Easy but Frustrating Problem

**alibond07** · March 7th 2010, 02:28 PM

A circular dartboard is divided into 20 equal sectors, one of which is shown in the diagram. O is the centre of the circle. The areas for scoring double and treble are marked A and B respectively.

Find the ratio area A:area B in the form n:1, giving n correct to 1 d.p

OM:99mm
MN:8mm
OP:162mm
PQ:8mm

Thanks guys

**Krahl** · March 7th 2010, 02:58 PM

the circle with radius OM has area $\pi 99^2$
the circle with radius ON has area $\pi (99+8)^2$
the circle with radius OP has area $\pi 162^2$
the circle with radius OQ has area $\pi (162+8)^2$

the sector OM has area $\frac{\pi 99^2}{20}$
the sector ON has area $\frac{\pi (99+8)^2}{20}$
the sector OP has area $\frac{\pi 162^2}{20}$
the sector OQ has area $\frac{\pi (162+8)^2}{20}$

**Soroban** · March 7th 2010, 07:03 PM

Hello, alibond07!

You need to know the area of a circle and some common sense.
Exactly where is your difficulty?

A circular dartboard is divided into 20 equal sectors,
one of which is shown in the diagram. O is the centre of the circle.
The areas for scoring double and treble are marked A and B respectively.

Find the ratio (area A) : (area B) in the form n : 1, giving n correct to 1 d.p

OM: 99mm . MN: 8mm . OP: 162mm . PQ: 8mm

This is a diagram of the radii:

Code:

      : - - - - 107 - - - :
      : - - -99 - - :
                      (A)             (B)
      O ----------- M --- N ------- P --- Q

      : - - - - - - - 162 - - - - - :
      : - - - - - - - - 170 - - - - - - - :

The circle with radius $OQ = 170$ has area: . $\pi(170^2)$

The circle with radius $OP = 162$ has area: . $\pi(162^2)$

. . The area of ring $A$ is: . $28,\!900\pi - 26,\!244\pi \:=\:2656\pi$

The circle with radius $ON = 107$ has area: . $\pi(107^2)$

The circle with radius $OM = 99$ has area: . $\pi(99^2)$

. . The area of ring $B$ is: . $11,\!449\pi - 9801\pi \:=\:1648\pi$

Hence: . $\frac{\text{ring A}}{\text{ring B}} \:=\:\frac{2656\pi}{1648\pi} \;=\;1.611650485$

Therefore: . $\text{(area A)} : \text{(area B)} \;\approx\;1.6:1$

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