Geometry and Trigonometry

QUESTION 1 [2 marks] Easy

Find the area of a rectangle with length 12 and width 7.

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Area $=12\times7=84$.

QUESTION 2 [2 marks] Easy

Find the volume of a rectangular prism with length 5, width 4, and height 3.

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Volume $=5\times4\times3=60$.

QUESTION 3 [3 marks] Medium

A circle has radius 6. Find its area in terms of $\pi$.

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Area $=\pi r^2=\pi(6)^2=36\pi$.

QUESTION 4 [3 marks] Medium

A cylinder has radius 3 and height 10. Find its volume in terms of $\pi$.

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Volume $=\pi r^2h=\pi(3)^2(10)=\pi(9)(10)=90\pi$.

QUESTION 5 [4 marks] Hard

A cone has a volume of $100\pi$ cubic units and a height of 12 units. Find the radius of the cone's base.

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Volume of a cone: $V=\dfrac{1}{3}\pi r^2h$. Substituting: $100\pi=\dfrac{1}{3}\pi r^2(12)=4\pi r^2$. Dividing both sides by $4\pi$: $r^2=25$, so $r=5$.

QUESTION 6 [5 marks] Hard

A cylindrical water tank has a radius of 4 feet and needs to hold $480\pi$ cubic feet of water. What must the height of the tank be, in feet?

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$V=\pi r^2h \Rightarrow 480\pi=\pi(4)^2h=16\pi h$. Dividing both sides by $16\pi$: $h=\dfrac{480}{16}=30$ feet.

QUESTION 7 [5 marks] Hard

A sphere has a volume of $\dfrac{500\pi}{3}$ cubic units. Find its radius.

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$V=\dfrac{4}{3}\pi r^3 \Rightarrow \dfrac{500\pi}{3}=\dfrac{4}{3}\pi r^3$. Multiplying both sides by $\dfrac{3}{4\pi}$: $r^3=\dfrac{500}{4}=125$, so $r=\sqrt[3]{125}=5$.

Area and volume 7 questions