Exercise 35 Page 820

Let's analyze the given solid. It can be viewed as a quadrangular prism with the base in a shape of a trapezoid and a height of h= 28 centimeters.

We are asked to find the surface area of the prism S. S= P h+2 BHere, P is the perimeter of the base and B is the area of the base. The base is a trapezoid with bases b_1=20 cm and b_2=13 cm, and the height is \textcolor{darkviolet}{h_\text{base}}=\textcolor{darkviolet}{21} cm. Let's find its area B using the formula for the area of a trapezoid.

B=\dfrac{1}{2}\textcolor{darkviolet}{h_\text{base}}(\textcolor{darkorange}{b_1}+\textcolor{darkorange}{b_2})

SubstituteValues

Substitute values

B=1/2(21)(20+13)

▼

Simplify right-hand side

AddTerms

Add terms

B=1/2(21)(33)

Multiply

B=1/2* 693

MoveRightFacToNumOne

1/b* a = a/b

B=693/2

CalcQuot

Calculate quotient

B=346.5

Therefore, the area of the base is B=346.5 square centimeters. Now, let's find the perimeter of the base.

Notice that AB=20-13=7 centimeters and AC=21 centimeters. Let's use the Pythagorean Theorem for right △ ABC to find BC.

AB^2+AC^2=BC^2

SubstituteII

AB= 7, AC= 21

7^2+ 21^2=BC^2

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Solve for BC

CalcPow

Calculate power

49+441=BC^2

AddTerms

Add terms

490=BC^2

RearrangeEqn

Rearrange equation

BC^2=490

SqrtEqn

sqrt(LHS)=sqrt(RHS)

sqrt(BC^2)=sqrt(490)

SqrtPowToNumber

sqrt(a^2)=a

BC=sqrt(490)

SplitIntoFactors

Split into factors

BC=sqrt(49* 10)

SqrtProd

sqrt(a* b)=sqrt(a)*sqrt(b)

BC=sqrt(49)*sqrt(10)

CalcRoot

Calculate root

BC=7sqrt(10)

UseCalc

Use a calculator

BC=22.135943...

RoundDec

Round to 2 decimal place(s)

BC≈ 22.14

Now, let's find the perimeter of the base. P=20+21+13+22.14= 76.14 Finally, let's substitute known values into the formula for the surface area of the given solid.

S= P h+2 B

SubstituteValues

Substitute values

S=( 76.14)( 28)+2( 346.5)

▼

Evaluate right-hand side

Multiply

S=2131.92+693

AddTerms

Add terms

S=2824.92

RoundDec