find the radius and height of a cylindrical soda can with a volume of 21in^3 that minimize the surface area.
radius -- r
height --- h
given: πr^2 h = 21
h = 21/(πr^2)
Surface area (SA) = 2πr^2 + 2πrh
= 2πr^2 + 2πr(21/(πr^2)
= 2πr^2 + 42/r
d(SA)/dr = 4πr - 42/r^2
= 0 for a max/min of SA
4πr = 42/r^2
r^3 = 42/(4π)
r = (42/(4π)^(1/3) = 1.495163.. or appr 1.5
then h = 21/(πr^2) = 2.99027 or appr 3
notice the h = 2r
To find the radius and height of a cylindrical soda can with a volume of 21in^3 that minimizes the surface area, we can use calculus and optimization techniques.
Let's start by establishing the formulas for the volume and surface area of a cylinder:
Volume of a cylinder = π * r^2 * h
Surface Area of a cylinder = 2 * π * r^2 + 2 * π * r * h
Given that the volume of the soda can is 21in^3, we can write the equation for the volume:
π * r^2 * h = 21
To minimize the surface area, we need to minimize the following equation:
2 * π * r^2 + 2 * π * r * h
To simplify the problem, we can solve the volume equation for h in terms of r and substitute it into the surface area equation.
First, rearrange the volume equation:
h = 21 / (π * r^2)
Substitute this expression for h into the surface area equation:
2 * π * r^2 + 2 * π * r * (21 / (π * r^2))
Simplifying further:
2 * π * r^2 + (42 / r)
Now, we have a simplified expression for the surface area in terms of r only. To find the minimum surface area, we need to take the derivative of this expression with respect to r and set it equal to zero.
d(surface area) / d(r) = 4πr - 42/r^2
Setting the derivative equal to zero:
4πr - 42/r^2 = 0
Multiplying through by r^2:
4πr^3 - 42 = 0
Therefore, we have:
4πr^3 = 42
Dividing by 4π:
r^3 = 42 / (4π)
Simplifying further:
r^3 = 42 / (2π)
To solve for r, calculate the cube root of both sides:
r = (42 / (2π))^(1/3)
Now, we can substitute the value of r back into the equation for h:
h = 21 / (π * r^2)
h = 21 / (π * ((42 / (2π))^(1/3))^2)
Simplifying further:
h = 21 / (π * (42 / (2π))^(2/3))
Finally, calculate the value of h using a calculator.
By solving these equations, you will find the values of the radius and height that minimize the surface area of the cylindrical soda can with a volume of 21in^3.