Final answer:
To design a one-liter oil can shaped like a right circular cylinder, we need to determine the dimensions that will use the least material. We can find these dimensions by minimizing the surface area of the cylinder using calculus techniques. The dimensions that use the least material are approximately r ≈ 4.24 cm and h ≈ 13.24 cm.
Step-by-step explanation:
To design a one-liter oil can shaped like a right circular cylinder, we need to determine the dimensions that will use the least material. The volume of the cylinder can be calculated using the formula V = πr²h, where V is the volume, r is the radius, and h is the height.
Since we want the can to hold one liter, which is equivalent to 1000 cm³, we can set up the equation 1000 = πr²h. Now, we need to minimize the surface area of the cylinder, which is given by the formula A = 2πrh + 2πr². To find the dimensions that minimize the surface area, we need to use calculus techniques.
First, we need to express the surface area in terms of a single variable. Since we have two variables, r and h, we can express h in terms of r using the equation 1000 = πr²h. Solving for h, we get h = 1000 / (πr²).
Next, substitute the expression for h into the formula for the surface area. A = 2πr(1000 / (πr²)) + 2πr². Simplifying, we get A = 2000/r + 2πr².
To find the dimensions that minimize the surface area, we need to find where the derivative of A with respect to r equals zero. Differentiating A with respect to r, we get dA/dr = -2000/r² + 4πr.
Setting dA/dr equal to zero, we get -2000/r² + 4πr = 0. Multiply both sides by r², we get -2000 + 4πr³ = 0. Solving for r, we get r = (∛(2000 / (4π))).
Since we want to minimize the surface area, we need to find the value of h that corresponds to this value of r. Substituting r into the equation h = 1000 / (πr²), we get h = 1000 / (π(∛(2000 / (4π)))).
Therefore, the dimensions of the oil can that use the least material to hold one liter are approximately r ≈ 4.24 cm and h ≈ 13.24 cm.