302 algebra De mystif ieD
EXAMPLE
A rectangle is twice as long as it is wide. If the length is decreased by
4 inches and its width is decreased by 3 inches, the area is decreased by
88 square inches. Find the original dimensions.
The area formula for a rectangle is A = lw. Let A represent the original area;
l, the original length; and w, the original width. We know that the original
length is twice the original width, so l = 2w and A = lw becomes A = (2w)w =
2 w^2. The new length is l – 4 = 2w – 4 and the new width is w – 3, so the new
area is (2w – 4)(w – 3). But the new area is also 88 square inches less than
the old area, so A – 88 represents the new area, also. We then have for the
new area, A – 88 = (2w – 4)(w – 3). But the A can be replaced with 2w^2. We
now have the equation 2w^2 – 88 = (2w –^ 4)(w – 3), an equation with one
unknown.
288243
2882 6412
2
22
www
wwww
−= −−
−= −−+
()()
(UsetheFOOILmethod
oneachside
.)
28 ww^22 −= 82 −+ 10 ww 12 ( 2 2 ccancels.)
−−
−=−
−
−
=
=
12 12
100 10
100
10
10
w
w
w
The width of the original rectangle is 10 inches and its length is 2w = 2(10) =
20 inches.
A square’s length is increased by 3 cm, which causes the area of the
new square to increase by 33 cm^2. What is the length of the original
square?
A square’s length and width are the same, so the area formula for the
square is Al=⋅=ll^2. Let l represent the original length. The new length
is l + 3. The original area is A = l^2 and its new area is (l + 3)^2. The new
area is also the original area plus 33, so (l + 3)^2 = new area = A + 33 =
l^2 + 33.
EXAMPLE
A rectangle is twice as long as it is wide. If the length is decreased by