If there is a cylinder that has a volume of 100 ft, what would be the DIMENSIONS of the cylinder to have to LEAST surface area?
Mackenzie, this problem is way too advanced for you. I never had this type of problem until I took a course in Calculus. I don't understand why your teacher assigned it, perhaps your parents can ask that.
OK! It was in our book but my teacher did say it would be a challange. Can you just explain it one more time?!
I cant explain it any better for you than I did. It is a calculus question.
The alternative, graphing, requires a graphing calculator and knowledge how to use it.
Sorry.
Alight! Thanks anyway!!!! I kind of get it but do you think any one else can explain it?
To determine the dimensions of a cylinder with the least surface area for a given volume, we need to use calculus and optimization techniques.
Let's denote the radius of the cylinder as "r" and the height of the cylinder as "h". The formula for the volume of a cylinder is V = πr^2h, where π is a constant.
In this case, we are given that the volume (V) is 100 ft^3. Therefore, we have the equation:
100 = πr^2h (Equation 1)
The surface area (A) of a cylinder is given by the formula:
A = 2πr^2 + 2πrh (Equation 2)
Our goal is to minimize the surface area (A) while maintaining the given volume (V). To do this, we need to express Equation 2 in terms of a single variable, either r or h, using Equation 1.
Rearranging Equation 1 to solve for h, we have:
h = 100 / (πr^2) (Equation 3)
Substituting Equation 3 into Equation 2, we get:
A = 2πr^2 + 2πr * (100 / (πr^2))
Simplifying further, we have:
A = 2πr^2 + 200 / r
Now, since we want to find the dimensions of the cylinder that result in the least surface area, we need to find the critical points of A. This can be done by taking the derivative of A with respect to r, setting it equal to zero, and solving for r.
dA/dr = 4πr - 200 / r^2 = 0
4πr = 200 / r^2
r^3 = 50 / π
r ≈ (50 / π)^(1/3)
Once you determine the value of r, substitute it into Equation 3 to find the corresponding value of h. These values of r and h will give you the dimensions of the cylinder with the least surface area for a given volume of 100 ft^3.