A cylindrical can is to be made to hold 1 L (1000 cm3 ) of oil. Find the dimensions of the radius and height that will minimize
the cost of the metal to manufacture the can. [Hint: Minimize A = 2πr2 + 2πrh subject to a constraint on the volume.]
1 answer:
Answer:
r=5.42cm
h=10.83cm
Step-by-step explanation:
Hello!
To solve this problem please use the following steps, the complete procedure is attached.
1. Raise the equation for the volume of a cylinder and solve for the height (H)
2. Use the equation for the surface area of a cylinder, and replace the height using the equation found in step 1.
3. Derive the equation with respect to the radius and equal it to zero, then solve for the radius of the cylinder. Remember that the objective is to find that with the minimum surface area, the required volume is generated.
4. Once the radius is found, find the height with the equation from step 1
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Step-by-step explanation:
The function is infinitely differentiable, so it satisfies a infinite number of differential equations. The required answer depends on your previous part, so I will describe a general procedure to obtain the equations.
Using rules of differentiation, we obtain that . Differentiate again to obtain,
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This can repeated infinitely, so it is possible to obtain a differential equation of order n. The key is to differentiate the required number of times and write the equation in terms of x.