Exercise 27 Page 475

We want to find the length of the safety rail and the area of the deck.

First, we will use the formula for the length of an arc to find the length of the safety rail.

s = θ r

In this formula,

r

represents the radius of the deck and

θ

is the measure of the angle in radians. Since the given angle measure is expressed in degrees, we will convert it to radians. This will involve using the conversion factor from degrees to radians.

\frac{π radians}{1 8 0 degrees}

Let's find radian measure of

9 0^{\circ} .

90 degrees (\frac{π radians}{1 8 0 degrees})

▼

Simplify

CrossCommonFac

Cross out common factors

90 degrees (\frac{π radians}{1 8 0 degrees})

SimpQuot

Simplify quotient

90 (\frac{π radians}{1 8 0})

MoveLeftFacToNum

$a \cdot \frac{b}{c} = \frac{a \cdot b}{c}$

\frac{9 0 π radians}{1 8 0}

MovePartNumRight

$\frac{a \cdot b}{c} = \frac{a}{c} \cdot b$

\frac{9 0 π}{1 8 0} radians

ReduceFrac

$\frac{a}{b} = \frac{a / 9 0}{b / 9 0}$

\frac{π}{2} radians

Now we will substitute

10

for

r

and

\frac{π}{2}

for

θ

in the formula for the length of an arc.

s = θ r

SubstituteII

$θ = \frac{π}{2}$ , $r = 10$

s = (\frac{π}{2}) (10)

▼

Evaluate

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

s = \frac{1 0 π}{2}

CalcQuot

Calculate quotient

s = 5 π

UseCalc

Use a calculator

s = 15.707963 \dots

RoundDec

Round to $2$ decimal place(s)

s \approx 15.71

The length of the safety rail is about

15.71

yards. Finally, we will find the area of the deck using the formula for the area of a sector of a circle.

A = \frac{1}{2} r^{2} θ

Again, we will substitute

10

for

r

and

\frac{π}{2}

for

θ .

A = \frac{1}{2} r^{2} θ

SubstituteII

$θ = \frac{π}{2}$ , $r = 10$

A = \frac{1}{2} (10)^{2} (\frac{π}{2})

▼

Evaluate

CalcPow

Calculate power

A = \frac{1}{2} (100) (\frac{π}{2})

MoveRightFacToNumOne

$\frac{1}{b} \cdot a = \frac{a}{b}$

A = \frac{1 0 0}{2} (\frac{π}{2})

MultFrac

Multiply fractions

A = \frac{1 0 0 π}{4}