1.8, Problem 37: A lidless cardboard box is to be made with a volume of 4 m3
. Find the
dimensions of the box that requires the least amount of cardboard.
Solution: If the dimensions of our box are x, y, and z, then we’re seeking to minimize
A(x, y, z) = xy + 2xz + 2yz subject to the constraint that xyz = 4. Our first step is to make
the first function a function of just 2 variables. From xyz = 4, we see z = 4/xy, and if we substitute
this into A(x, y, z), we obtain a new function A(x, y) = xy + 8/y + 8/x. Since we’re optimizing
something, we want to calculate the critical points, which occur when Ax = Ay = 0 or either Ax
or Ay is undefined. If Ax or Ay is undefined, then x = 0 or y = 0, which means xyz = 4 can’t
hold. So, we calculate when Ax = 0 = Ay. Ax = y − 8/x2 = 0 and Ay = x − 8/y2 = 0. From
these, we obtain x
2y = 8 = xy2
. This forces x = y = 2, which forces z = 1. Calculating second
derivatives and applying the second derivative test, we see that (x, y) = (2, 2) is a local minimum
for A(x, y). To show it’s an absolute minimum, first notice that A(x, y) is defined for all choices
of x and y that are positive (if x and y are arbitrarily large, you can still make z REALLY small
so that xyz = 4 still). Therefore, the domain is NOT a closed and bounded region (it’s neither
closed nor bounded), so you can’t apply the Extreme Value Theorem. However, you can salvage
something: observe what happens to A(x, y) as x → 0, as y → 0, as x → ∞, and y → ∞. In each
of these cases, at least one of the variables must go to ∞, meaning that A(x, y) goes to ∞. Thus,
moving away from (2, 2) forces A(x, y) to increase, and so (2, 2) is an absolute minimum for A(x, y).
Yes, I think the answer is y = 7. The line x = 4 runs vertical (because all of the points in that line have an x-value of 4) so any line that is perpendicular to it has to be horizontal. Any line that is horizontal is in the form y = (some number). In order for the new line to run through the point (5,7) it would have to have the same y-value as that point, which is 7. So the new line is y = 7. Hope this helps :) (P.S. try graphing them both if you need a visual.)
Firstly we check variables to make sure we understand.
y = dollars
x = pounds
On the scale, it says 1.126 lb. Which means 1.126 pounds.
And so, x = 1.126
y = 2x + 3
y = 2(1.126) + 3
y = 2.252 + 3
y = 2.252
The graph for part a and park b. It is underneath. I wrote 1.1, because it is rounded down from 1.126 lb, which is part b. The one with one line and no dots is part a.