MATH 131 Quiz #4
June 23, 2014
Print Last Name: SOLUTION
First Name: SOLUTION
Maximum Points: 10
Grade: 10
1) A farmer has 1,000 ft of fencing and wants to enclose a rectangular area and then
divide it into four equally sized pens with fencing parallel to one side of the rectangle.
What is the largest possible total area of the four pens?
a) Draw a diagram illustrating the general situation. Introduce notation and label
the diagram with your symbols.
x
y
y
y
y
y
x
b) Write an expression for the total area.
A = xy
c) Use the given information to write an equation that relates the variables.
P = 1000 = 2x + 5y
So x = 500 − 25 y.
d) Use parts (b) and (c) to find the maximum area the farmer can enclose.
From (b) and (c),
5
5
A = 500 − y y = 500y − y 2
2
2
Then
dA
= A0 = 500 − 5y
dy
And A0 = 0 whenever y = 500/5 = 100. Which means that x = 500 − 52 (100) = 250.
Then the Area is
A = 250(100) = 250, 000f t2
2) Suppose the revenue a manufacturer receives for producing x tablets is
R(x) = −4x3 + 600x2 + 3000
How many tablets should the manufacturer produce in order to maximize their revenue?
SOLUTION: First, take a derivative and find the critical points:
R0 (x) = −12x2 + 1200x
R0 (x) = 12x(−x + 100) = 0 ⇐⇒ x = 0 or x = 100
Now if we look at the sign or R0 , we will see that between 0 and 100, R0 (x) > 0, and
then for x > 100, R0 (x) < 0. Therefore, x = 100 is a maximum for this function.
So the manufacturer should produce 100 tablets to maximize their revenue, which we
can calculate to be
R(100) = $2, 003, 000