Notes
Volume and surface area
- Prism and cylinder: $V = Bh$. Pyramid and cone: $\displaystyle V = \frac{1}{3}Bh$. Sphere: $\displaystyle V = \frac{4}{3}\pi r^3$.
- Sphere: $SA = 4\pi r^2$. Cylinder: $SA = 2\pi r^2 + 2\pi r h$. Cone: $SA = \pi r^2 + \pi r\ell$ with $\ell^2 = r^2 + h^2$.
Cross-sections and solids of revolution
- A plane slice of a sphere is a circle. A slice of a cylinder parallel to its bases is a circle; through its axis it is a rectangle.
- Spinning a rectangle about a side makes a cylinder. Spinning a right triangle about a leg makes a cone.
Modeling
- Density $\displaystyle = \frac{\text{mass}}{\text{volume}}$, so mass $=$ density $\times$ volume.
- Break composite shapes into pieces. If lengths scale by $k$, areas scale by $k^2$ and volumes by $k^3$.
Topics in this unit
Each topic has its own short notes and practice.
Practice
$\displaystyle \frac{1}{3}\pi(36)(5) = 60\pi$.
$4\pi(49) = 196\pi$.
Slices parallel to the base of a cone are circles.
Density $\displaystyle = \frac{1930}{100} = 19.3$.
The solid is a cylinder with $r = 3$ and $h = 7$: $\pi(9)(7) = 63\pi$.
$(2l)(2w)(2h) = 8lwh$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A water tank is a cylinder with radius 2 m and height 5 m.
(a) Find the volume of the tank in cubic meters.
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$V = \pi r^2 h = \pi(4)(5) = 20\pi \approx 62.8 \text{ m}^3$.
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(b) Water has a density of $1000 \text{ kg/m}^3$. What is the mass of the water when the tank is full?
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Mass $= 1000 \cdot 20\pi = 20000\pi \approx 62{,}800$ kg.
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(c) The side and top of the tank (not the bottom) will be painted. Find the area to be painted.
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Lateral area $= 2\pi r h = 2\pi(2)(5) = 20\pi$. Top $= \pi r^2 = 4\pi$. Total $= 24\pi \approx 75.4 \text{ m}^2$.
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