Exercise 37 Page 820

Let's analyze the given composite solid.

The composite solid consists of two parts.

• A half of a cylinder with a diameter of 12 centimeters and a height of 12 centimeters. The radius of the cylinder is r= 122=6 centimeters.
• A cube with a side length of s=12 centimeters.The surface area of the composite solid is equal to the surface area of 5 out of 6 faces of the cube, and the half of the surface area of the cylinder. \begin{gathered} S_\text{solid}=\textcolor{darkorange}{S_\text{cube part}}+\textcolor{darkviolet}{S_\text{half-cylinder}} \end{gathered} Since each side of the cube is a square with a side length of s=12 centimeters, let's use the formula for the area of a square to find one part of the surface area of the composite solid, \textcolor{darkorange}{S_\text{cube part}}.
  \textcolor{darkorange}{S_\text{cube part}}=5\cdot A_\text{square}

  Substitute

  A_\text{square}={\color{#0000FF}{s^2}}

  \textcolor{darkorange}{S_\text{cube part}}=5\cdot s^2

  Substitute

  s= 12

  \textcolor{darkorange}{S_\text{cube part}}=5\cdot ({\color{#009600}{12}})^2

  CalcPowProd

  Calculate power and product

  \textcolor{darkorange}{S_\text{cube part}}=\textcolor{darkorange}{720}

  Now, let's use the formula for the surface area of a cylinder.

  \textcolor{darkviolet}{S_\text{half-cylinder}}=\dfrac{1}{2}\cdot S_\text{cylinder}

  Substitute

  S_\text{cylinder}={\color{#0000FF}{2\pi rh+2\pi r^2}}

  \textcolor{darkviolet}{S_\text{half-cylinder}}=\dfrac{1}{2}\cdot\left(2\pi rh+2\pi r^2\right)

  ▼

  Simplify right-hand side

  FactorOut

  Factor out 2

  \textcolor{darkviolet}{S_\text{half-cylinder}}=\dfrac{1}{2}\cdot 2\left(\pi rh+\pi r^2\right)

  MoveRightFacToNumOne

  1/b* a = a/b

  \textcolor{darkviolet}{S_\text{half-cylinder}}=\dfrac{2}{2}\cdot \left(\pi rh+\pi r^2\right)

  CalcQuot

  Calculate quotient

  \textcolor{darkviolet}{S_\text{half-cylinder}}=1\cdot \left(\pi rh+\pi r^2\right)

  OneMult

  1* a=a

  \textcolor{darkviolet}{S_\text{half-cylinder}}=\pi rh+\pi r^2

  ▼

  Substitute values and evaluate

  SubstituteII

  r= 6, h= 12

  \textcolor{darkviolet}{S_\text{half-cylinder}}=\pi({\color{#0000FF}{6}})({\color{#009600}{12}})+\pi({\color{#0000FF}{6}})^2

  CalcPowProd

  Calculate power and product

  \textcolor{darkviolet}{S_\text{half-cylinder}}=72\pi+36\pi

  AddTerms

  Add terms

  \textcolor{darkviolet}{S_\text{half-cylinder}}=108\pi

  UseCalc

  Use a calculator

  \textcolor{darkviolet}{S_\text{half-cylinder}}=339.292006\ldots

  RoundDec

  Round to 1 decimal place(s)

  \textcolor{darkviolet}{S_\text{half-cylinder}}\approx \textcolor{darkviolet}{339.3}

  Finally, let's add \textcolor{darkorange}{S_\text{cube part}} and \textcolor{darkviolet}{S_\text{half-cylinder}} to find the surface area of the composite solid, S_\text{solid}.

  S_\text{solid}=\textcolor{darkorange}{S_\text{cube part}}+\textcolor{darkviolet}{S_\text{half-cylinder}}

  SubstituteII

  \textcolor{darkorange}{S_\text{cube part}}={\color{#0000FF}{\textcolor{darkorange}{720}}}, \textcolor{darkviolet}{S_\text{half-cylinder}}={\color{#009600}{\textcolor{darkviolet}{339.3}}}

  S_\text{solid}=\textcolor{darkorange}{720}+\textcolor{darkviolet}{339.3}

  AddTerms

  Add terms

  S_\text{solid}=1059.3

  This tells us that the surface area of the given solid is about 1059.3 cubic centimeters.