Final answer:
To minimize the cost of constructing the cylindrical drum, we need to minimize its surface area. We can do this by finding the dimensions that satisfy the given volume while minimizing the surface area. Using mathematical formulas and optimization techniques, we find that the drum should have a radius of approximately 5.02 yards and a height of 30 yards to be constructed at minimum cost.
Step-by-step explanation:
To minimize the cost of constructing the cylindrical drum, we must consider the areas of the top, bottom, and side wall of the drum. Let's assume the radius of the drum is r and the height is h.
The volume of the drum is given as 954 cubic yards, which can be expressed as a formula:
V = πr^2h = 954
To minimize the cost, we need to minimize the surface area, which is the sum of the top, bottom, and side wall areas:
S = A(top) + A(bottom) + A(side wall)
Note that the top and bottom of the drum have the same area, as they both have the same radius:
A(top) = A(bottom) = πr^2
The side wall of the drum can be represented as a rectangle, whose length is the circumference of the top/bottom and height is the height of the drum:
A(side wall) = 2πrh
Combining these formulas, we can express the surface area in terms of r and h:
S = 2(πr^2) + 2πrh
Now, we need to express h in terms of r using the volume formula:
πr^2h = 954
h = 954 / (πr^2)
Substituting this value of h into the surface area formula, we get:
S = 2(πr^2) + 2πr(954 / (πr^2))
Simplifying the equation gives:
S = 2(πr^2) + 1908 / r
To minimize the cost, we can take the derivative of S with respect to r:
dS/dr = 4πr - 1908 / r^2
Setting this derivative equal to zero and solving for r gives:
4πr - 1908 / r^2 = 0
4πr^3 - 1908 = 0
r^3 = 1908 / (4π)
r ≈ 5.02 yards
Now, substituting this value of r back into the volume formula, we can solve for h:
π(5.02)^2h = 954
h ≈ 30 yards
Therefore, the dimensions that will minimize the cost of constructing the cylindrical drum are approximately r = 5.02 yards and h = 30 yards.