Basic Calculus Performance Task 2
SOLVING OPTIMIZATION PROBLEMS USING CALCULUS
5. The sum of two positive numbers is 30. Find the numbers if the sum of
their squares is a minimum.
Important Notes:
1) We can denote the problem as x + y = 30, wherein we will minimize
x2+y2
. [Where x is the first number and y is the second number].
2) Since
x+y=30
, then
y=30−x .
a. What is the objective? Let it be 𝑆(𝑥).
The objective is the sum of the squares of the two positive numbers. We are
required to find the numbers that will minimize
S
.
b. What variable are you going to control? Let it be 𝑥.
Let
x
be the first number. Let it be the control variable. Let
30−x
be the
second number (Notice that y is the second number on “Important Notes”
number 1).
c. What function accurately models this problem?
Our model is
S
(
x
)
=x2+
(
30−x
)
2=x2+900−60 x+x2=2x2−60 x+900
. Let it be
continuous over [0, 30].
d. What are the numbers?
Finding its critical points.
S'
(
x
)
=
(
2
)
2x2−1−60 x1−1+900=4x−60
4x−60=0→ x=15
Evalu
ating P at 0,
15
, 30
x 0
15
30
S(x) 900
Max
450
Min
900
Max
To minimize S, the value of the first number should be
15
while the value of
the first number should be (
30−15
)
¿15
Therefore, the first and second number are
15
and
15
.
e. What is the minimum sum of their squares?
The minimum sum of the squares of the first number and the second
number is
P
(
15
)
=2(15)2−60
(
15
)
+900=2
(
225
)
−900+900 =450
Alternative Method: Using
x2+y2
x2+y2=(15)2+(15 )2=225+225=450
