Exercise 36 Page 820

Let's analyze the given composite solid. We assume that there is a rectangle in the base, not a parallelogram. Otherwise, we have too little data to solve the exercise.

The composite solid consists of two parts.

• A half of a cylinder with a diameter of 6 inches and a height of 15 inches. Therefore, the radius of the cylinder is r= 62=3 inches.
• A rectangular prism that is 6 inches by 15 inches by 4 inches.The surface area of the composite solid is equal to the surface area of 5 out of 6 faces of the rectangular prism and half of the surface area of the cylinder. \begin{gathered} S_\text{solid}=\textcolor{darkorange}{S_\text{rect. part}}+\textcolor{darkviolet}{S_\text{half-cylinder}} \end{gathered} The surface area of a rectangular prism is made of two rectangles 6 inches by 4 inches, two rectangles 15 inches by 4 inches, and one rectangle 6 inches by 15 inches. Now, let's find its surface area. \begin{aligned} \textcolor{darkorange}{S_\text{rect. part}}&=2\cdot{\color{#0000FF}{6}}\cdot{\color{#0000FF}{4}}+2\cdot{\color{#009600}{15}}\cdot{\color{#009600}{4}}+1\cdot{\color{#FF0000}{6}}\cdot{\color{#FF0000}{15}} \\ &=2\cdot{\color{#0000FF}{24}}+2\cdot{\color{#009600}{60}}+1\cdot{\color{#FF0000}{90}} \\ &=48+120+90=258 \end{aligned} Then use the formula for the surface area of a cylinder.
  \textcolor{darkviolet}{S_\text{half-cylinder}}=\dfrac{1}{2}\cdot A_\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= 3, h= 15

  \textcolor{darkviolet}{S_\text{half-cylinder}}=\pi({\color{#0000FF}{3}})({\color{#009600}{15}})+\pi({\color{#0000FF}{3}})^2

  CalcPowProd

  Calculate power and product

  \textcolor{darkviolet}{S_\text{half-cylinder}}=45\pi+9\pi

  AddTerms

  Add terms

  \textcolor{darkviolet}{S_\text{half-cylinder}}=54\pi

  UseCalc

  Use a calculator

  \textcolor{darkviolet}{S_\text{half-cylinder}}=169.646003\ldots

  RoundDec

  Round to 1 decimal place(s)

  \textcolor{darkviolet}{S_\text{half-cylinder}}\approx \textcolor{darkviolet}{169.6}
  Finally, let's add \textcolor{darkorange}{S_\text{rect. 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{rect. part}}+\textcolor{darkviolet}{S_\text{half-cylinder}}

  SubstituteII

  \textcolor{darkorange}{S_\text{rect. part}}={\color{#0000FF}{\textcolor{darkorange}{258}}}, \textcolor{darkviolet}{S_\text{half-cylinder}}={\color{#009600}{\textcolor{darkviolet}{169.6}}}

  S_\text{solid}=\textcolor{darkorange}{258}+\textcolor{darkviolet}{169.6}

  AddTerms

  Add terms

  S_\text{solid}=427.6
  This tells us that the surface area of the given solid is about 427.6 cubic inches.