How do you setup the equation for this?
Thanky ou!
Find the ratio of the surface area of a cylinder to the volume of the cylinder if the height remains at 10 cm. Simplify this ratio and let f(x) = the simplified ratio. Let r be the radius of the cylinder.
a) Find the average rate of change as the radius increases from 2 to 5 cm.
b) Is the ratio increasing of decreasing? justify answer based on part a.
It is not clear if the surface area includes the top and bottom. I will assume not
then
area = 2 pi r h
volume = pi r^2 h
ratio = 2 /r
well that is too easy, try including top and bottom
area = 2 pi r h + 2 pi r^2
volume = pi r^2 h
ratio = 2/r + 2/h
= 2/r + .2 if h = 10
To set up the equation for finding the ratio of the surface area of a cylinder to its volume, we need to recall the formulas for surface area and volume of a cylinder.
The formula for the surface area (S) of a cylinder is given by:
S = 2πrh + 2πr^2
The formula for the volume (V) of a cylinder is given by:
V = πr^2h
In the given scenario, the height remains constant at 10 cm, and we need to find the ratio of the surface area to the volume. Let's denote this ratio as f(x).
So, the surface area of the cylinder with height 10 cm can be written as:
S = 2π(10)(x) + 2πx^2
Simplifying, we get: S = 20πx + 2πx^2
And the volume of the cylinder with height 10 cm can be written as:
V = πx^2(10)
Simplifying, we get: V = 10πx^2
Now, to find the simplified ratio f(x), we divide the surface area (S) by the volume (V):
f(x) = (20πx + 2πx^2) / (10πx^2)
Simplifying this expression, we get:
f(x) = (2x + x^2) / x^2
Now, let's move on to part (a) and (b) of the question.
a) To find the average rate of change as the radius increases from 2 to 5 cm, we need to evaluate the difference quotient for f(x). The average rate of change is the difference in the values of f(x) divided by the difference in the values of x.
Using the formula for f(x) = (2x + x^2) / x^2, we can calculate the average rate of change:
(f(5) - f(2)) / (5 - 2)
Substituting the values into the formula, we get:
(f(5) = (2(5) + (5)^2) / (5)^2 = 27/5
(f(2) = (2(2) + (2)^2) / (2)^2 = 3/2
Calculating further, we get:
(27/5 - 3/2) / (5 - 2) = 11/10
Therefore, the average rate of change as the radius increases from 2 to 5 cm is 11/10.
b) To determine if the ratio f(x) is increasing or decreasing, we analyze the average rate of change calculated in part (a). If the average rate of change is positive, it means that the ratio is increasing. Likewise, if the average rate of change is negative, the ratio is decreasing.
In our case, the average rate of change is 11/10, which is positive. Hence, the ratio f(x) is increasing.
To summarize:
The equation for the ratio of the surface area to the volume is f(x) = (2x + x^2) / x^2.
The average rate of change as the radius increases from 2 to 5 cm is 11/10.
The ratio f(x) is increasing based on the positive average rate of change.