V_{\text{hemi}}=\frac{1}{2}\cdot\frac{4}{3}\pi r^3=\frac{2}{3}\pi(10)^3=\frac{2000\pi}{3}=2094.4\text{ m}^3$$

$$A_{\text{curved}}=\frac{1}{2}\cdot4\pi r^2=2\pi(10)^2=200\pi=628.3\text{ m}^2$$

Final answers: V = 2094.4 m³, curved area = 628.3 m².

Answer

Total volume = cylinder + hemisphere. Radius = 2 m.

$$V_{\text{cyl}}=\pi(2)^2(10)=40\pi=125.66\text{ m}^3$$

$$V_{\text{hemi}}=\frac{2}{3}\pi(2)^3=\frac{16\pi}{3}=16.76\text{ m}^3$$

$$V_{\text{total}}=125.66+16.76=142.42\text{ m}^3$$

Final answer: $142.42\text{ m}^3$.

Answer

The ice cream fills the cylindrical portion only. The conical bottom holds the stick/wafer, not ice cream. Actually — let me restate: find the total internal volume = cylinder + cone.

$$V_{\text{cyl}}=\pi(3)^2(8)=72\pi\text{ cm}^3$$

$$V_{\text{cone}}=\frac{1}{3}\pi(3)^2(5)=15\pi\text{ cm}^3$$

$$V_{\text{total}}=72\pi+15\pi=87\pi=273.3\text{ cm}^3$$

Final answer: $273.3\text{ cm}^3$.

Answer

A sphere inscribed in a cube touches all six faces. Its diameter equals the side length, so $r=5$ cm.

$$V_{\text{sphere}}=\frac{4}{3}\pi(5)^3=\frac{500\pi}{3}=523.6\text{ cm}^3$$

$$V_{\text{cube}}=10^3=1000\text{ cm}^3$$

$$V_{\text{gap}}=1000-523.6=476.4\text{ cm}^3$$

The sphere fills only $\approx52.4\%$ of the cube. Final answers: V (sphere) = 523.6 cm³, V (gap) = 476.4 cm³.

Answer

Volume of one ball: $V_{\text{ball}}=\frac{4}{3}\pi(2)^3=\frac{32\pi}{3}=33.51\text{ cm}^3$.

$$V_{\text{container}}=\pi(10)^2(30)=3000\pi=9424.8\text{ cm}^3$$

Total volume of balls at 64% packing efficiency:

$$V_{\text{balls}}=0.64\times9424.8=6031.9\text{ cm}^3$$

$$n=\frac{6031.9}{33.51}\approx180\text{ balls}$$

Final answer: approximately 180 balls, total ball volume ≈ 6031.9 cm³.

Question 2

Sphere Volume and Area

A sphere has diameter 6 m. Find its volume and surface area.

Question 3

★ Finding Radius from Volume

A spherical water tank has a volume of 500 m³. Find its radius.

Question 4

★ Volume and Curved Area of a Hemisphere

A hemispherical dome has a radius of 10 m. Find (a) the volume enclosed and (b) the curved surface area of the dome.

Question 5

★★ Capsule: Cylinder with Two Hemispheres

A grain silo consists of a cylinder (diameter 4 m, height 10 m) with a hemispherical roof on top. Find the total volume of the silo.

Question 6

★★★ Cone Removed from Cylinder (Ice Cream Problem)

An ice cream cup is a cylinder of radius 3 cm and height 8 cm with a conical bottom of the same radius and height 5 cm. Find the volume of ice cream the cup can hold.

Question 7

★★★ Sphere Inscribed in a Cube

A solid sphere is inscribed in a cube of side length 10 cm. Find (a) the volume of the sphere and (b) the volume of space between the sphere and the cube walls.

Question 8

★★★ How Many Steel Balls Fit?

A cylindrical container with diameter 20 cm and height 30 cm is to be filled with solid steel balls each of diameter 4 cm. Assuming random packing at 64% efficiency, how many balls fit and what is their total volume?