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).
Answer:
To convert a quadratic from y = ax2 + bx + c form to vertex form, y = a(x - h)2+ k, you use the process of completing the square.
Step-by-step explanation:
Answer:
7x² - 3x - 4
Step-by-step explanation:
<u>Step 1: Write expression</u>
(6x² - x + 3) + (-2x + x² - 7)
<u>Step 2: Combine like terms (x²)</u>
7x² - x + 3 - 2x - 7
<u>Step 3: Combine like terms (x)</u>
7x² - 3x + 3 - 7
<u>Step 4: Combine like terms (constants)</u>
7x² - 3x - 4
Answer:
- 2 1/10
Step-by-step explanation:
-2/5 -x -9/10
Let x = 4/5
-2/5 - 4/5 - 9/10
Combine like terms
-6/5 - 9/10
Get a common denominator of 10
-6/5 * 2/2 - 9/10
-12/10 - 9/10
-21/ 10
Change this to a mixed number
10 goes into 21 2 times with 1 left over
-2 1/10
