Chapter 12.3, Problem 7PSB

a.

To determine

To calculate: The total area of cone with sides of triangles as 12 and 13.

Answer to Problem 7PSB

The total area of cone is $90\pi$ .

Explanation of Solution

Given information:

Height of cone AB = 12,

Slant height of cone AC = 13.

Formula used:

Lateral Area of cone $=\frac{1}{2}\times C\times l$

C = Circumference of cone.

l = length of cone.

Circumference of cone:

  $C=2\pi r$

r = radius of cone

The below theorem is used:

Pythagoras theorem states that “In a right angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides”.

  

In right angle triangle,

  $a^2+b^2\text{= }{c}^2$

Area of a circle: $A=\pi r^2$

r = radius of circle

Calculation:

  

The radius r can be calculated by applying Pythagoras Theorem.

In right angled triangle ABC , we get

  $\begin{array}{l}{\left(AC\right)}^2={\left(AB\right)}^2+{\left(BC\right)}^2\\ {\left(13\right)}^2={\left(12\right)}^2+{\left(r\right)}^2\\ r^2=169-144\\ r^2=25\\ r=\sqrt{25}=5\end{array}$

Lateral Area of cone $=\frac{1}{2}\times C\times l$

Circumference of cone:

  $C=2\pi r=2\pi \times 5=10\pi$

Lateral Area of cone $=\frac{1}{2}\times 10\pi \times 13$

Lateral Area of cone $=65\pi$

Area of base: $A=\pi r^2=\pi {\left(5\right)}^2=25\pi$

Total Area = Lateral Area + Area of base

Total Area $=65\pi +25\pi$

Total Area $=90\pi$

b.

To determine

To find: The total area of cylinder with height of 8 and slant height of 10.

Answer to Problem 7PSB

The total area of cylinder is $66\pi$ .

Explanation of Solution

Given information:

Height of cylinder PQ = 8,

Slant height of cylinder PR = 10.

Formula used:

Lateral Area of cylinder $=C\times h$

C = Circumference of cylinder.

h = height of cylinder.

Circumference of cylinder:

  $C=2\pi r$

r = radius of cylinder

The below theorem is used:

Pythagoras theorem states that “In a right angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides”.

  

In right angle triangle,

  $a^2+b^2\text{= }{c}^2$

Area of a circle: $A=\pi r^2$

r = radius of circle

Calculation:

  

The radius r can be calculated by applying Pythagoras Theorem.

In right angled triangle PQR , we get

  $\begin{array}{l}{\left(PR\right)}^2={\left(PQ\right)}^2+{\left(QR\right)}^2\\ {\left(10\right)}^2={\left(8\right)}^2+{\left(QR\right)}^2\\ {\left(QR\right)}^2=100-64\\ {\left(QR\right)}^2=36\\ QR=\sqrt{36}=6\\ r=\frac{QR}{2}=\frac{6}{2}=3\end{array}$

Lateral Area of cylinder $=C\times h$

Circumference of cylinder:

  $C=2\pi r=2\pi \times 3=6\pi$

Lateral Area of cylinder $=6\pi \times 8$

Lateral Area of cylinder $=48\pi$

Area of base: $A=\pi r^2=\pi {\left(3\right)}^2=9\pi$

Total Area = Lateral Area +2( Area of base)

Total Area $=48\pi +\left(2\times 9\pi \right)$

Total Area $=48\pi +18\pi$

Total Area $=66\pi$

c.

To determine

To find: The total area of cone with height as 8 and radius as 15.

Answer to Problem 7PSB

The area of cone is $480\pi$ .

Explanation of Solution

Given information:

Height of cone XZ = 8,

Radius of cone r = 15.

Formula used:

The below theorems are used:

Two tangent theorem states that if two tangent segments are drawn to one circle from the same external point, then they are congruent.

Pythagoras theorem states that “In a right angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides”.

  

In right angle triangle,

  $a^2+b^2\text{= }{c}^2$

Area of triangle: $A=\frac{1}{2}\times b\times h$

b = base of triangle

h = height of triangle Area of a circle: $A=\pi r^2$

r = radius of circle

Calculation:

  

The side XY can be calculated by applying Pythagoras Theorem.

In right angled triangle XZY , we get

  $\begin{array}{l}{\left(XY\right)}^2={\left(XZ\right)}^2+{\left(ZY\right)}^2\\ {\left(XY\right)}^2={\left(8\right)}^2+{\left(15\right)}^2\\ {\left(XY\right)}^2=64+225\\ {\left(XY\right)}^2=289\\ XY=\sqrt{289}=17\end{array}$

Lateral Area of cone $=\frac{1}{2}\times C\times l$

Circumference of cone:

  $C=2\pi r=2\pi \times 15=30\pi$

Lateral Area of cone $=\frac{1}{2}\times 30\pi \times 13$

Lateral Area of cone $=255\pi$

Area of base: $A=\pi r^2=\pi {\left(15\right)}^2=225\pi$

Total Area = Lateral Area + Area of base

Total Area $=255\pi +225\pi$

Total Area $=480\pi$