1) Fint the point on the line y=5x+2 that is closest to the origin. 2)Find the d

Question

1) Fint the point on the line y=5x+2 that is closest to the origin.

2)Find the dimensions of a rectangle with area 1,728 m2 whose perimeter is as small as possible. (If both values are the same number, enter it into both blanks.)

3) Find two numbers whose difference is 162 and whose product is a minimum

If two numbers have a difference of 162, and one of them is x + 162, then the other is

4)We must maximize the area A = xy = x(56 x) = 56x x2, where 0 < x < 256

Solving 0 = A'(x) gives us x =

Explanation / Answer

Ans)

1)  We can use "The Method Of Least Squares" to solve this problem.

Let (x, y) be a point on the line, y = 5x + 2.

Then the distance squared, f(x), from a point on the line to the origin is:

f(x) = (x - 0)^2 + (5x + 2 - 0)^2 = x^2 + 25x^2 + 20x + 4 = 26x^2 + 20x + 4.

Now minimizing the distance squared is the same as minimizing the distance. Taking the derivative and setting it equal to zero gives:

f '(x) = 52x + 20 = 0

52x = -20

x = -20 / 52 = -5/13

Solving for y = 5x + 2 = 5(-5/13) + 2 = -25/13 + 26/13 = 1/13

Therefore, (-5/13, 1/13) is the point on the line that is closest to the origin.

The result is a minimum because the second derivative, f ''(x) = 52, is positve       

2) It's pretty well known that a rectangle of given area will have minimum perimeter when it is square. (Equivalently, a rectangle of given perimeter will have maximum area when it is square.)

You can see that as follows:
Let the dimensions be x and y, and let the area, A = xy, be fixed. The perimeter is
P = 2(x + y) = 2(x + A/x)
dP/dx = 2(1 - A/x²) = 0
A = x² = xy
x = y
Meaning that it's a square. So its dimensions are
1728 m * 1728 m
and because 1728 = 12³ = (2²•3)³ = 2•3³
1728 = 2³•33 = 243

and the square is (243 m) on a side.            

3) Let x+162 be the larger number
Let x be the smaller number
The difference between the two is 162.

The product would be x(x+162) = x² + 162x.

If you take the derivative, you get 2x + 162.

Set this to zero:
2x + 162 = 0
2x = -162
x = -81
, the numbers are 81 and -81. The product is -6561 which is minimum.

4) A = xy = x(56 x) = 56x x2

dA = 56 -2x = 0 that gives x=28 that gives A = 28(56-28) = 28^2 = 784

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