5.5 Answer ExplanationsDS160SO- If xy = -^6 ,^ what^ is^ the^ value of^ xy(x^ + y)?
( 1 ) X - Y=^5c (^2 ) xy2 =^180S16464- The circular base of an above-ground swimming pool
lies in a level yard and just touches two straight sides
of a fence at points A and B, as shown in the figure
above. Point C is on the ground where the two sides of
the fence meet. How far from the center of the pool's
base is point A?
(1) The base has area 250 square feet.
(2) The center of the base is 20 feet from point C.Geometry
Let Q be the center of the pool's base and r be
the distance from Q to A, as shown in the figure
below.Algebra
By substituting -6 as the value of xy, the
question can be simplified to "What is the value
of-6(x + y) ?"(1) Adding y to both sides of x - y =^5 gives
x = y + 5. When y + 5 is substituted for x in
the equation xy = -6, the equation yields(y+ (^5) )y=—6, or I +Sy+ 6 = 0. Factoring
the left side of this equation gives
(y + 2)(y + 3) = 0. Thus,y may have a value
of-2 or -3. Since a unique value of y is
not determined, neither the value of x nor
the value of xy can be determined; NOT
sufficient.
(^2 )
=—
Since xy = (xy)y and (^) 寸= 18 , it follows
that (xy)y = 18. When -6 is substituted
for xy, this equation yields -6y = 18, and
hence y 3. Since y = -3 and xy = -6, it
follows that -3x = -6, or x = 2. Therefore,
the value of x + y, and hence the value of
劝(x+ y) = -^6 (x + y) can be^ determined;
SUFFICIENT.
Th e correct answer 1s B·，
statement 2 alone is sufficient.
DS05519
- [y] denotes the greatest integer less than or equal toy.
Is d < 1?
Since A is a point on the circular base, QA is a
radius (r) of the base.
(^1 ) Since the formula for the area of a circle is
area=正，thisinformation can be stated as
250=正or尸=r; SUFFICIENT.
冗
(2) Since也is tangent to the base, f:..QAC is
a right triangle. It is given that QC= 20,
but there is not enough information to use
the Pythagorean theorem to determine the
length of伽NOT sufficient.Th e correct answer 1s A·， (^)
statement 1 alone is sufficient.
(1) d = y-[y]
(2) [d] =^0
Algebra ·-一心
(^1 ) It is given d = y—[y]. If y is an integer, then
y = [y], and thus y-[y] = 0, which is less
than 1. If y is not an integer, then
y lies between two consecutive integers,
the smaller of which is equal to [y]. Since
each of these two consecutive integers is at
a distance ofless than 1 from y, it follows
that [y] is at a distance ofless than 1 from y,
or y -[y] < 1. Thus, regardless of whether y
is an integer or y is not an integer, it can be
determined that d < l; SUFFICIENT.