Exercise 30 Page 836

Let's analyze the given composite solid.

The composite solid consists of three parts.

• Two halves of a cylinder with a diameter of 4 inches and a height of 8 inches.
• One rectangular prism with a length of l= 5 inches, a width of w= 4 inches, and a height of h= 8 inches.
  The volume of a rectangular prism is equal to the product of its dimensions.
  V_\text{prism} = {\color{#0000FF}{\ell}} {\color{#009600}{w}} {\color{#FF0000}{h}}

  ▼

  Substitute values and evaluate

  SubstituteValues

  Substitute values

  V_\text{prism} = ({\color{#0000FF}{5}})({\color{#009600}{4}})({\color{#FF0000}{8}})

  Multiply

  Multiply

  V_\text{prism} = 160
  Therefore, the volume of the rectangular prism is \textcolor{darkviolet}{V_\text{prism}}=\textcolor{darkviolet}{160} cubic inches. Now we will find the volumes of two halves of a cylinder. We will find it by calculating the volume of one full cylinder.
  
  Since the diameter of the base is 4 inches, its radius is r= 42=2 inches. The height of the cylinder is 8 inches. Let's use the formula for the volume of a cylinder.
  V_\text{cylinder}=\pi r^2h

  ▼

  Substitute values and evaluate

  SubstituteII

  r= 2, h= 8

  V_\text{cylinder}=\pi ({\color{#0000FF}{2}})^2({\color{#009600}{8}})

  CalcPowProd

  Calculate power and product

  V_\text{cylinder}=32\pi

  UseCalc

  Use a calculator

  V_\text{cylinder}=100.530964\ldots

  RoundDec

  Round to 1 decimal place(s)

  V_\text{cylinder}\approx 100.5
  Therefore, the volume of the full cylinder is about 100.5 cubic inches. Now, let's add the volume of the cylinder and the volume of the rectangular prism to get the volume of the composite solid.
  V_\text{solid}=\textcolor{darkorange}{V_\text{cylinder}}+\textcolor{darkviolet}{V_\text{prism}}

  SubstituteII

  \textcolor{darkorange}{V_\text{cylinder}}={\color{#0000FF}{\textcolor{darkorange}{100.5}}}, \textcolor{darkviolet}{V_\text{prism}}={\color{#009600}{\textcolor{darkviolet}{160}}}

  V_\text{solid}=\textcolor{darkorange}{100.5}+\textcolor{darkviolet}{160}

  AddTerms

  Add terms

  V_\text{solid}=260.5
  Finally, we get that the volume of the composite solid is about 260.5 cubic inches.