Question

The amount of freight transported by rail in the u.s was about 580 billion ton-miles in 1960 and has been increasing at a rate of 2.32% per year since then.a. write a function representing the amount of freight, in billions of ton-miles, transported annually. ( 1960 = year 0 )b. graph the functionc. in what year would you predict that the number of ton-miles would have exceeded or would exceed 1 trillion (1,000 billion)?

Answer 1

The amount of freight transported by rail was 580 billion ton-miles in 1960.

The amount increased at a rate of 2.32% per year.

a. The growth in freight transport with respect to the passing years can be expressed as exponential growth. The general form of this equation is

`y=a(1+r)^x`

Where

y represents the final number after "x" periods of time

a represents the initial number at x=0

r represents the growth rate, expressed as a decimal value

x represents the number of time intervals that have passed

For this example, the initial amount is a=580 billion

The growth rate is r=2.32/100=0.0232

You can determine the equation as follows:

`y=580(1+0.0232)^x`

b. This function is of exponential increase, its domain is all real numbers and its range is all positive real numbers, you can symbolize it as y > 0, it's increasing, and it has a horizontal asymptote as x approaches -∞

To graph it, you have to determine at least two points of the graph, I will determine 3

For x=-100

`\begin{gathered} y=580(1+0.0232)^(-100) \\ y=58.53 \end{gathered}`

For x=0

`\begin{gathered} y=580(1+0.0232)^0 \\ y=580 \end{gathered}`

For x=10

`\begin{gathered} y=580(1+0.0232)^(10) \\ y=729.51 \end{gathered}`

The points are

(-100,58.53)

(0,580)

(10,729.51)

Plot the points and then draw an increasing line from -∞ to +∞

c. To determine the time when the freight transported will exceed 1000 billion tom-miles, you can use the graph, just determine the value of x, for which y=1000

This value is approximate x=23.57

This means that counting from 1960 will take 23 and a half years to reach over a trillion ton-miles transported.

Add the number of years to the initial year to determine the date:

`1960+23=1983`

By 1983 the number of ton-miles transported will exceed the trillion.