Chapter 12, Problem 20RP

To determine

To calculate: The volume and total surface area of right cylindrical piece.

Answer to Problem 20RP

The volume is $271.5$ and total surface area is $632.25$ .

Explanation of Solution

Given information:

A rectangle with dimensions 10 and 30. Measure of arc is ${60}^{\circ }$ .

Formula used:

Area of equilateral triangle: $A=\frac{s^2}{4}\sqrt{3}$

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

r = radius of circle

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

l = length of rectangle

w= width of rectangle Area of cylinder $=C\times h$

C = Circumference of cylinder and h = height of cylinder

Volume of prism $=B\times h$

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

Calculation:

  

To find B , find the area of the shaded segment. The inscribed triangle will be equilateral, so the radii will be 10 because all sides of an equilateral triangles are congruent.

  $\begin{array}{l}{A}_{sector}=\left(\frac{measureofarc}{360^o}\right)\pi r^2\\ =(\frac{60^o}{360^o})\pi {(}^1\\ =\frac{1}{6}\times 100\pi \\ =\frac{50}{3}\pi \end{array}$

  $\begin{array}{l}{A}_{Equilateraltriangle}=\frac{s^2}{4}\sqrt{3}\\ A_{Equilateraltriangle}=\frac{{(10)}^2}{4}\sqrt{3}\\ A_{Equilateraltriangle}=\frac{100}{4}\sqrt{3}\\ A_{Equilateraltriangle}=25\sqrt{3}\\ B=A_{sector}-A_{Equilateraltriangle}=\frac{50}{3}\pi -25\sqrt{3}=52.35-43.30=9.05\end{array}$

Area of rectangle $=l\times w=10\times 30=300$

Lateral area of cylinder $=C\times h$

  $\begin{array}{l}C=2\pi r=2\pi \times 10=20\pi \\ h=30\end{array}$

Lateral area of cylinder $=20\pi \times 30=600\pi =1884.95$

Total Area = 2B + Area of rectangle + $\left(\frac{measureofarc}{360^o}\right)\times$ Lateral area of cylinder

Total Area $=\left(2\left(9.05\right)\right)+300+\left(\left(\frac{{60}^{\circ }}{{360}^{\circ }}\right)\times 1884.95\right)$

Total Area $=18.1+300+314.15$

Total Area $=632.25$

Volume of prism $=B\times h$

  $\begin{array}{l}B=9.05,\\ h=30.\end{array}$

Volume of prism $=9.05\times 30$

Volume of prism $=271.5$