Surface Areas and Volumes: questions with answers

From the outside the glass looks like a full cylinder and appears to hold $\pi r^2 h = 135\pi$. The bulge occupies $\frac{2}{3}\pi r^3 = 18\pi$ of that space, so the liquid can only fill $135\pi - 18\pi = 117\pi$. $135\pi$ is the apparent capacity, $153\pi$ adds the bulge instead of taking it out, and $99\pi$ takes it out as a whole sphere.

The jar's radius is $2$ cm. Its height is $3$ diameters: $h = 2 \times 3 \times 2 = 12$ cm. The jar's volume is $\pi r^2 h = 48\pi\ \text{cm}^3$. The balls occupy $3 \times \frac{4}{3}\pi(2)^3 = 32\pi\ \text{cm}^3$. Therefore, the air volume is $48\pi - 32\pi = 16\pi\ \text{cm}^3$. The value $32\pi$ gives the balls' volume, while $48\pi$ gives the whole jar's volume.

The glass appears to hold $\pi r^2 h = 1620\pi$ and actually holds $1134\pi$; the shortfall is exactly the space the bulge occupies, $\frac{2}{3}\pi r^3 = 486\pi$. $1134\pi$ is what the glass DOES hold, not what it loses; $1620\pi$ is the apparent capacity; $972\pi$ makes the bulge a whole sphere.

Each hemisphere contributes its RADIUS to the total height, and there are two of them, so the cylinder's own height is $35 - 14 = 21$ cm. $35$ is the height of the whole solid, not of the cylinder. $28$ takes the wrong number of caps off, and $14$ takes off one too many.

Only curved surfaces show, so $2\pi r h + 4\pi r^2 = 7524$. This gives the cylinder's own length $h = 15$ cm. Each hemisphere adds $21$ cm more at its end, so the total length is $57$ cm. $15$ is the cylinder's own length kept as if it were the whole solid. $99$ adds a diameter at each end instead of a radius, and $36$ adds a radius at one end only.

The capsule is $42$ mm across, so $r = 21$ mm, and each hemisphere takes $21$ mm of the total length: the cylinder's own length is $72 - 2 \times 21 = 30$ mm. Only curved surfaces are seen, so the area is $2\pi r h + 2 \times 2\pi r^2 = 9504$. Using the whole $72$ mm as the cylinder's length gives $15048$; adding the two circles where the parts meet gives $12276$; counting only one hemisphere gives $6732$.

Three surfaces are left: the cylinder's curved side $2\pi r h = 2376$, its base circle $\pi r^2 = 693/2$ (the cavity opens at the TOP), and the cavity's own curved wall $\pi r l$, where $l = \sqrt{ (21/2)^2 + (36)^2 } = 75/2$ cm. That gives $3960$. $1485$ subtracts the cavity's wall, $8613/2$ keeps the top circle as well, and $7821/2$ uses the cavity's vertical height where its slant height belongs.

The cylinder's curved side has area $3036\ \text{cm}^2$. One flat circular end remains, and one curved hemispherical surface is exposed. The remaining flat area is $1386\ \text{cm}^2$, and the curved cavity area is $2772\ \text{cm}^2$. Adding these exposed areas gives $7194\ \text{cm}^2$. The answer $1650$ subtracts the curved cavity area instead of adding it. The answer $8580$ also counts the circular faces removed by scooping. The answer $5808$ gives the original cylinder's total surface area.

Each circular opening has diameter $6$ cm. Each row holds $36 \div 6 = 6$ openings, and there are $6$ rows. The greatest number in this square grid is $6 \times 6 = 36$. Treating the stated diameter as the radius incorrectly gives an opening diameter of $12$ cm and a total of $9$ hollows.

A volume is built from three lengths, so $125 : 27$ is the CUBE of the ratio of the radii, which is therefore $5 : 3$. A surface area is built from two lengths, so the areas are in the ratio $5 : 3$ squared $= 25 : 9$. Copying $125 : 27$ across scales an area like a volume, and $5 : 3$ scales it like a length. $9 : 25$ reverses the pair.

The formula $2\pi r h$ is written in the RADIUS, and $14$ cm is the diameter, so $r = \frac{ 14 }{2}$. That gives $2 \times \frac{22}{7} \times \frac{ 14 }{2} \times 12 = 528$. Putting the diameter straight into the formula doubles it to $1056$. $836$ is the TOTAL surface area, with the two end circles added in, and $1848$ is the volume.

The slant height is the HYPOTENUSE of the right triangle whose other two sides are the radius and the vertical height, so the missing side is $\sqrt{ 34^2 - 30^2 } = 16$ cm. $34$ is the slant height itself, which is always the longest of the three and can never be one of the other two. $4$ subtracts the two given lengths and $64$ adds them.

The roof's radius is half the shed's width, $\frac{ 14 }{2}$ m, and its length is $30$ m. Add the cuboid and half-cylinder volumes: $2100 + 2310 = 4410\ \text{m}^3$. The answer $210$ comes from subtracting the smaller volume from the larger instead of adding them. Both parts contain air, so their volumes must be added. The answer $6720$ uses a whole cylinder for the roof. The answer $11340$ uses the whole width as the roof's radius.

The hemisphere's volume is $\frac{2}{3}\times 3.14\times 3^3 = 56.52\ \text{cm}^3$. Subtract this from the solid's volume to get the cone's volume: $197.82 - 56.52 = 141.30\ \text{cm}^3$. Using $\frac{1}{3}\pi r^2h = 141.30$ gives $h = 15$ cm. Omitting the cone formula's factor $\frac{1}{3}$ gives the incorrect height $5$ cm.

The capacity is the sum of the cylindrical and hemispherical volumes: $3956.40 + 452.16 = 4408.56\ \text{cm}^3$. No volume is subtracted at the join. Subtracting the hemisphere gives $3504.24\ \text{cm}^3$. Using a whole sphere instead of a hemisphere gives $4860.72\ \text{cm}^3$. Counting only the cylinder gives $3956.40\ \text{cm}^3$.