PRECALCULUS:GRAPHICAL,...-NASTA ED.
10th Edition
ISBN: 9780134672090
Author: Demana
Publisher: PEARSON
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Expert Solution & Answer
Chapter 3.1, Problem 53E
Solution
The year when the population of Columbus, Ohio, and Austin Texas are equal.
The population of Columbus, Ohio, and Austin Texas are equal in 2009 .
Given information:
The table is shown below:
| City | 1990 Population | 2015 Population |
| Austin, Texas | 465,648 | 931,830 |
| Columbus, Ohio | 632,945 | 850,106 |
Concept Used:
Let a and b be real number constants. An exponential function in x is a function that can be written in the form f(x)=abx where a is nonzero, b is positive, and b≠1 . The constant a is the initial value of f (the value at x=0 ), and b is the base.
Calculation:
Let P1(t) be the population of Austin, Texas t years after July 1, 1990.
P1(t)=P0⋅bt1 where P0 is the initial population of Austin, Texas and b is the base.
The initial population of Austin, Texas is P0=465,648 .
The value of t is 2015−1990=25 .
The population of Austin, Texas after 25 years is P1(25)=931,830 .
Substitute 25 for t , 931,830 for P1(25) and 465,648 for P0 in P1(t)=P0⋅bt .
P1(25)=465,648⋅b25931,830=465,648b25b=25√931,830465,648≈1.0281374
The population of Austin, Texas P1(t) is 465,648⋅(1.0281374)t
Let P2(t) be the population of Columbus, Ohio t years after July 1, 1990.
P2(t)=p0⋅bt where p0 is the initial population of Columbus, Ohio and b is the base.
The initial population of Columbus, Ohio is p0=632,945 .
The population of Columbus, Ohio after 25 years is P2(25)=850,106 .
Substitute 25 for t , 850,106 for P2(25) and 632,945 for p0 in P2(t)=p0⋅bt .
P2(25)=632,945⋅b25850,106=632945b25b=25√850106632945≈1.0118689
The population of Columbus, Ohio P2(t) is 632,945⋅(1.0118689)t
Ti-83 Calculator:
- Press the button Y= and then write Y1=465,648⋅(1.0281374)^(X) and Y2=632,945⋅(1.0118689)^(X) .
- Graph the functions. The intersection must show on the screen.
- Press 2ND and then CALC (with the TRACE button). Select ‘5: intersect’.
- At the ‘First curve?’ prompt, use the up and down arrows to select one function. Press the ENTER button.
- At the ‘Second curve?’ prompt, use the up and down arrows to select the second function. Press the ENTER button. Note, if you are only graphing two equations, they will both be automatically selected during steps 4 and 5.
- At the ‘Guess?’ prompt, use the left and right arrows to move the cursor near the intersection and push the ENTER button. The coordinates of the intersection will be at the bottom of the screen!
This population model intersects is about 19.245≈19 year.
So, according to this model, the population of Columbus, Ohio and Austin Texas are equal after 1990+19=2009 .
Chapter 3 Solutions
PRECALCULUS:GRAPHICAL,...-NASTA ED.
Ch. 3.1 - Prob. 1QRCh. 3.1 - Prob. 2QRCh. 3.1 - Prob. 3QRCh. 3.1 - Prob. 4QRCh. 3.1 - Prob. 5QRCh. 3.1 - Prob. 6QRCh. 3.1 - Prob. 7QRCh. 3.1 - Prob. 8QRCh. 3.1 - Prob. 9QRCh. 3.1 - Prob. 10QR
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