Transcript File

Functions as Real
World Models
Functions as Real World Models
Many of the processes studied in the physical
and social sciences involves understanding how one
quantity is related to another quantity. Finding the
function that describes the dependence of one
quantity to another is called modeling. Modeling
real world problems especially those that require
optimization is one of the important applications of
the study of functions.
Functions as Real World Models
Example 2.4.1.
Emma and Brandt drive away from a campground at
right angles to each other. Emma’s speed is 65 kph
and Brandon’s is 55 kph.
a) Express the distance between the cars as a
function of time.
b) Find the domain of the function.
Functions as Real World Models
Illustration:
distance  rate  time
Emma’s
direction
camp
ground
Distance between Emma
and Brandt at time t
Brandon’s
direction
Functions as Real World Models
Solution:
a. Suppose 1 hr has gone by. At that time,
Emma has traveled 65 km and Brandt has
traveled 55 km.
We can use the Pythagorean theorem to find
the distance between them.
This distance would be the length of the
hypotenuse of a triangle with legs measuring 65
km and 55 km.
Functions as Real World Models
Solution:
(continuation)
After 2 hours, the triangle’s legs would measure
130 km and 110 km. Observe that the distances
will always be changing. We make a drawing
and let t be the time in hours that Emma and
Brandt have been driving since leaving the
campground.
Functions as Real World Models
Illustration:
After t hours, Emma has traveled 65t km and
Brandt 55t km.
Functions as Real World Models
Solution:
(continuation)
Using the Pythagorean theorem:
d t 2  65t 2  55t 2 ,
Because distance must be nonnegative, we need
consider only the positive square root when solving
for d(t):
2
2
d t  
65 t   55 t 
Thus, d(t) = 85.15t, t ≥ 0.
Functions as Real World Models
Solution:
b. Since the time traveled, t, must be
nonnegative, the domain is the set of nonnegative
real numbers [0,) .
Functions as Real World Models
Example 2.4.2
A rectangular field is to be fenced along the bank of a river,
and no fence is required along the river. The material for the
fence costs PhP8 per running foot for the two ends and
PhP12 for running foot for the side parallel to the river;
PhP3600 worth of fence is to be used.
a. Let x be the length of an end; express the number of
square feet in the area of the field as a function of x
b. What is the domain of the resulting function?
Functions as Real World Models
Illustration:
x
Given:
The material for the fence costs PhP8 per
running foot for the two ends and PhP12 for
running foot for the side parallel to the river;
PhP3600 worth of fence is to be used.
Functions as Real World Models
Solution
a. Let y be the length of the side of the field parallel to the
river and A square feet be the area of the field.
Then because the cost of the material for each end is PhP8
per running foot and the length of an end is x feet , the
total cost of the fence for each end is 8x pesos. Similarly,
the total cost of the fence for the third side is 12y pesos.
We then have
8x + 8x + 12y =3600. (1)
Functions as Real World Models
Solution:
(continuation)
To express A in terms of a single variable, we first
solve equation (1) for y in terms of x.
12 y  3600 16 x
4
y  300  x
3
We substitute this value of y in the equation ,
yielding as a function of x, and
4
A( x )  x( 300  x )
3
Functions as Real World Models
Solution:
b. Both x and y must be nonnegative. The smallest
value that x can assume is 0. The smallest value that
y can assume is 0, and when y=0, we obtain from
equation (1) x=225. Thus is 225 is the largest. And
[0,225] is the domain of A as a function of x.
Functions as Real World Models
Example 2.4.3 A buko pie store can produce buko
pie at a cost of PhP95 per piece. It is estimated that
if the selling price of the buko pie is x pesos, then the
number of buko pie that are sold each day is 1000-x.
a. Express the daily profit of the store as a function of
x.
b. Use the result in a) to determine the daily profit ,
given that the selling price is PhP160.
Functions as Real World Models
Solution:
a. The profit P(x) can be obtained by subtracting the
total cost C(x) from the total revenue, R(x).
The total revenue is the product of the selling
price and the number of buko pie sold in a day. So
R(x)= x(100-x) . On the other hand, the total cost is the
product of cost per buko pie and the number of buko
pie sold in a day. Equivalently, C(x) = 95(1000-x).
Functions as Real World Models
Solution:
(continuation)
Hence,
P(x) = R(x) – C(x)
P(x) = x(1000 - x) – 95(1000 - x)
P(x) = (1000 - x)(x - 95)
Functions as Real World Models
Solution:
b.
P(160) = (1000 -160)(160 - 95)
P(160) = (840)(65) = PhP54600
Before we proceed with more examples let us define
the following relationships.
Functions as Real World Models
Definition 1. Directly Proportional
A variable y is said to be directly proportional
to a variable x if
y = kx,
where k is a nonzero constant.
More generally, a variable y is said to be
directly proportional to the nth power of x if
y = kxn
where k is a nonzero constant.
Functions as Real World Models
Example 2.4.4.
The approximate weight of a person’s muscles is
directly proportional to his or her body weight.
a. Express the number of kilograms in the
approximate muscle weight of a person as a function
of the person’s body weight, given that a person
weighs 68 kg has muscles weighing approximately 27
kg.
b. Find the approximate muscle weight of a person
weighing 50 kg.
Functions as Real World Models
Solution:
a. Let x kg be the approximate muscle weight of a
person having a body weight of kg.
Then f(x) = kx. Because a person of body weight 68
kg has muscles weighing approximately 27 kg,
x = 68 and f(x) = 27. Then
27 = k(68)
27
x
Thus, f ( x ) 
68
27
k
68
Functions as Real World Models
Solution:
27
x the approximate muscle
b. Since f ( x ) 
68
weight of a person weighing 50 kg is
27
f (50) 
(50)
68
f (50)  19.85kg
Functions as Real World Models
Definition 2. Inversely Proportional
A variable y is said to be inversely proportional to a
variable x if y  k
x
where k is a nonzero constant.
More generally, a variable y is said to be inversely
proportional to the nth power of x if
k
y n
x
where k is a nonzero constant.
Functions as Real World Models
Example 2.4.5
For an electric cable of fixed length, the resistance is
inversely proportional to the square of the diameter of the
cable.
a. Given that a cable having the fixed length is ½ cm in
diameter and has a resistance of 1 ohm, express the
number of ohms in the resistance as a function of the
number of centimeters in the diameter.
b. What is the resistance of a cable having the fixed length
and a diameter of 2/3 cm?
Functions as Real World Models
Solution:
a. Let f(x)ohms be the resistance of a cable having the
k
f
(
x
)

fixed length with x cm in diameter. Then
. If
x2
a cable having the fixed length is ½ cm in diameter and
has a resistance of 1 ohm,
k
then we obtain 1 
. Then k = ¼.
2
1
2
1
Thus, f ( x) 
 
4x
2
Functions as Real World Models
Solution :
9
2
b. If x  then f ( 2 / 3)  .
3
16
Therefore, the resistance of a cable having a fixed
length and a diameter of 2/3 cm is 9/16 ohm.
Functions as Real World Models
Definition 3. Jointly Proportional
A variable z is said to be jointly proportional to
variable x and y if
z = kxy
where k is a nonzero constant.
More generally, a variable z is said to be jointly
proportional to the nth power of x and the mth
n m
power of y if
z  kx y
where k is a nonzero constant.
TIME TO THINK
1. A right circular cylinder of height h and radius r is
inscribed in a right circular cone with a height of 10
ft and a base with radius 6 ft.
a. Express the height h of the cylinder as a function
of r.
b. Express the volume V of the cylinder as a function
of r.
c. Express the volume V of the cylinder as a function
of h.
TIME TO THINK
2. An open-top box with a square base is to be
constructed from two materials, one for the bottom
and one for the sides. The volume of a box is to be 9
cubic feet. The cost of the material for the bottom is
Php4 per square foot, and the cost of the material
for the sides is Php3 per square foot.
a. Determine a model for the cost of the box as a
function of its height h. What is the domain of the
function?
b. Which will be the most expensive to construct, a
box with a height of 1 foot, 2 feet, or 3 feet?