Part II: The Answers
322
Answers
401–500
Using y = –25 and x = 50 + y gives you x = 25, so the product is
P = ( )(− ) = −
25 25
625
You can verify that y = –25 gives you a minimum by using the first derivative test.
499.
20, 20
Let x and y be the two positive numbers so that xy = 400. The function that you want to
minimize, the sum, is given by
S x y
x y
( , ) = +
You can write the product in terms of one variable:
xy
y
x
=
=
400
400
Then substitute the value of y into the sum function:
S x
x
x
x
x
( ) = +
= +
−
400
400
1
Next, find the derivative of the function:
′
S x
x
x
x
x
( ) = −
= −
=
−
−
1 400
1 400
400
2
2
2
2
Setting the derivative equal to zero gives you x
2
– 400 = 0, and keeping only the positive
solution, you have x = 20. You can verify that this value gives you a minimum by using
the first derivative test.
Using x = 20 and xy = 400 gives you y = 20.
500.
15 m × 15 m
If you let x and y be the side lengths of the rectangle, you have the following equation
for the perimeter:
2
2
60
x
y
+
=
The function that you want to maximize, the area, is given by
A x y
xy
( , ) =
You can write this as a function of one variable by using 2x + 2y = 60 to get y = 30 – x.
Substituting the value of y into the area equation gives you
A x
x
x
x x
( )
(
)
=
−
=
−
30
30
2