Exercise 51 Page 937

We are given a diagram and asked to find the area of the shaded region. Let's take a look at the diagram.

To find the area of the shaded region, we will subtract the area of the square from the area of the circle. To do so, let's first find the area of the square. We can see from the diagram that the length of the side of the square is

10 m .

Recall the formula for the area

S

of a square with side length

a .

S = a^{2}

By substituting

10

for

a

in this formula, we will find the area of the square.

S = a^{2}

Substitute

$a = 10$

S = 10^{2}

CalcPow

Calculate power

S = 100

The area of the square is

100

square meters. To find the area of the circle, note that the diagonal of the square is the diameter of the circle.

Since the length of the diagonal of a square is

2

times the length of the side of the square, the diameter of our circle is

102 m .

The radius of a circle is half of its diameter, so the radius of our circle is

52 m .

Recall the formula for the area

C

of a circle with radius

r .

C = π r^{2}

By substituting

52

for

r

in this formula, we will find the area of the circle.

C = π r^{2}

Substitute

$r = 52$

C = π (52)^{2}

▼

Evaluate

PowProd

$(a \cdot b)^{m} = a^{m} \cdot b^{m}$

C = π \cdot 5^{2} (2)^{2}

CalcPow

Calculate power

C = π \cdot 25 2^{2}

PowSqrt

$(a)^{2} = a$

C = π \cdot 25 \cdot 2

Multiply

C = 50 π

Now we can calculate the area of the shaded region by subtracting the area

S

of the square from the area

C

of the circle.

Area of the Shaded Region = C - S

SubstituteII

$C = 50 π$ , $S = 100$

Area of the Shaded Region = 50 π - 100

UseCalc

Use a calculator

Area of the Shaded Region = 57.079633 \dots

RoundDec

Round to $1$ decimal place(s)

Area of the Shaded Region = 57.1

The area of the shaded region rounded to the nearest tenth is

57.1 m^{2} .

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