Campbell’s Soup Company requires that is its tomato soup containers have a capacity of 64 in^3 , have the shape of a circular cylinder, and be made of aluminum. Determine the radius and height of the container that requires the least amount of metal.
v = πr^2h = 64, so h = 64/(πr^2)
a = 2πr^2 + 2πrh
= 2πr^2 + 128/r
da/dr = 4πr - 128/r^2
= (4πr^3-128)/r^2
Now set da/dr=0 to find minimum area.
To determine the radius and height of the container that requires the least amount of metal, we need to find the dimensions that minimize the surface area of the cylinder.
The surface area of a cylinder can be determined using the formula: A = 2πr² + 2πrh, where r is the radius and h is the height.
To find the least amount of metal required, we need to find the minimum value of this surface area function.
Step 1: Find the volume of the cylinder.
The volume of the cylinder is given as 64 in³. The formula for the volume of a cylinder is V = πr²h. Substituting the given volume, we get 64 = πr²h.
Step 2: Simplify the equation to express height in terms of radius.
Rearranging the above equation, we get h = 64 / (πr²).
Step 3: Substitute the expression for height in terms of radius into the surface area formula.
Substituting h = 64 / (πr²) into the surface area formula A = 2πr² + 2πrh, we get:
A = 2πr² + 2πr (64 / (πr²)).
Simplifying this, we get A = 2πr² + 128 / r.
Step 4: Find the derivative of the surface area function.
To find the minimum value, we need to find where the derivative of the surface area function is equal to zero. Differentiating A with respect to r, we get:
dA/dr = 4πr - 128 / r².
Step 5: Set the derivative equal to zero and solve for r.
Setting dA/dr = 0, we get:
4πr - 128 / r² = 0. Multiplying through by r², we have:
4πr³ - 128 = 0.
Simplifying further, we get:
r³ = 32 / π.
Taking the cube root of both sides gives us:
r = (32 / π)^(1/3).
Step 6: Find the corresponding height.
Substituting the value of r into the equation for height in terms of radius (h = 64 / (πr²)), we can determine the corresponding height.
That's how you can find the dimensions of the container that requires the least amount of metal.