A right circular cylinder is to be designed to hold 750 cm3 of processed milk, and to use a minimal amount of material in its construction. Find the dimensions for the container.

To find the dimensions of the container, we need to consider the volume of a right circular cylinder and minimize the amount of material used in its construction.

The volume V of a right circular cylinder is given by the formula V = πr^2h, where r is the radius of the base and h is the height of the cylinder.

In this case, we are given that the volume of the cylinder should be 750 cm^3. So we have the equation:

750 = πr^2h

To minimize the amount of material used, we can attempt to minimize the surface area of the cylinder. The surface area A of a right circular cylinder is given by the formula A = 2πrh + 2πr^2.

We can use the equation for the volume to isolate one of the variables, and then substitute it into the equation for the surface area to get a single variable equation.

From the equation for the volume:
750 = πr^2h

We can isolate h:
h = 750 / (πr^2)

Now, substitute this value of h into the equation for the surface area:
A = 2πr(750 / (πr^2)) + 2πr^2

Simplifying the equation:
A = (1500 / r) + 2πr^2

To minimize the surface area, we need to find the critical points of this equation. We can do this by taking the derivative of A with respect to r, setting it equal to zero, and solving for r.

dA/dr = -1500/r^2 + 4πr

Setting dA/dr = 0 and solving for r:
0 = -1500/r^2 + 4πr
1500/r^2 = 4πr
1500 = 4πr^3
r^3 = 375/π
r ≈ 4.26 cm

Now, substitute this value of r back into the equation for h to find the height:
h = 750 / (π(4.26)^2)
h ≈ 13.20 cm

Therefore, the dimensions of the container that will hold 750 cm^3 of processed milk while using a minimal amount of material are:
Radius ≈ 4.26 cm
Height ≈ 13.20 cm