Exercise 48 Page 847

A combination hopper can be modeled by the following composite solid.

It consists of the following three solids.

• A bottom cone with a radius of r= 9 feet, a slant length of l_1, and a height of h_1=28-5-12-2= 9 feet.
• A middle cylinder with aradius of r= 9 feet and a height of h_2= 12 feet.
• A top cone with a radius of r= 9 feet, a slant length of l_3, and a height of h_3= 5 feet.
  Now, let's find the lengths of the slants of the cones. Analyze the following two right triangles.
  

  Now, let's use the Pythagorean Theorem for the left triangle.

  r^2+ h^2_1= l^2_1

  SubstituteII

  r= 9, h_1= 9

  9^2+ 9^2= l^2_1

  ▼

  Solve for l_1

  CalcPow

  Calculate power

  81+81= l^2_1

  AddTerms

  Add terms

  162= l^2_1

  RearrangeEqn

  Rearrange equation

  l^2_1=162

  SqrtEqn

  sqrt(LHS)=sqrt(RHS)

  sqrt(l^2_1)=sqrt(162)

  SqrtPowToNumber

  sqrt(a^2)=a

  l_1=sqrt(162)

  SplitIntoFactors

  Split into factors

  l_1=sqrt(81* 2)

  SqrtProd

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

  l_1=sqrt(81)*sqrt(2)

  CalcRoot

  Calculate root

  l_1=9sqrt(2)

  Now, let's use the Pythagorean Theorem for the right triangle.

  r^2+ h^2_3= l^2_3

  SubstituteII

  r= 9, h_3= 5

  9^2+ 5^2= l^2_3

  ▼

  Solve for l_3

  CalcPow

  Calculate power

  81+25= l^2_3

  AddTerms

  Add terms

  106= l^2_3

  RearrangeEqn

  Rearrange equation

  l^2_3=106

  SqrtEqn

  sqrt(LHS)=sqrt(RHS)

  sqrt(l_3^2)=sqrt(106)

  SqrtPowToNumber

  sqrt(a^2)=a

  l_3=sqrt(106)

  Next, we will use the formula for the lateral area of a right circular cone and for the lateral area of a cylinder.

  Solid	Bottom Cone	Middle Cylinder	Top Cone
  Radius	r= 9	r= 9	r= 9
  Height	h_1= 9	h_2= 12	h_3= 5
  Slant	l_1= 9sqrt(2)	-	l_2= sqrt(106)
  Lateral Area	S_\text{bottom}=\pi {\color{#0000FF}{r}}{\color{#FF0000}{\ell_1}}	S_\text{middle}=2\pi {\color{#0000FF}{r}}{\color{#009600}{h_2}}	S_\text{top}=\pi {\color{#0000FF}{r}}{\color{#FF0000}{\ell_3}}
  	\textcolor{darkorange}{S_\text{bottom}}=\pi({\color{#0000FF}{9}})({\color{#FF0000}{9\sqrt{2}}})=\textcolor{darkorange}{81\pi\sqrt{2}}	\textcolor{darkviolet}{S_\text{middle}}=2\pi ({\color{#0000FF}{9}})({\color{#009600}{12}})=\textcolor{darkviolet}{216\pi}	{\color{#FF0000}{S_\text{top}}}=\pi ({\color{#0000FF}{9}})({\color{#FF0000}{\sqrt{106}}})={\color{#FF0000}{9\pi\sqrt{106}}}

  Finally, to find the surface area of the composite solid we will add the lateral areas of the three smaller solids. Then we will round the answer to the nearest square foot.

  S_\text{solid}=\textcolor{darkorange}{S_\text{bottom}}+\textcolor{darkviolet}{S_\text{middle}}+{\color{#FF0000}{S_\text{top}}}

  ▼

  Substitute values and evaluate

  SubstituteValues

  Substitute values

  S_\text{solid}=\textcolor{darkorange}{81\pi\sqrt{2}}+\textcolor{darkviolet}{216\pi}+{\color{#FF0000}{9\pi\sqrt{106}}}

  FactorOut

  Factor out π

  S_\text{solid}=\left(81\sqrt{2}+216+9\sqrt{106}\right)\pi

  ▼

  Round to nearest integer

  UseCalc

  Use a calculator

  S_\text{solid}=1329.559615\ldots

  RoundInt

  Round to nearest integer

  S_\text{solid}\approx 1330

  Finally, we find that the surface area of the combination hopper is exactly (81sqrt(2)+216+9sqrt(106))π square feet, which is about 1330 square feet.