Exercise 31 Page &nbsp; 853

r= 7, h= 11

7^2+ 11^2= l^2

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Solve for l

CalcPow

Calculate power

49+121= l^2

Add terms

170= l^2

RearrangeEqn

Rearrange equation

l^2=170

SqrtEqn

sqrt(LHS)=sqrt(RHS)

sqrt(l^2)=sqrt(170)

SqrtPowToNumber

sqrt(a^2)=a

l=sqrt(170)

Use a calculator

l=13.0384048...

Round to 2 decimal place(s)

l≈ 13.04

Let's use the formula for the surface area of a sphere and for the surface area of a cone.

Solid	Top Part	Bottom Part
Radius	r= 7	r= 7
Slant	-	l= 13.04
Surface Area	A_\text{top}=\dfrac{1}{2}\cdot 4\pi {\color{#0000FF}{r}}^2=2\pi {\color{#0000FF}{r}}^2	A_\text{bottom}=\pi r {\color{#FF0000}{\ell}}
Surface Area	\textcolor{darkorange}{A_\text{top}}=2\pi ({\color{#0000FF}{7}})^2=\textcolor{darkorange}{98\pi}	\textcolor{darkviolet}{A_\text{bottom}}=\pi ({\color{#0000FF}{7}})({\color{#FF0000}{13.04}})=\textcolor{darkviolet}{91.28\pi}

Now, let's add the surface areas of the Top Part and the Bottom Part to find the area of the composite solid, A_\text{solid}. We will also round the answer to the nearest tenth as we are asked to do.

A_\text{solid}=\textcolor{darkorange}{A_\text{top}}+\textcolor{darkviolet}{A_\text{bottom}}

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Substitute values and evaluate

\textcolor{darkorange}{A_\text{top}}={\color{#0000FF}{\textcolor{darkorange}{98\pi}}}, \textcolor{darkviolet}{A_\text{bottom}}={\color{#009600}{\textcolor{darkviolet}{91.28\pi}}}

A_\text{solid}=\textcolor{darkorange}{98\pi}+\textcolor{darkviolet}{91.28\pi}

Add terms

A_\text{solid}=189.28\pi

Use a calculator

A_\text{solid}=594.640657\ldots

Round to 1 decimal place(s)

A_\text{solid}\approx 594.6

We found that the area of the spinning top is about 594.6 square centimeters.

Volume

The volume of the spinning top is equal to the volume of the hemisphere and the volume of the cone. Let's use the formula for the volume of a sphere and for the volume of a cone.

Solid	Top Part	Bottom Part
Radius	r= 7	r= 7
Height	-	h= 11
Volume	V_\text{top}=\dfrac{1}{2}\cdot\dfrac{4}{3}\pi {\color{#0000FF}{r}}^3=\dfrac{2}{3}\pi {\color{#0000FF}{r}}^3	V_\text{bottom}=\dfrac{1}{3}\pi {\color{#0000FF}{r}}^2{\color{#009600}{h}}
Volume	\textcolor{darkorange}{V_\text{top}}=\dfrac{2}{3}\pi ({\color{#0000FF}{7}})^3=\textcolor{darkorange}{\dfrac{686\pi}{3}}	\textcolor{darkviolet}{V_\text{bottom}}=\dfrac{1}{3}\pi ({\color{#0000FF}{{\color{#0000FF}{7}}}})^2({\color{#009600}{11}})=\textcolor{darkviolet}{\dfrac{539\pi}{3}}

Now, let's add the volumes of the Top Part and Bottom Part to find the volume of the composite solid, V_\text{solid}. Just like before, we will round the volume of the solid to the nearest tenth.

V_\text{solid}=\textcolor{darkorange}{V_\text{top}}+\textcolor{darkviolet}{V_\text{bottom}}

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Substitute values and evaluate

\textcolor{darkorange}{V_\text{top}}={\color{#0000FF}{\textcolor{darkorange}{\dfrac{686\pi}{3}}}}, \textcolor{darkviolet}{V_\text{bottom}}={\color{#009600}{\textcolor{darkviolet}{\dfrac{539\pi}{3}}}}

V_\text{solid}=\textcolor{darkorange}{\dfrac{686\pi}{3}}+\textcolor{darkviolet}{\dfrac{539\pi}{3}}

AddFrac

Add fractions

V_\text{solid}=\dfrac{1225\pi}{3}

Use a calculator

V_\text{solid}=1282.817000\ldots

Round to 1 decimal place(s)

V_\text{solid}\approx 1282.8

We found that the volume of the spinning top is about 1282.8 cubic centimeters.

The manufacturer of the top makes a model with dimensions that are one-half of the dimensions of the top from Part A. Let's analyze the new model.

It consists of two parts.

Top Part: a hemisphere with a radius of r= 72= 3.5 centimeters.
Bottom Part: a right cone with a radius of r= 3.5 centimeters and a height of h= 5.5 centimeters.

We are asked to find the surface area and the volume of the composite solid.

Surface Area

The surface area of the spinning top is equal to the area of the hemisphere and the lateral area of the cone. First, let's use the Pythagorean Theorem for the right triangle to find the slant height of the cone, l.

r^2+ h^2= l^2

r= 3.5, h= 5.5

3.5^2+ 5.5^2= l^2

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Solve for l

CalcPow

Calculate power

12.25+30.25= l^2

Add terms

42.5= l^2

RearrangeEqn

Rearrange equation

l^2=42.5

SqrtEqn

sqrt(LHS)=sqrt(RHS)

sqrt(l^2)=sqrt(42.5)

SqrtPowToNumber

sqrt(a^2)=a

l=sqrt(42.5)

Use a calculator

l=6.519202...

Round to 2 decimal place(s)

l≈ 6.52

Let's use the formula for the surface area of a sphere and for the surface area of a cone.

Solid	Top Part	Bottom Part
Radius	r= 3.5	r= 3.5
Slant	-	l= 6.52
Surface Area	A_\text{top}=\dfrac{1}{2}\cdot 4\pi {\color{#0000FF}{r}}^2=2\pi {\color{#0000FF}{r}}^2	A_\text{bottom}=\pi r {\color{#FF0000}{\ell}}
Surface Area	\textcolor{darkorange}{A_\text{top}}=2\pi ({\color{#0000FF}{3.5}})^2=\textcolor{darkorange}{24.5\pi}	\textcolor{darkviolet}{A_\text{bottom}}=\pi ({\color{#0000FF}{3.5}})({\color{#FF0000}{6.52}})=\textcolor{darkviolet}{22.82\pi}

Now, let's add the surface areas of Top Part and of Bottom Part to find the area of the composite solid, A_\text{solid}. We will also round the answer to the nearest tenth as we are asked to do.

A_\text{solid}=\textcolor{darkorange}{A_\text{top}}+\textcolor{darkviolet}{A_\text{bottom}}

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Substitute values and evaluate

\textcolor{darkorange}{A_\text{top}}={\color{#0000FF}{\textcolor{darkorange}{24.5\pi}}}, \textcolor{darkviolet}{A_\text{bottom}}={\color{#009600}{\textcolor{darkviolet}{22.82\pi}}}

A_\text{solid}=\textcolor{darkorange}{24.5\pi}+\textcolor{darkviolet}{22.82\pi}

Add terms

A_\text{solid}=47.32\pi

Use a calculator

A_\text{solid}=148.66016\ldots

Round to 1 decimal place(s)

A_\text{solid}\approx 148.7

We found that the area of the spinning top is about 148.7 square centimeters.

Volume

The volume of the spinning top is equal to the volume of the hemisphere and the volume of the cone. Let's use the formula for the volume of a sphere and for the volume of a cone.

Solid	Top Part	Bottom Part
Radius	r= 3.5	r= 3.5
Height	-	h= 5.5
Volume	V_\text{top}=\dfrac{1}{2}\cdot\dfrac{4}{3}\pi {\color{#0000FF}{r}}^3=\dfrac{2}{3}\pi {\color{#0000FF}{r}}^3	V_\text{bottom}=\dfrac{1}{3}\pi {\color{#0000FF}{r}}^2{\color{#009600}{h}}
Volume	\textcolor{darkorange}{V_\text{top}}=\dfrac{2}{3}\pi ({\color{#0000FF}{3.5}})^3=\textcolor{darkorange}{28.58\pi}	\textcolor{darkviolet}{V_\text{bottom}}=\dfrac{1}{3}\pi ({\color{#0000FF}{{\color{#0000FF}{3.5}}}})^2({\color{#009600}{5.5}})=\textcolor{darkviolet}{22.46\pi}

Now, let's add the volumes of Top Part and of Bottom Part to find the volume of the composite solid, V_\text{solid}. Just like before, we will round the volume of the solid to the nearest tenth.

V_\text{solid}=\textcolor{darkorange}{V_\text{top}}+\textcolor{darkviolet}{V_\text{bottom}}

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Substitute values and evaluate

\textcolor{darkorange}{V_\text{top}}={\color{#0000FF}{\textcolor{darkorange}{28.58\pi}}}, \textcolor{darkviolet}{V_\text{bottom}}={\color{#009600}{\textcolor{darkviolet}{22.46\pi}}}

V_\text{solid}=\textcolor{darkorange}{28.58\pi}+\textcolor{darkviolet}{22.46\pi}

Add terms

V_\text{solid}=51.04\pi

Use a calculator

V_\text{solid}=160.346889\ldots