Can you help with these Calc questions? Please I really need help for them

Question

1. Find two positive numbers whose product is 49 and whose sum is a minimum. (If both values are the same number, enter it into both blanks.)

_____ (smaller number)

_____ (larger number)

2.What is the maximum vertical distance between the line y = x + 56 and the parabola y = x² for −7 ≤ x ≤ 8?

3. A model used for the yield Y of an agricultural crop as a function of the nitrogen level N in the soil (measured in appropriate units) is Y = (k * N)/ 49 + N^2. where k is a positive constant. What nitrogen level gives the best yield?

N =  ____

4. A farmer wants to fence an area of 37.5 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. What should the lengths of the sides of the rectangular field be so as to minimize the cost of the fence?

_____ ft (smaller value)

_____ ft (larger value)

Answer 1

1) f(x) = x+y , xy = 49 --> f(x) = x+49/x

f '(x) = 1-49/x² , x = ±√49 , choose x = +7

y = 49/x = 7

smaller = 7

larger = 7

2) Vertical Distance is difference between the two functions

V(x) = x+56 - x², V '(x) = 1 -2x --> x = 1/2 --> V(1/2) = 56.25 <---Max Vertical distance

Boundary at -7 and 8 are less

3) I googled the equation, it is really y(n) = kn/(49+n²)

y '(n) = (k(49+n²) - 2kn²)/(49+n²) , Setting to zero, k(49+n²) -2kn² = 0 , n = 7

4) Let x and y be the sides of the rectangle. A = xy = 37.5 Million or y = 37.5/x

The length of the fences is f(x) = 2x + 3y = 2x+112.5/x

f '(x) = 2-112.5/x² --> x = √56.5 = 7.5 and y = 37.5/7.5 = 5 then scale by √Million

smaller = 5000 ft and larger = 7500 ft