A cheesemaker makes portions of cheese by cutting a cylindrical wheel of cheese into wedges. If the given wedge is 6 inches thick and its dimension is as given, then find its volume.

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Solution

Picturing the wedge as a part of the wheel of cheese, we find:

$R a d i u s o f t h e w h e e l = 10 i n ⟹ C i r c u m f e r e n c e o f t h e w h e e l = 2 π \times (r a d i u s) = 2 π \times 10 i n = 20 π i n$

$We know,$
$Arc length = Circumference \times \frac{θ °}{360 °}$
$\Rightarrow θ ° = \frac{Arc length of a circle}{Circumference} \times 360 °$

$L e n g t h o f t h e a r c o f t h e c h e e s e w e d g e = \frac{10 π}{3} i n$

$A n g l e o f t h e w e d g e = \frac{L e n g t h o f t h e a r c}{C i r c u m f e r e n c e o f t h e w h e e l} \times 360 ° = (\frac{10 π}{3} \div 20 π) \times 360 ° = (\frac{10 π}{3} \times \frac{1}{20 π}) \times 360 ° = \frac{1}{6} \times 360 ° = 60 °$

Now, the volume of the wedge can be calculated as:

$Volume of the wedge$ $= \frac{Angle of wedge}{Total angle} \times Volume of wheel$

We know that the wheel of cheese is a cylinder with a height of 6 inches.

$⟹ V o l u m e o f t h e w h e e l = π (r a d i u s)^{2} \times h = π \times 10^{2} \times 6 = π \times 100 \times 6 = 600 π c u i n$

$⟹ V o l u m e o f t h e w e d g e = \frac{60 °}{360 °} \times 600 = \frac{600}{6} = 100 c u i n$

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