Among all pairs of numbers (x,y) such that 4x+2y=30,find the pair for witch the sum of square, x^2+y^2 is minimum

x = 6, y =3, and the minimum = 45

Step-by-step explanation:

There are plenty of ways to solve this problem.

The Lagranian multiplier method is chosen (it is simple to apply in this case).

Given: A = x^2 + y^2

Find: x and y that satisfy 4x + 2y = 30 or 4x + 2y - 30 = 0 and A is minimum.

This is the same as finding x, y, and lambda that make the following L minimum.

L= x^2 + y^2 + lambda(4x + 2y - 30)

To find x, y, and lambda, we take the derivative of L with respect to them, set the derivative to 0, and solve the equations.

L'(x) = 2x + 4(lambda) = 0    (1)

L'(y) = 2y + 2(lambda) = 0    (2)

L'(lambda) = 4x + 2y - 30 = 0   (3)

Multiply (2) by 2 and subtract to (1), we have:

4y - 2x = 0  (*)

Combine (*) with (3) we have:

4y - 2x = 0

4x + 2y - 30 = 0

Multiply the 1st equation by 2 and add to the 2nd equation:

=>  10y - 30 = 0

=> y = 3

=> x = 4y/2 = 4(3)/2 = 6 (as (*))

=> lambda = -y = -3  (as (2))

Substitute x = 6, y = 3, and lambda = -3 into L, or substitute x = 6 and y = 3 into A to get the minimum.

Both ways would lead to the same answer.

Check:

L = x^2 + y^2 + lambda(4x + 2y - 30)

  = 6^2 + 3^2 + (-3)[4(6) + 2(3) - 30]

  = 36 + 9 + (-3)(0)

  = 45

A = x^2 + y^2

   = 36 + 9

   = 45

Hope this helps!

Maria had 2 yards of trim when she started.

Step-by-step explanation:

Step 1:

All of the sides of the triangle is 22 inches long each.

The total length of the sides of the triangle

So the total length of the trim for the banner is 66 inches long.

If Maria still has a foot of trim left, she had 6 inches of the trim left.

The length of the trim when she started inches.

Step 2:

Now we need to convert the 72 inches into yards.

We have 1 inch = 0.0277778 yards.

So 72 inches

So Maria had 2 yards of trim when she started.