A cylindrical can with no top is to be made to hold 1 liter of liquid. Find the dimensions (radius and height) of the can that will minimize the surface area.
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To minimize the surface area, we need to find the dimensions that minimize the lateral area of the cylindrical can.
Let's denote the radius of the can by r and the height of the can by h.
The lateral area of a cylindrical can is given by the formula A = 2πrh, where r is the radius and h is the height.
We are given that the can needs to hold 1 liter of liquid, which is equivalent to 1000 cm³.
The volume of a cylindrical can is given by the formula V = πr²h. Since the can needs to hold 1 liter of liquid, we have V = πr²h = 1000 cm³.
We can solve for h in terms of r by rearranging the volume equation: h = 1000 / (πr²).
Substituting this expression for h into the lateral area formula, we get A = 2πr(1000 / (πr²)) = 2000 / r.
To find the dimensions that minimize the surface area, we need to find the value of r that minimizes the expression A = 2000 / r.
We can do this by taking the derivative of A with respect to r and setting it equal to 0:
dA/dr = -2000 / r² = 0.
Solving this equation, we find that r² = 2000. Taking the square root of both sides, we get r = √2000 ≈ 44.72 cm.
Substituting this value of r back into the equation for h = 1000 / (πr²), we find h ≈ 1000 / (π(2000)) ≈ 0.159 cm.
Therefore, the dimensions that will minimize the surface area of the can are approximately r = 44.72 cm and h ≈ 0.159 cm.
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