A cylindrical tank is to have a capacity of 1000 m³. It is to fit into a foundry that is 12 m wide with a height of 11 m.

The base of the tank will cost half as much as the top. The metal for the side of the tank will cost four fifths as much as the top.

An equation that can be used to model the cost of the tank is C(r) = (3/2) pi(r)^2 + 1600/r. Determine what radius value will give a minimum cost. You must prove this value gives a minimum.

well, we know that thee will be a min or max when dC/dr = 0

C' = 3πr - 1600/r^2 = (3πr^3-1600)/r^2
C'=0 when r = ∛(1600/(3π)) ≈ 5.537
Now just check the sign of C" to see whether it's a min or a max

To determine the radius value that will give a minimum cost, we need to find the derivative of the cost function and set it equal to zero. Let's go through the steps:

1. The cost function is given as C(r) = (3/2)πr^2 + 1600/r, where r is the radius of the cylindrical tank.

2. To find the derivative of C(r), we can differentiate each term separately. The derivative of (3/2)πr^2 with respect to r is (3/2)π(2r), which simplifies to 3πr. The derivative of 1600/r with respect to r is -1600/r^2.

3. Now we can set the derivative equal to zero and solve for r:
3πr - 1600/r^2 = 0

4. To simplify the equation, we can multiply both sides by r^2 to get rid of the fraction:
3πr^3 - 1600 = 0

5. Rearrange the equation to isolate r:
3πr^3 = 1600
r^3 = 1600 / (3π)
r = (1600 / (3π))^(1/3)

6. Using a calculator, evaluate the right-hand side of the equation to find the numerical value of r.

To prove that the value of r we found gives a minimum, we need to perform a second derivative test. This involves taking the second derivative of the cost function and examining its value at the critical point.

1. The second derivative of C(r) can be found by differentiating the derivative we found earlier. The derivative of 3πr is 3π, and the derivative of -1600/r^2 is 3200/r^3.

2. Combine the derivatives to get the second derivative:
C''(r) = 3π + 3200/r^3

3. Substitute the critical point value of r into the second derivative:
C''((1600 / (3π))^(1/3)) = 3π + 3200 / ((1600 / (3π))^(1/3))^3

4. Evaluate this expression using a calculator.

If the second derivative is positive, it means the cost function has a minimum at the critical point. If it is negative, the cost function has a maximum. If the second derivative is zero, the test is inconclusive and further analysis is needed.

By following these steps, you can determine the radius value that will give the minimum cost for the cylindrical tank and prove that it is indeed a minimum.