a company manufactures large cylindrical drums.the bottom and sides are made from a metal that costs $4.00 a square foot, while the reinforced lid costs $6.00 a square foot. ind thedmensions ofa drm that hasa volume of 10cubic feet and minizes the total cost
i mean find the dmensions of a drum that has a volume of 10 cubic feet and minizes the total cost
let the radius be r ft and the height h ft
πr^2h = 10
h = 10/(πr^2)
cost = 6πr^2 + 4πr^2 + 4(2πrh)
= 10πr§2 + 8πr(10/(πr^2))
= 10πr^2 + 80/r
d(cost)/dr = 20πr - 80/r^2 = 0 for max/min of cost
I got r = appr. 1.084 , h = 2.71
To find the dimensions of the drum that minimize the total cost, we need to determine the relationships between the cost, volume, and dimensions of the drum.
Let's assume the height of the drum is "h" and the radius of the base is "r". Since it is cylindrical, we can find the volume using the formula:
Volume of a cylinder = π * r^2 * h
Given that the volume of the drum is 10 cubic feet, we can use this equation:
10 = π * r^2 * h
Now, let's determine the surface area of the bottom and sides, which will help us calculate the cost. The surface area of the bottom is simply the area of a circle, while the surface area of the sides can be calculated as the circumference of the base (2πr) multiplied by the height (h):
Surface area of the bottom = π * r^2
Surface area of the sides = 2πr * h
The total cost of manufacturing the drum will be the cost of the metal for the bottom and sides, plus the cost of the metal for the lid. Let's denote the cost of the bottom and sides as "C1" and the cost of the lid as "C2".
To calculate the total cost, we need to multiply the surface area of the bottom and sides by the cost per square foot of the metal used for them, and add it to the surface area of the lid multiplied by the cost per square foot of the metal used for the lid:
Total cost = (C1 * surface area of bottom and sides) + (C2 * surface area of lid)
Now, let's substitute the formulas for the surface area of the bottom, sides, and lid into the total cost equation:
Total cost = (C1 * (π * r^2 + 2πr * h)) + (C2 * surface area of lid)
We can simplify this equation by factoring out π:
Total cost = π * (C1 * r^2 + 2C1 * r * h + C2 * surface area of lid)
Since we want to minimize the total cost, we can rearrange the equation to isolate one variable. Let's isolate "h":
Total cost = π * (C1 * r^2 + 2C1 * r * h + C2 * surface area of lid)
Total cost = π * (C1 * r^2 + C2 * surface area of lid) + 2π * C1 * r * h
Total cost - π * (C1 * r^2 + C2 * surface area of lid) = 2π * C1 * r * h
(Total cost - π * (C1 * r^2 + C2 * surface area of lid)) / (2π * C1 * r) = h
Now we have an equation for "h" in terms of "r" and the given values. We can substitute this value of "h" into the volume equation to obtain a equation with only "r":
10 = π * r^2 * ((Total cost - π * (C1 * r^2 + C2 * surface area of lid)) / (2π * C1 * r))
Simplifying further:
10 = ((Total cost - π * (C1 * r^2 + C2 * surface area of lid)) / (2 * C1 * r))
Now we have an equation that relates the volume of the drum to the variables "r" and the given values. To find the value of "r" that minimizes the total cost, we can substitute the known values for the cost per square foot and solve for "r". Once "r" is found, we can substitute it back into the equation for "h" to determine the height.
Solving this equation is the next step to find the dimensions of the drum that minimize the total cost.