With a circular dartboard, the probability is the ratio between the area of entire circular board to the area of the desired target, the inner yellow circle.

P (Hitting Yellow) = \frac{A _{Yellow Circle}}{A _{Entire Board}}

Since our answers need to be in centimeters, recall that

1

meter is the same as

100

centimeters. Let's use this to find the area of the whole circular board.

A_{Board} = π r^{2}

Substitute

$r = 100$

A_{Board} = π (100)^{2}

CalcPow

Calculate power

A_{Board} = π (10000)

CommutativePropMult

Commutative Property of Multiplication

A_{Board} = 10000 π

Now, let's put this area in our probability equation. Since we are interested in the radius of the yellow circle, we can use the formula for the area of the yellow circle.

P (Hitting Yellow) = \frac{A _{Yellow Circle}}{A _{Board}} = \frac{A _{Yellow Circle}}{1 0 0 0 0}

We are interested in finding the radius of the yellow circle. Therefore, let us also substitute the formula for the area of a circle for the area of the yellow circle.

P (Hitting Yellow) = \frac{A _{Yellow Circle}}{1 0 0 0 0} = \frac{π r _{Yellow Circle}^{2}}{1 0 0 0 0}

Now, we can solve this equation for the radius of the yellow circle.

P (Yellow) = \frac{π r _{Yellow}^{2}}{1 0 0 0 0 π}

▼

Solve for

r_{Yellow}

DivEqn

$LHS / (π) = RHS / (π)$

P (Yellow) = \frac{r _{Yellow}^{2}}{1 0 0 0 0}

MultEqn

$LHS \cdot 10000 = RHS \cdot 10000$

10000 P (Yellow) = r_{Yellow}^{2}

SqrtEqn

$LHS = RHS$

\pm 10000 P (Yellow) = r_{Yellow}

SqrtProd

$a \cdot b = a \cdot b$

\pm 10000 P (Yellow) = r_{Yellow}

CalcRoot

Calculate root

\pm 100 P (Yellow) = r_{Yellow}

RearrangeEqn

Rearrange equation

r_{Yellow} = \pm 100 P (Yellow)

We can disregard the negative root since we are working with positive distances. Let's input our equation into our calculator. We can press

Y =

and enter our function.

Now let's look at the table of values. Make sure to change the table setup, so that the $Δ Tbl = 0.1 .$ We can press $2nd$ then $GRAPH$ to see the table of values.

From here, we can see all the probabilities between $0.2$ and $0.8$ on one screen. We can scroll down to get our last value of $y$ when $x = 1 .$

Now, we can express our radius $y$ that would generate all six answers $x$ for the probability of a dart landing in the yellow circle. We can round each answer to the nearest centimeter.

$P (Hitting Yellow)$	Radius of Yellow Region (cm)
a. $0.2$	$45$
b. $0.4$	$63$
c. $0.5$	$71$
d. $0.6$	$77$
e. $0.8$	$89$
f. $1.0$	$100$

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