Mr. Smith would like to enclose a rectangular field that has an area of 1000 square feet. What is the minimum amount of fencing he will need if he only needs to use it on 3 sides since he can use the side of the barn for the fourth side.
P=2W+L
LW=1000
P=2W+1000/W
dP/dw=2-1000/W^2=0
solve for W first, then L, then Perimeter (length of fencing)
L=22.360
W=44.722
To find the minimum amount of fencing Mr. Smith will need, we can first find the dimensions of the rectangular field.
Let's assume the length of the field is L feet, and the width is W feet. Since the area of the field is given as 1000 square feet, we have the equation:
L * W = 1000
Since Mr. Smith only needs to enclose three sides of the field, we can calculate the perimeter of the rectangle using the formula:
Perimeter = 2 * (length + width)
Since one of the sides is the barn, we won't include it when calculating the perimeter. So, the minimal amount of fencing he needs is the perimeter of the remaining three sides.
Now, let's solve for L and W in the equation L * W = 1000. Here are a few potential dimensions:
L = 1, W = 1000
L = 2, W = 500
L = 4, W = 250
L = 5, W = 200
L = 8, W = 125
L = 10, W = 100
L = 20, W = 50
L = 25, W = 40
L = 40, W = 25
L = 50, W = 20
L = 100, W = 10
L = 125, W = 8
L = 200, W = 5
L = 250, W = 4
L = 500, W = 2
L = 1000, W = 1
Now, calculate the perimeter for each set of dimensions using the formula:
Perimeter = 2 * (length + width)
Perimeter for L = 1, W = 1000: Perimeter = 2 * (1 + 1000) = 2002
Perimeter for L = 2, W = 500: Perimeter = 2 * (2 + 500) = 1004
Perimeter for L = 4, W = 250: Perimeter = 2 * (4 + 250) = 508
Perimeter for L = 5, W = 200: Perimeter = 2 * (5 + 200) = 410
Perimeter for L = 8, W = 125: Perimeter = 2 * (8 + 125) = 266
Perimeter for L = 10, W = 100: Perimeter = 2 * (10 + 100) = 220
Perimeter for L = 20, W = 50: Perimeter = 2 * (20 + 50) = 140
Perimeter for L = 25, W = 40: Perimeter = 2 * (25 + 40) = 130
Perimeter for L = 40, W = 25: Perimeter = 2 * (40 + 25) = 130
Perimeter for L = 50, W = 20: Perimeter = 2 * (50 + 20) = 140
Perimeter for L = 100, W = 10: Perimeter = 2 * (100 + 10) = 220
Perimeter for L = 125, W = 8: Perimeter = 2 * (125 + 8) = 266
Perimeter for L = 200, W = 5: Perimeter = 2 * (200 + 5) = 410
Perimeter for L = 250, W = 4: Perimeter = 2 * (250 + 4) = 508
Perimeter for L = 500, W = 2: Perimeter = 2 * (500 + 2) = 1004
Perimeter for L = 1000, W = 1: Perimeter = 2 * (1000 + 1) = 2002
From the calculations, we can see that the minimum amount of fencing required is 130 feet, which occurs when the dimensions of the rectangular field are L = 25 feet and W = 40 feet.