Basic Calculus Performance Task 2
SOLVING OPTIMIZATION PROBLEMS USING CALCULUS
1) Find two positive numbers whose sum is 20 if the product of the first
number and the square of the second number is to be a maximum.
Important Notes:
1) We can denote the problem as x + y = 20, wherein we will maximize
x y2
.
2) Since
x+y=20
, then
y=20โx .
a. What is the objective? Let it be ๐(๐ฅ).
The objective is the product of the first number and the square of the
second number.
We are required to find the numbers that will maximize ๐.
b. What variable are you going to control? Let it be ๐ฅ.
Let
x
be the second number. Let it be the control variable. Let
20โx
be the
first number.
c. What function accurately models this problem?
Our model is
P
(
x
)
=
(
20โx
)
x2=20 x2โx3. Let it be continuous
[0,20]
d. What are the numbers?
Finding its critical points.
P ' (x)=
(
2
)
20 x2โ1โ
(
3
)
x3โ1=40 xโ3x2
40 xโ3x2=0โ x=0,40
3
Evalu
ating P at 0,
40
3
, 20
x 0
40
3
20
P(x) 0
Min
32000
27
Max
0
Min
To maximize ๐, the value of the second number should be
40
3
while the
value of the first number should be
(
20โ40
3
)
=
60โ40
3=20
3
Therefore, the first and second number are
20
3
and
40
3
, respectively.
