Download Quiz IV - U.I.U.C. Math

Quiz IV
Section 2.5-2.7 & 3.3
September 28, 2006 2pm
Problem 1. A rectangular garden of area 75 square feet is to be surrounded on
three sides by a brick wall costing $10 per foot and on one side by a fence costing
$5 per foot. Find the dimensions of the garden such that the cost of materials
is miminized. Make sure to justify why your solution gives the minimal cost.
Solution. We have a rectangle with sides x and y. We know that the area of
this rectangle is going to be x · y. The problem tells us that the area is 75 feet,
so we get the constraint equation:
x · y = 75
Now the quantity we want to optimize is going to be the cost to make our fence.
So, if we have brick on three sides of the equation, say one side of length y and
both sides of length x, the cost for these sides of the garden will cost 10(y + 2x)
and the remaining side will be fence and hence have cost 5y. Thus we have the
objective equation:
C = 10(y + 2x) + 5y = 15y + 20x
We can solve the constraint equation for one of the variables to get:
x=
75
y
Thus, we get the cost equation in terms of one variable:
75
C(y) = 15y + 20
y
We now take the derivative and set it equal to zero.
−75
C 0 (y) = 15 + 20
=0
y2
1
Now, if we multiply both sides by y 2 , we see we have
15y 2 − 20(75) = 0
15y 2 = 1500
y 2 = 100
y = 10
To check that this is indeed a value of y that gives us a minimum, we need
to take the second derivative of our cost function.
C 00 (y) = 3000y −3
C 00 (10) = 3000(10−3 ) = 3 > 0
Now, we need values of both x and y, thus as x =
75
y ,
we get x = 7.5 and y = 10.
Problem 2. Determine the slope of the graph of
xy + y 3 = 14
where x = 3 and y = 2.
Solution. We want the slope of the graph, that is
to the equation.
dy
dx .
So, we need to apply
d
d
xy + y 3 =
14
dx
dx
d d 3
d x y+x
y +
y =0
dx
dx
dx
dy
dy
y+x
+ 3y 2
=0
dx
dx
dy
3y 2 + x = −y
dx
dy
−y
= 2
dx
3y + x
So, we want the slope of the tangent line at the point (3, 2), so
dy −2
=
dx (3,2)
3(2)2 + 3
=
2
−2
15
d
dx