You were assigned to construct

qn
open-top box with a square base
from
two materials, one
for
the
bottom and one
for
the sides. The volume of a box is 78 cubic inches. The cost of the material for
the
bottom is Php 4 per square inch, while the cost of the material
for
the sides is Php 3 per square inch.

(a) Determine a function C for the cost of constructing the box as function of the side r, of the base.
(b) What is the cost of constructing the box rt x is 2 inches? 3.5 inches?
(c) What should be the dimension of the box that will give the minimum cost of constructing it?
Justify
your answer.

ok, looks like you are saying that you have a box with a square base of r inches each and a height of x inches

volume = r^2 x = 78
so x = 78/r^2

cost = 4(area of base) + 3(4 sides)
= 4r^2 + 12 rx
= 4r^2 + 12(78/r^2)(r) = 4r^2 + 936/r

b) makes no sense to me, since the volume is to be 78 inches^3, your given case does not give me that volume.

c)
d(cost)/dr = 8r - 936/r^2
= 0 for a max/min
936/r^2 = 8r
r^3 = 117
r = appr 4.891
x = appr 3.261

min cost = 4(4.891) + 936/4.891
= appr 210.936.. <------ minimum cost

To solve this problem, we need to consider the following:

1. Volume of the box: The volume of a box with a square base is given by V = r^2 * h, where r is the length of the side of the square base and h is the height of the box.
2. Cost of the material for the bottom: The cost for the material of the bottom is Php 4 per square inch.
3. Cost of the material for the sides: The cost for the material of the sides is Php 3 per square inch.
4. Cost function: We need to determine a cost function C(r) that represents the cost of constructing the box as a function of the side length r of the base.

Now, let's address each part of the question:

(a) To determine the cost function C(r), we need to consider the surface area of both the bottom and the sides of the box.

The surface area of the bottom is simply the area of a square, which is A_bottom = r^2.

The surface area of the sides is the sum of the four sides of the box, which can be calculated as A_sides = 4rh.

Therefore, the total cost function C(r) can be expressed as follows:
C(r) = (4 * r^2 * Php 4) + (4 * r * h * Php 3)

Since we are given the volume of the box as 78 cubic inches, we can express the height h in terms of r using the volume formula:
78 = r^2 * h
h = 78 / r^2

Substituting this value of h into the cost function, we get:
C(r) = (4 * r^2 * Php 4) + (4 * r * (78 / r^2) * Php 3)
Simplifying further, we have:
C(r) = 16r^2 + 936 / r

(b) To find the cost of constructing the box for a particular value of r, we can substitute the value of r into the cost function C(r).

For r = 2 inches:
C(2) = 16(2^2) + 936 / 2 = 64 + 468 = Php 532

For r = 3.5 inches:
C(3.5) = 16(3.5^2) + 936 / 3.5 = 196 + 267.43 = Php 463.43

(c) To determine the dimensions of the box that will result in the minimum cost of construction, we need to find the minimum of the cost function C(r).

Take the derivative of C(r) with respect to r and set it equal to zero to find critical points.
C'(r) = 32r - 936 / r^2 = 0

Simplifying further, we get:
32r^3 - 936 = 0
32r^3 = 936
r^3 = 936 / 32
r^3 = 29.25
r ≈ 3.088

We can use this value of r to find the corresponding cost:
C(3.088) = 16(3.088^2) + 936 / 3.088 ≈ Php 456.92

Therefore, the dimensions of the box that will give the minimum cost of construction are approximately: r ≈ 3.088 inches for the side length and h ≈ 6.01 inches for the height.