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:
After simplifying we get (x,y) as (1,3).
Step-by-step explanation:
Given:
,
We need to use elimination method to solve the and simplify the equations.
Solution;
Let ⇒ equation 1
Also Let 
⇒ equation 2
Now by solving the equation we get;
first we will Add equation 2 from equation 1 we get;
Now Dividing both side by 5 using division property of equality we get;
Now Substituting the vale of x in equation 1 we get;
subtracting both side by 1 using subtraction property of equality we get;
Now Dividing both side by 7 using division property of equality we get;
Hence we can say that, After simplifying we get (x,y) as (1,3).
Check the picture below.
now, keep in mind that ship B is going at 20kph, thus from noon to 4pm, is 4 hours, so it has travelled by then 20 * 4 or 80 kilometers, thus b = 80.
whilst the ship B is moving north, the distance "a" is not really changing, and thus is a constant, that matters because the derivative of a constant is 0.
Answer:
<h3>Figure 1</h3>
- Perimeter of base = 5 + 5 + 8 = 18 ft
- Base area = 1/2(8)(3) = 12 ft²
<u>Surface area:</u>
- S = 18*7 + 2*12 = 150 ft²
<h3>Figure 2</h3>
<u>Surface area of cube:</u>
- S = 6a² = 6(2.5)² = 37.5 m²
<u>Surface area of prism:</u>
- S = 2(11 + 9)(7) = 280 m²
<u>Overlapping area:</u>
<u>Surface area of composite figure:</u>
- S = 280 + 37.5 - 2(6.25) = 305 m²
