A 10 cm x 12 cm rectangular sheet that is used to make a box with open top is to be lined with cushion. if the cushion material for the sides costs four times per square centimeter as that of material for the bottom, find the dimensions of the box if the cost is to be minimized.
I assume this is the usual corner-cut problem. So, if the cuts have length x, then the cost of the material is
c = (12-2x)(10-2x) + 4(2x(10-2x)+2x(12-2x))
Hmm. I guess x" corners are not cut and the rest folded up.
To minimize the cost, we need to minimize the surface area of the box while maintaining the volume of the box.
Let's assume the height of the box is "h" cm.
The dimensions of the base of the box will be h cm x 10 cm.
The dimensions of the side of the box (rectangular sheet) will be h cm x 12 cm.
The area of the bottom of the box is given by 10 cm x 10 cm = 100 cm².
The area of the side of the box is given by 2(h cm x 10 cm) + 2(h cm x 12 cm) = 20h + 24h = 44h cm².
The total surface area of the box (including the bottom and sides) is given by 100 cm² + 44h cm² = 144h + 100 cm².
The volume of the box is given by V(cm³) = (h cm x 10 cm x 12 cm) = 120h cm³.
Now, we need to express the cost in terms of h.
Let the cost per square centimeter for the bottom be x cents.
The cost per square centimeter for the sides will then be 4x cents.
The total cost for the bottom is x * 100 cm² = 100x cents.
The total cost for the sides is 4x * (44h cm²) = 176hx cents.
Therefore, the total cost for the box is (100x + 176hx) cents.
Since the volume of the box needs to be maintained, we can express h in terms of V:
h = V/120.
Substituting this value into the cost equation we get:
Cost = 100x + 176hx
Cost = 100x + 176x(V/120)
Simplifying further:
Cost = 100x + (176/120)hx
Now, to minimize the cost, we can take the derivative of the cost equation with respect to h and set it equal to zero:
d(Cost)/dh = 0
0 = (176/120)x - (176/120)V/120
Simplifying further:
(176/120)x = (176/120)V/120
x = V/120
Since we need to find the dimensions of the box that minimize the cost, we know that the cost per square centimeter for the bottom (x) should be equal to V/120.
Therefore, we have:
x = V/120
x = 120h/120
x = h
So, the cost per square centimeter for the bottom should be equal to the height of the box.
Now, we can substitute x = h back into the cost equation:
Cost = 100x + 176hx
Cost = 100h + 176h^2
To find the minimum cost, we need to find the value of h that minimizes the Cost function.
To do this, we can take the derivative of the Cost function with respect to h and set it equal to zero:
d(Cost)/dh = 0
100 + 352h = 0
352h = -100
h = -100/352
h = -0.2841
Since the height of the box cannot be negative, we discard this solution.
Therefore, there is no minimum cost for this problem, as the height of the box must be positive.
To find the dimensions of the box that minimize the cost, we need to consider the cost of the cushion material for the sides and the bottom.
Let's start by finding the cost of the cushion material for the bottom. The area of the bottom is the product of its length and width, which is 10 cm * 12 cm = 120 cm^2.
Let's assume the cost of the cushion material for the bottom is x dollars per square centimeter. Therefore, the cost of the cushion material for the bottom can be calculated as follows:
Cost of bottom = x dollars/cm^2 * 120 cm^2 = 120x dollars
Now, let's find the cost of the cushion material for the sides. Since the box has an open top, the sides will be in the shape of rectangular prisms with dimensions, height, and width.
Let's assume the height of the box is h centimeters. The width of each side of the box (excluding the open top) will be equal to the height (h) and the length will be equal to the perimeter of the bottom (2 * width + 2 * length).
The perimeter of the bottom is given by: 2 * (10 cm + 12 cm) = 2 * 22 cm = 44 cm.
Therefore, the length of each side of the box is 44 cm and the width is h cm.
The area of each side is given by: Area of each side = length * width = 44 cm * h cm = 44h cm^2
The total cost of the cushion material for the sides can be calculated as follows:
Cost of sides = 4 times cost of bottom
Cost of sides = 4 * 120x dollars = 480x dollars
The total cost for the box is the sum of the cost of the bottom and the cost of the sides:
Total cost = Cost of bottom + Cost of sides
Total cost = 120x dollars + 480x dollars
= 600x dollars
Now, to minimize the total cost, we need to find the value of x that minimizes the equation "Total cost = 600x dollars".
Since x is the cost per square centimeter, it cannot be negative. Therefore, the minimum value of x that makes sense in this context is 0.
So, when x = 0, the total cost is also 0. However, this means that we are not using any cushion material, which doesn't fulfill the requirement of lining the box with cushion material.
Therefore, we need to find the maximum value of x for which the total cost is minimized.
Since the total cost is in direct proportion to the value of x (total cost = 600x dollars), the maximum value of x that minimizes the total cost is when x = 0.
Therefore, the dimensions of the box with minimized cost are:
Length = 10 cm
Width = 12 cm
Height = Any positive value, as long as it makes sense in the context of the problem.