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Word Lesson: Volume and Surface Area of Cylinders
In order to solve problems which require application of the volume and surface area for cylinders, it is necessary to
 
 
Volume: V = pr2h
 
Surface Area: S = 2pr2 + 2prh
 

 
 
 
Notice that these factored versions of the surface area formula are sometimes easier to use:
 
S = 2pr2 + 2prh = 2p(r2 + rh) = 2pr(r + h)
 
 
A typical problem involving the volume or surface area of a cylinder gives us the volume or height and/or radius of the cylinder. We then need to calculate the unknown quantities based on the information given about the others.
 
Suppose that the height of a cylinder is 30 cm and its volume is 750p cm3. Find its radius and surface area.
 
To get started, we need to organize as much of the given information as possible into a known formula. Since the volume (750p cm3) and height (30 cm) of the cylinder are given, we will start with the equation for volume.
 
V = pr2h = 30pr 2
V = 750p
750p = 30pr2
r = 5 cm
 
Now we will find the surface area of the cylinder using our values for the radius and height.
 
S = 2pr2 + 2prh
S = 2p(25) + 2p(5)(30)
S = 50p + 300 p
S = 350p cm2

Examples
Example Trumpet The height of a cylinder is 9 inches and the volume is 144p cubic inches. Find the radius and surface area of this cylinder.
What is your answer?
 
Example Trumpet In a cylinder, r = 7 and surface area S = 210p. Find the height and volume of the cylinder.
What is your answer?
 
Example Trumpet A cylinder has a height of 10 and a surface area of 78p. Find the radius and volume of this cylinder.
What is your answer?
 

Examples
Example Trumpet A cylinder has a radius of 8 inches and a height of 12 inches. What are the volume and surface area of this cylinder?
  1. V = 768p and S = 320p
  2. V = 96p and S = 208p
  3. V = 1152p and S = 480p
What is your answer?
 
Example Trumpet A cylinder has a height of 11 and a volume of V = 275p. What are the radius and surface area of this cylinder?
  1. r = 25 and S = 1850p
  2. r = 5 and S = 160p
  3. r = 5 and S = 352p
What is your answer?
 
Example Trumpet A cylinder has a height of 7 and a surface area S = 88p. What are the radius and volume of this cylinder?
  1. r = 4 and V = 112p
  2. r = 11 and V = 847p
  3. r 12.57 and V 1106.29p
What is your answer?
 

This type of word problem involves the use of known formulas. Usually we used one of the formulas for either volume or surface area, depending on what information was given, to solve for the radius or height of the cylinder. In one case, when the height and surface area are given, we were required to solve a quadratic equation for the value of r. If that expression had not factored easily, we would have had to used the quadratic formula.
 
Once values of both r and h were known, the volume and surface area formulas were used. These problems require careful arithmetic since it is often difficult to determine whether or not our answers are reasonable. Notice that the surface area formula can be written in more than one way. Using more than one of these can serve as a way to check answers or make a calculation easier.
M Ransom

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