Exercise 1.1
1. World Population A low-projection scenario of world population for the years 1995-2150 by the United Nations is given the function y = 0.36x² +38.52x+5822.86, where x is the number of the years after 1990 and the world population is measured in millions of people.
a. Graph this function for x =0 to x =120.
b. What will be the world population in 2010 if the projections made using this model are accurate?
2. Flight of a Ball If a ball is thrown upward at 96 feet per second from the top of a building that is 100 feet high, the height of the ball can be modeled by S=100+96-16t² feet, where t is the number of seconds after the ball is thrown.
a. Describe the graph of the model.
b. Find the t-coordinate and S-coordinate of the vertex of the graph of this quadratic function.
c. Explain the meaning of the coordinates of the vertex for this model.
3. Satellite Dishes The number of U.S. schools with satellite dishes between 1992and 1997 can be modeled by the function S(x) =-0.877x² + 10.867x - 17.041 thousand schools, where x is the number of years after 1990. Did the number of the schools with satellite dishes reach a maximum or a minimum value between 1992 and 1997? If so, give the maximum or minimum number of dishes.
4. World population A low-projection scenario of the world population for 1995-2150 by the United nation is given by the function y=-0.36x² + 38.52x + 5822.86, where x is the number of the years after 1990 and the world population is measured in millions of people.
a. Find the input and output at the vertex of the graph of the model.
b. Interpret the values from part (a).
c. For what years after 1995 does this model predict that the population will increase?
Exercise 1.2
1. Flight of a Ball If a ball is thrown upward at 96 feet per second from the top of a building that is 100 feet high, the height of the ball can be modeled by S=100+96t-16t² feet, where t is the number of seconds after the ball is thrown. How long after the ball is thrown is the height 228 feet?
2. Velocity of Blood Because of friction from the walls of an artery, the velocity of a blood corpuscle in an artery is greatest at the center of the artery and decreases as the distance r from the center increases. The velocity of the blood in the artery can be modeled by the function.
V = k (R²-r²)
Where R is the radius of the artery and k is a constant that is determined by the pressure, viscosity of the blood, and the length of the artery. In the case where k = 2 and R = 0.1 centimeter, the velocity is v = 2(0.01-r²) centimeters per second cm/sec).
a. What distance r would give a velocity of 0.02 cm/sec?
b. What distance r would give a velocity of 0.015 cm/sec?
c. What distance r would give a velocity of 0 cm/sec? Where is the blood corpuscle?
3. Student Aid Federal funds providing for student financial assistance through programs funded by congressional appropriations between 1980 and 2000 can be described by the model.
f (x) = 9.03x² = 99.970x + 3645.90
million dollars where x is the number of years after 1980.
a. Use the model to compare the federal aid available to students in 1990 and 2000.
b. When did the available federal aid reach and exceed $ 8,000,000,000?
4. World Population One projection of the world population by the United Nation (a low-projection scenario) is given in the table below. This data can be modeled by y = -0.36x² + 38.52x +5822.86 million people, where x is the number of years from 1990. In what year past 1990 does this model predict the world population will first reach 6,702,000,000?
|
Year
|
Projected Population (millions)
|
|
1995
|
5666
|
|
2000
|
6028
|
|
2025
|
7275
|
|
2050
|
7343
|
|
2075
|
6402
|
|
2100
|
5153
|
|
2125
|
4074
|
|
2150
|
3236
|