A box has a bottom with one edge 7 times as long as the other. If the box has no top and the volume is fixed at V, what dimensions minimize the surface area?
I agree with to FredR, up until
0 = 14b - (16v/7)/b^2
The next step should be
b^3 = 16v/(14*7)= (8/49)v
b = 2*(v/49)^1/3 = 0.5466 v^1/3
Then a = 7b = 3.8264 v^1/3
and c = v/(7b^2) = 0.4781 v^1/3
let the 3 dimensions be
a, b, and c where a and b are the bottom edges and c is the height.
v is the volume
s is the surface area
a = 7b
v = abc
= 7b^2c
or rearranging,
c = v/7b^2
s = ab + 2ac + 2bc
= 7b^2 + 14bc + 2bc
= 7b^2 + 16bc
= 7b^2 + 16bv/7b^2
= 7b^2 + (v16/7)/b
set the 1st derivative equal to zero,
0 = 14b - (v16/7)/b^2
(v16/7) = 15b^3
b=(v16/(7*15))^(1/3)
what are the complete dimensions?
Oops. Thanks for the correction.
To find the dimensions of the box that minimize the surface area, we need to first express the surface area as a function of the dimensions.
Let's assume the dimensions of the bottom of the box are x and 7x (since one edge is 7 times as long as the other).
The height of the box (h) can be determined by the volume (V) of the box, which is given as a fixed value.
The volume of a box is calculated by multiplying its base area by its height, so we have:
V = x * 7x * h
V = 7x^2h
Now, let's express the surface area of the box:
Surface area = 2 * (base area) + (side area)
The base area is given by the product of the lengths of the two sides:
Base area = x * 7x = 7x^2
The side area is found by multiplying the perimeter (P) of the base by the height (h):
Perimeter (P) = 2 * (x + 7x) + (x + 7x)
P = 18x
Side area = P * h = 18x * h
Now, let's express the surface area in terms of x and h:
Surface area = 2 * (7x^2) + (18x * h)
Surface area = 14x^2 + 18xh
To minimize the surface area, we need to find values of x and h that minimize the surface area function.
To do this, we can take partial derivatives of the surface area function with respect to both x and h and set them equal to zero. Solving the resulting equations will give us the values of x and h that minimize the surface area.
d(Surface area)/dx = 28x + 18h = 0
d(Surface area)/dh = 18x = 0
From the first equation, we can solve for h in terms of x:
h = -28x/18 = -14x/9
Substituting this value of h into the second equation, we get:
18x = 0
Since x cannot be equal to zero (as it represents the length of a side), we can conclude that x = 0, which is not possible in this context.
Therefore, there is no minimum for the surface area function.