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Math Help - Need Help: Applied Optimization

  1. #1
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    Need Help: Applied Optimization

    1. A commercial cherry grower estimates from past records that if 51 trees are planted per acre, each tree will yield 31 pounds of cherries for a growing season. Each additional tree per acre (up to 20) results in a decrease in yield per tree of 1 pound. How many trees per acre should be planted to maximize yield per acre, and what is the maximum yield?

    2. A parcel delivery service will deliver a package only if the length plus the girth (distance around) does not exceed 108 inches. Find the maximum volume of a rectangular box with square ends that satisfies the delivery company's requirements.

    3. A fence is to be built to enclose a rectangular area of 270 square feet. The fence along three sides is to be made of material that costs 4 dollars per foot, and the material for the fourth side costs 16 dollars per foot. Find the length L and width W (with W<=L) of the enclosure that is most economical to construct.

    4. 1 pt) Let Q=(03) and R=(108) be given points in the plane. We want to find the point P=(x0) on the x-axis such that the sum of distances PQ+PR is as small as possible. (Before proceeding with this problem, draw a picture!)
    To solve this problem, we need to minimize the following function of x:
    f(x)=
    over the closed interval [ab] where a= and b=.
    We find that f(x) has only one critical number in the interval at x=
    where f(x) has value
    Since this is smaller than the values of f(x) at the two endpoints, we conclude that this is the minimal sum of distances.

    Any help is appreciated. Thanks!

    -Brad
    Last edited by CaptainBlack; March 7th 2007 at 10:21 PM.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by lmao View Post
    1. A commercial cherry grower estimates from past records that if 51 trees are planted per acre, each tree will yield 31 pounds of cherries for a growing season. Each additional tree per acre (up to 20) results in a decrease in yield per tree of 1 pound. How many trees per acre should be planted to maximize yield per acre, and what is the maximum yield?
    From what you are told you will find that the yeild per tree:

    y = 31 - (t-51), for t<=71

    where t is the number of trees per acre. therefor the total yield per arcre:

    Y= t*y = 31*t - t*(t-51) = 72*t - t^2

    This takes a maximum at the point where dY/dt=0, or an end point of the interval.

    dY/dt = 72 - 2t,

    so dY/dt = 0, gives us t=36 with a yeild of 1296 pounds per acre, we also
    check the end point t=71, to get yeilds per acre of : 71.

    So the maximum yeild is 1296 pounds per arce at a tree density of 36 trees pre acre.

    RonL
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