You are creating a rectangular garden and want to surround it with brick edging.

Question

You are creating a rectangular garden and want to surround it with brick edging. You will divide it into 2 halves with this edging so veggies are on one side and flowers are on the other. You only have 120 feet worth of brick edging, and want to enclose the most area possible. Determine the dimensions(length and width) that will maximize the total area of the garden and state that maximum area

A. Write equations or functions relating these varaibles and using the given information

B. Develop a function of one variable that describes the area of the garden and use calculus to locate the critical value(s)

C. Check to find the critical value that is the maximum's location(show calculus work of some type)

D. What are the best dimensions for the garden ____ by ____

E. What is the garden area obtained by using these dimensions

Explanation / Answer

1 . let x is half of length and y is width

thus

2. area = 2x*y = 2xy

given = 2 (2x+y) = 120

2x+y = 120/2 = 60

y = (60-2x)

3. area = 2xy =    2x*(60-2x)

take derivative

da/dx = 2x *-2 + (60-2x)*2 = 0 => -8x+120 = 0

-4x + 120 -4x = 0

120 = 8x

x= 120/8 = 15

y = 60-2x = 60 - 2*15 = 60-30 = 30

4. take double derivative

d^a/dx^2 = -8 < 0

so at x = 15 it has max value.

5. 2x by y = 2*15 by 30 = 30 by 30

garden area = 30 by 30

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