A rectangular pen for a pet is 7 feet longer than it is wide?
A rectangular pen for a pet is 7 feet longer than it is wide. Give
possible values for the width W of the pen if its area must be
between 198 and 540 sq. ft. inclusively.
The width of the pen ranges from smaller value of the width, ft,
to the larger value of the width, ft, inclusively.
l = w + 7
a = l * w = (w + 7) * w = w^2 + 7w
For the smaller pen, w^2 + 7w = 198
w^2 + 7w - 198 = 0
-b ± √(b² - 4*a*c)
w = -------------------------
2a
-7 ± √(7² - 4*1*(-198))
w = -------------------------------
2*1
-7 ± √(49 - (-792))
w = --------------------------
2
-7 ± √(841)
w = -----------------
2
-7 ± 29
w = -------------
2
22 -36
w = --- or -----
2 2
w = 11 or -18
For the larger pen, w^2 + 7w = 540
w^2 + 7w - 540 = 0
-b ± √(b² - 4*a*c)
w = -------------------------
2a
-7 ± √(7² - 4*1*(-540))
w = --------------------------------
2*1
-7 ± √(49 - (-2160))
w = ----------------------------
2
-7 ± √(2209)
w = ------------------
2
-7 ± 47
w = -------------
2
40 -54
w = --- or -----
2 2
w = 20 or -27
It's impossible to have a width that is
less than zero, so we can discard the two
negative values.
The width can range from 11 ft to 20 ft.