Kevin K. answered • 05/07/15
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The problem discusses two conditions about the field: the area and the length of fencing. First, the entire field has an area of 1,000,000 ft^2. Since we don't know the dimensions yet, let's call those x and y. Since it's a rectangle, the area equation would be:
1,000,000 = xy
Next for the length of fencing, we will have the four sides of the rectangle as well as the length down the middle. Since we don't know which way is divided, it actually won't matter which we pick (x or y). Let's use x. So the fence around the field is 2x + 2y and if we divide it along the x direction, we'll add an extra x. Our length equation will be:
L = 2x + 2y + x or L = 3x + 2y
Since we want to optimize L, that's the equation we need to get in terms of just one variable (either one, but I'll do in terms of x).
From our area equation, y = 1,000,000/x, so we plug that into our L equation:
L = 3x + 2(1,000,000/x) or L = 3x + 2,000,000/x
Whenever we want to optimize a function, we need to take the derivative and set it equal to 0.
L ' = 3 - 2,000,000/x2 = 0
Now just solve for x:
3 = 2,000,000/x2
x2 = 2,000,000/3
x = 1000sqrt2 / sqrt3 = 1000sqrt6 / 3 (Note we use the positive sqrt since x is a length and can't be negative.)
Now we just plug back into our A equation to find y:
y = 1,000,000/(1000sqrt6 / 3) = 3000/sqrt6 = 500sqrt6
And we have our dimensions!
1000sqrt6/3 by 500sqrt6
Finally, we can find our length L
L = 3x + 2y = 3(1000sqrt6/3) + 2(500sqrt6) = 1000sqrt6 + 1000sqrt6 = 2000sqrt6 = about 4899 ft
[ We know this is a minimum by using the second derivative test. L '' = 4,000,000/x^3, which will be positive for all x > 0. If the second derivative is positive, L is concave up, which means our critical point is a minimum].
