field is rect. w/ semicircle on end; hay in rect, rye in end

[methodical]

1)(a) A farmer wants to clear a field in the shape of a rectangular plot with semicircular plot adjoined to one side of the rectangle. The rectangular plot is to be planted in hay,which will yield a profit at 5 cents per sq ft; and the semicircular plot is to be planted in rye, which will yield a profit at 6 cents per sq ft. If the perimeter of the field is to be 1600 ft, how should the farmer lay out the field to earn most money?

(b) If the price of rye increases so that a field planted with rye would yield 10 cents per sq ft. How should the farmer lay out the field now?

This is what I did so far----------

1)(a) simi circle area = (1/2)?r²
= (1/2)?(w²/4)
=(1/8)?w²

1600 = 2l+w+(1/2)2?r
1600 = 2l+w+(1/2)2?(w/2)
1600 = 2l+w+?w/2
1600 - 2l = w(1+?/2)
w = (1600 - 2l)/(1+?/2) ---------------(1)

profit P = 5lw + 6((1/8)?w²)
P = 5lw + (3/4)?w²

Combine with (1)
P = 5l[(1600 - 2l)/(1+?/2)] + (3/4)?[(1600 - 2l)/(1+?/2)]²

I knowI am suppose to Solve l and then w and Find P' and set it to 0 but how when I have more one than one unknown variable .

b) 10 cents per sq ft for rye :
profit P = 5lw +10((1/8)?w²)
P = 5l[(1600 - 2l)/(1+?/2)] + (5/4)?[(1600 - 2l)/(1+?/2)]²
Find P'
and set P' = 0
Solve l and then w. Same problem with above.

 

[mmm4444bot]

... P = 5l[(1600 - 2l)/(1+?/2)] + (3/4)?[(1600 - 2l)/(1+?/2)]² ...

... Find P' and set it to 0 but how when I have more one than one [variable] ...

Hello Methodical:

I see only one variable in your profit function P(l).

Find P'(l).

(I'm working this exercise to find P'(w), instead; we can then compare to see if we get the same result.)

Thank you for showing your work; it's nice to deal with people who possess sufficient intelligence to follow instructions.

Cheers,

~ Mark

 

[mmm4444bot]

Here is my result for part (a), rounded to one-tenth of a foot.

w = diameter of semicircle

w = 491.3 feet

l = 168.5 feet

And you?