Question

ms Callahan is building a new dog pen for her dog. she has 24feet of fencing. the pen needs to be a rectangle. what is the length and width of the diffrent sizes of pens she can make with the amount of fencing

Answer 1

If Ms. Callahan has 24 feet of fencing, and she is building a pen, the PERIMETER of the pen must be 24 feet.  The perimeter is basically the distance around a figure.  The perimeter of a rectangle is equal to length plus width plus length plus width, AKA l+w+l+w, or P=2l+2w.  In a rectangle, two pairs of sides are of equal length--so the two lengths and the two widths must be equal.

So, the formula is P=2l+2w.  P, the perimeter, is 24, so 24=2l+2w.  Let's try some values for l and see what we get for w.  If the length is 1, l=1.  24=(2*1)+2w.  24=2+2w.  22=2w.  w=11.  So if length is 1 foot, width is 11 feet.

What if l=2?  24=(2*2)+2w.  24=4+2w.  2w=20.  w=10.  If l=2, w=10.  And l=3?  24=(2*3)+2w.  24=6+2w.  18=2w.  w=9.  If l=3, w=9.  Do you see a pattern?  Every time we add 1 to l, we subtract 1 from w.  So if l=4, w=8.  If l=5, w=7.  If l=6, w=6.  Here, we start getting similar answers: if l=7, w=5.  If l=8, w=4.  Since we already know these values work, it doesn't matter whether we call it length or width.  So, our answers are below.

Answer: Ms Callahan can make a pen with a length of 1 foot and a width of 11 feet, a length of 2 feet and a width of 10 feet, a length of 3 feet and a width of 9 feet, a length of 4 feet and a width of 8 feet, a length of 5 feet and a width of 7 feet, or a length of 6 feet and a width of 6 feet.