Exercise 34 Page 82

To find the area of a shaded region, subtract the area of the white region from the area of the whole.

A=3x^2 - 21

Practice makes perfect

When we want to find the area of a shaded region, we need to subtract the area of the smaller white region from the area of the whole region. Let's start with the area of the whole region. It's length is 2x+5 and width is 2x-5.

A = l * w

SubstituteII

l= 2x+5, w= 2x-5

A = ( 2x+5)( 2x-5)

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Simplify right-hand side

ExpandDiffSquares

(a+b)(a-b)=a^2-b^2

A = (2x)^2 - (5)^2

CalcPow

Calculate power

A = 4x^2 - 25

Now, let's find the area of the white region with its length x+2 and width x-2.

A = l * w

SubstituteII

l= x+2, w= x-2

A = ( x+2)( x-2)

▼

Simplify right-hand side

ExpandDiffSquares

(a+b)(a-b)=a^2-b^2

A = (x)^2 - (2)^2

CalcPow

Calculate power

A = x^2 - 4

From here, we can subtract the white area from the whole area. cc A_(Shaded Region)&= &A_(Whole)&-& A_(White) & & ⇓ && ⇓ &=& (4x^2 - 25) &-& (x^2 - 4) Let's simplify that expression.

A = (4x^2 - 25) - (x^2 - 4)

▼

Simplify right-hand side

Distr

Distribute - 1

A = 4x^2-25-x^2+4

CommutativePropAdd

Commutative Property of Addition

A = 4x^2-x^2 -25+4

SubTerms

Subtract terms

A = 3x^2 - 25+4