A craftsman wants to make a cylindrical jewelry box that has volume, V, equal to 60 cubic inches.
He will make the base and side of the box out of a metal that costs 50 cents per square inch. The lid of the box will be made from a metal with a more ornate finish which costs 150 cents per square inch.
-Rewrite your expression for the cost of the box in terms of the single variable r.
-Differentiate C with respect to r, to find the derivative dC / dr.
-Find the value of r for which we have a potential relative extreme point of C.
-What is the height of the box?
Volume:
V = pi*r^2*h = 60
h = 60/(pi*r^2)
Area:
Area of lid = pi*r^2
Area of Curved area + bottom = pi^r^2 + 2pi*r*h
Put the h we worked out in terms of r into the area equation.
Cost = 0.50*(pi*r^2) + 1.50*(pi^r^2 + 2pi*r*[60/(pi*r^2)]
Tidy up a little and then differenciate and put equal to zero (0 is the slope of the tangent to the graph where the maximum/minimum points occur)
See how you get on from there.
typo in the curved area.
Sould be pi*r^2 not pi^r^2
Same in the cost equation
To rewrite the expression for the cost of the box in terms of the single variable r, we need to determine the surface area of the base and side, as well as the surface area of the lid.
1. Surface area of the base and side:
The lateral surface area of a cylinder is given by the formula 2πrh, where r is the radius and h is the height. Since the base is included in the lateral surface area, the total surface area of the base and side is 2πrh + πr^2.
The cost of the base and side is 50 cents per square inch, so the cost for this part of the box is (2πrh + πr^2) * 0.50.
2. Surface area of the lid:
The lid is a disc with radius r. Therefore, the surface area of the lid is A = πr^2.
The cost of the lid is 150 cents per square inch, so the cost for this part of the box is A * 1.50.
The total cost of the box, C, is the sum of the costs for the base and side, as well as the cost for the lid: C = (2πrh + πr^2) * 0.50 + πr^2 * 1.50.
To differentiate C with respect to r, we will apply the differentiation rules:
dC / dr = (2πh + 2πr) * 0.50 + (2πr) * 1.50 = πh + 3πr.
To find the value of r for which we have a potential relative extreme point of C, we need to set the derivative equal to zero and solve for r:
πh + 3πr = 0
3πr = -πh
r = -h / 3.
Since r represents the radius, negative values are not meaningful in this context. Therefore, we can conclude that there are no potential relative extreme points for C.
To find the height of the box, we can use the expression for the volume of a cylinder:
V = πr^2h.
Given that the volume of the box is 60 cubic inches, we have:
60 = πr^2h.
Solving for h:
h = 60 / (πr^2).
1. To find the cost of the box in terms of the single variable r, we need to express the cost of different components based on their areas.
Let's define the radius of the base and side of the box as "r" and the height of the box as "h."
The cost of the base and side of the box is given by the area of the base and side multiplied by the cost per square inch, which is 50 cents:
Cost of base and side = (area of base + area of side) * 0.50
The area of the base of the cylinder is given by the formula:
Area of base = π * r^2
The area of the side of the cylinder is given by the product of the circumference of the base and the height:
Area of side = 2π * r * h
So the cost of the base and side is:
Cost of base and side = (π * r^2 + 2π * r * h) * 0.50
The cost of the lid of the box is given by the area of the lid multiplied by the cost per square inch, which is 150 cents:
Cost of lid = (area of lid) * 1.50
The area of the lid is the same as the area of the base:
Area of lid = π * r^2
So the cost of the lid is:
Cost of lid = (π * r^2) * 1.50
The total cost of the box is the sum of the cost of the base and side and the cost of the lid:
Cost of the box = (π * r^2 + 2π * r * h) * 0.50 + (π * r^2) * 1.50
2. To find the derivative dC / dr, we differentiate the expression for the cost of the box with respect to r while treating h as a constant:
dC / dr = d/d(r)[(π * r^2 + 2π * r * h) * 0.50 + (π * r^2) * 1.50]
Differentiating term by term, we get:
dC / dr = (2π * r + 2π * h) * 0.50 + (2π * r) * 1.50
Simplifying further, we have:
dC / dr = πr + πh + 3πr
dC / dr = 4πr + πh
3. To find the value of r for which we have a potential relative extreme point of C, we need to set the derivative equal to zero and solve for r:
4πr + πh = 0
Solving for r:
4πr = -πh
r = -h/4
This gives us a potential relative extreme point for C when r is equal to -h/4.
4. The height of the box, h, is not directly provided in the given information. Therefore, we cannot determine the height of the box with the given details.