Chapter 12.4, Problem 19PSC

To determine

To calculate: The volume and total surface area of the wedge cut from cylindrical block.

Answer to Problem 19PSC

The volume is $2094.35$ and total surface area is $1088.69$ .

Explanation of Solution

Given information:

Radius of circle r = 20,

Measure of arc $={40}^{\circ }$ ,

Height h $=15$

Formula used:

Volume of prism $=B\times h$ = Area of sector $\times h$

B = Base area of prism and h = height of prism

Area of sector $=\left(\frac{measureofarc}{360^o}\right)\pi r^2$

r = radius of circle

Surface area of cylinder = Area of sector $\times 2\pi rh$

r = radius of cylinder

h = height of cylinder

Area of rectangle: $A=l\times w$

l = length of rectangle

w= width of rectangle

Calculation:

r = 20,

Measure of arc $={40}^{\circ }$ ,

h $=15$ .

  

Volume of prism $=B\times h$

Volume of prism = Area of sector $\times h$

Volume of prism $=\left(\frac{{40}^{\circ }}{360^o}\right)\pi {\left(20\right)}^2\times 15$

Volume of prism $=\frac{2000}{3}\pi$

Volume of prism= $2094.35$

Total area = Surface area of cylinder + 2 (Area of sector + Area of rectangle)

Total area $=\left(\frac{measureofarc}{360^o}\right)2\pi rh$ + $2\left[\left(\frac{measureofarc}{360^o}\right)\pi r^2+\left(b\times h\right)\right]$

Total area $=\left(\frac{1}{9}\right)40\pi \times 15$$+2\left[\left(\frac{1}{9}\right)400\pi +\left(15\times 20\right)\right]$

Total area $=\frac{600}{9}\pi +\frac{800}{9}\pi +600$

Total area $=\frac{1400}{9}\pi +600$

Total area $=1088.69$