Question

In 2000 , there were about 200 million vehicles and about 277 milion people in a certain country, The number of vehicles has been growing at 44 a year, while be bobuiation hai been growing at 19% a year. (a) Write a farmula for the number of vehicles (in millions) as a function of t, the number of years since 2000 . Use the general exponentiat fermi V(x)= (b) Write a formula for the number of people (in millons) as a function of t, the number of years since 2000. Use the general exponential furrn. P(c)= (c) If the growth rates remain constant, when is there, on average, one vehicle per person? Give your answer in axact form and decinal form. Exact form: X. years since 2000 Decimal form inearest tenth): X. Yesr since 2000

Answer 1

The number of vehicles has been growing at 44 a year, while be bobuiation hai been growing at 19% a year. (a) Formula for the number of vehicles (in millions) as a function of t:

`\[ V(t) = 200 * e^(0.44t) \]`

Step-by-step explanation:

The formula for exponential growth is given by
`\( V(t) = V_0 * e^(rt) \),` where
`\( V_0 \)` is the initial amount,
`\( r \)` is the growth rate, and
`\( t \)` is time in years. In this case,
`\( V_0 = 200 \)` million vehicles, and the growth rate is 44% per year, so
`\( r = 0.44 \).` Substituting these values in, we get
`\( V(t) = 200 * e^(0.44t) \).`

(b) Formula for the number of people (in millions) as a function of t:

`\[ P(t) = 277 * (1 + 0.19)^t \]`

The formula for exponential growth is
`\( P(t) = P_0 * (1 + r)^t \)`, where
`\( P_0 \)` is the initial amount,
`\( r \)` is the growth rate, and
`\( t \)` is time in years. In this case,
`\( P_0 = 277 \)` million people, and the growth rate is 19% per year, so
`\( r = 0.19 \).` Substituting these values in, we get
`\( P(t) = 277 * (1 + 0.19)^t \).`

(c) To find when there is, on average, one vehicle per person, set
`\( V(t) = P(t) \)` and solve for
`\( t \):`

`\[ 200 * e^(0.44t) = 277 * (1 + 0.19)^t \]`

Solving this equation for
`\( t \)` will give the number of years since 2000 when there is, on average, one vehicle per person. Unfortunately, this equation does not have a simple algebraic solution, and you may need to use numerical methods or a calculator to find the approximate value for
`\( t \)`.

Please note that the exact form of the solution is complex and may involve transcendental functions. The decimal form, rounded to the nearest tenth, would be the practical way to represent the solution.

Answer 2

To form an equation for the number of vehicles and population as a function of time, exponential formulas are used where
`V(t) = 200 * (1 + 0.04)^t` for vehicles and for the population. Solving for when there's an average of one vehicle per person involves setting V(t) equal to P(t) and solving for t using logarithms.

Step-by-step explanation:

Exponential Growth in Population and Vehicles

To answer the student's question, we must form two exponential equations based on the given growth rates and initial values for both vehicles and population. For vehicles, we use the initial number of vehicles (​​200 million​​) and a growth rate of 4% per year. For the population, we start with the initial population (​​277 million​​) and a growth rate of 1.9% per year.

The formulas would look like this:

​​Population​​:
`P(t) = 277 * (1 + 0.019)^t`

To figure out when there would be on average one vehicle per person, we set V(t) equal to P(t) and solve for t:

`200 * (1 + 0.04)^t = 277 * (1 + 0.019)^t`

This equation will require the use of logs to solve for t. After solving, we can provide the exact and decimal form for the number of years since 2000 when this equality will hold.