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Big Ideas Math Algebra 1, 2015
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Big Ideas Math Algebra 1, 2015 View details
Chapter Test
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Exercise 16 Page 413

Practice makes perfect
a The area of a rectangle is calculated by multiplying the rectangle's length, l, with its width, w.
A= l w We are asked to find an expression for the width of the stage. We know that the width is x+16 and the area is x^2+27x+176. Let's substitute the given expression and find an expression for w.
A= l w
SubstituteExpressions

Substitute expressions

x^2+27x+176=( x+16)w
▼
Solve for w
WriteSum

Write as a sum

x^2+16x+11x+176=(x+16)w
FactorOut

Factor out x

x(x+16)+11x+176=(x+16)w
FactorOut

Factor out 11

x(x+16)+11(x+16)=(x+16)w
FactorOut

Factor out x+16

(x+16)(x+11)=(x+16)w
DivEqn

.LHS /(x+16).=.RHS /(x+16).

x+11=w
RearrangeEqn

Rearrange equation

w=x+11
Therefore, the width of the stage is x+11 feet.
b To determine the perimeter, add the length of all the sides of the stage.
P=2 l+2 w The length of the stage is l= x+16, and the width of the stage is w= x+11, as we know from Part A. Let's substitute the known expressions into the equation for the perimeter.
P=2 l+2 w
SubstituteII

l= x+16, w= x+11

P=2( x+16)+2( x+11)
▼
Simplify right-hand side
Distr

Distribute 2

P=2(x)+2(16)+2(x)+2(11)
Multiply

Multiply

P=2x+32+2x+22
AddTerms

Add terms

P=4x+54
The perimeter of the stage is 4x+54 feet.
c The area of a rectangle is calculated by multiplying the rectangle's length, l, with its width, w.
A= l wWe know that l= 2x feet and w= x+ 12 feet. Next, we will substitute these expressions into the formula for the area.
A= l w
SubstituteExpressions

Substitute expressions

A= 2x ( x+1/2)
Distr

Distribute 2x

A=2x(x)+2x(1/2)
Multiply

Multiply

A=2x^2+x
We are asked to find x if the area of the trapdoor is 10 square feet. Let's substitute 10 for A into the formula and solve it for x!
A= l w
SubstituteExpressions

Substitute expressions

10= 2x ( x+1/2)
▼
Solve for x
Distr

Distribute 2x

10=2x(x)+2x(1/2)
Multiply

Multiply

10=2x^2+x
SubEqn

LHS-10=RHS-10

0=2x^2+x-10
RearrangeEqn

Rearrange equation

2x^2+x-10=0
WriteSum

Write as a sum

2x^2-4x+5x-10=0
FactorOut

Factor out 2x

2x(x-2)+5x-10=0
FactorOut

Factor out 5

2x(x-2)+5(x-2)=0
FactorOut

Factor out x-2

(x-2)(2x+5)=0
ZeroProdProp

Use the Zero Product Property

lcx-2=0 & (I) 2x+5=0 & (II)
AddEqn

(I):LHS+2=RHS+2

lcx=2 & (I) 2x+5=0 & (II)
SubEqn

(II):LHS-5=RHS-5

lcx=2 & (I) 2x=- 5 & (II)
DivEqn

(II):.LHS /2.=.RHS /2.

lx=2 x=- 52
Since x=- 52 does not make sense in context with the problem, the only solution is x=2.
The
d From Part C we know that x=2 and the area of the trapdoor is 10 square feet. From Part A the area of the stage is A=x^2+27x+176. Let's find the area of the stage in our case by substituting x=2 into the formula.
A=x^2+27x+176
Substitute

x= 2

A=( 2)^2+27( 2)+176
▼
Simplify right-hand side
CalcPow

Calculate power

A=4+27(2)+176
Multiply

Multiply

A=4+54+176
AddTerms

Add terms

A=234
Notice that the area of the stage is 234 square feet and the area of the trapdoor is 10 square feet. Therefore, the stage is at least 20 times the area of the trapdoor. This means that the stage satisfies the magician's requirement.
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arrow_right Exercises p.413
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2
3
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5
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7
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Exercises 11 (Page 413)
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Exercises 16 (Page 413)
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