In a report issued by the U.S Department of Transportation in 2020, it was predicted that the number of passengers boarding planes in the United States would grow steadily from the current 0.7 billion boardings/year to 1.2 billion boardings/year in 2032.
a. Find a linear function f giving the projected boardings (in billions) in year t, where t=0 corresponds to 2012.
b. What is the projected annual rate of growth of boarding between 2012 and 2032?
c. How many boarding per year are projected for 2022?
a. 2032-2012=20
(0,0.7) (20,1.2)
1.2-0.7/20-0= 0.5/20 = 0.025
y-1.2=0.025(x-20)
y-12=0.025x-0.5
y-1.2+1.2=0.025x-0.5+1.2
y=0.025x+0.7
f(0)=0.025(0)+0.7=0.7
b. 2032-2012=20
f(20=0.025(20)+0.7=1.2
c. 2022-2012=10
f(10)= 0.025(10)+0.7 = 0.95
To find the linear function f that gives the projected boardings in billions in year t, where t=0 corresponds to 2012, you need to use the formula for a linear function: y = mx + b. In this case, y represents the projected boardings, x represents the number of years since 2012 (t), m represents the rate of growth, and b represents the initial number of boardings in 2012.
a. First, calculate the rate of growth (m). To do this, subtract the initial projected boardings in 2012 (0.7) from the final projected boardings in 2032 (1.2), and divide by the number of years (20): (1.2 - 0.7) / 20 = 0.025.
b. The projected annual rate of growth of boarding between 2012 and 2032 is 0.025 (or 2.5%).
c. To find the projected boardings for 2022, you need to substitute the value of x (10) into the linear function. Using the function f(x) = 0.025x + 0.7, plug in x = 10: f(10) = 0.025(10) + 0.7 = 0.95. Therefore, it is projected that there will be 0.95 billion boardings in 2022.