City urbanization City planners model the size of their city u...

Question

City urbanization City planners model the size of their city using the function A(t) = - 1/50t² + 2t +20, for 0 ≤ t ≤ 50, where A is measured in square miles and t is the number of years after 2010.

b. How fast will the city be growing when it reaches a size of 38 mi²?

c. Suppose the population density of the city remains constant from year to year at 1000 people mi². Determine the growth rate of the population in 2030.

Accepted Answer

Hi everyone. Let's take a look at this practice problem dealing with derivatives. This problem says a small town's expansion is modeled by the functions of T, which is equal to -1 divided by 40, multiplied by T2 + 3 T + 15, or 0 is less than or equal to T, which is less than or equal to 60, where S represents the area in square kilometers, and T is the number of years since 2000. If the population density remains steady at 800 people per square kilometer, what is the population growth rate in 2025? We're given 4 possible choices as our answers. For choice A, we have 2400 people per year, for choice B, we have 1900 people per year, for choice C, we have 1400 people per year, and for choice D, we have 2900 people per year. Now, we're asked to determine the population growth rate in the year 2025. And one way we can calculate our population growth rate is to multiply our population density by the rate at which our area is expanding. So to get the rate at which our area is expanding, we'll need to take a derivative of our area function, which is S of T. So, we're going to be looking for S T. Which is going to be equal to the derivative with respect to time, or T, in this case, of our functions of T. So, we'll take the derivative with respect to T of the quantity of minus 1 divided by 40, multiplied by T2 plus 3 T. Plus 15. So when you take a derivative of this expression, so this is going to be equal to. When you take the derivative of the first term, the minus 1 divided by 40, multiplied by T2d with respect to T, we're going to need to use our power rule for derivatives, so we'll multiply our coefficient by the exponent, which will give us minus 2 divided by 40, which is -1 divided by 20. Once we simplify that fraction, that's going to be multiplied by T raised to the 2 minus 1, which is T1, which we can write as just T. Now, for the next term, we're gonna be taking a derivative of 3 T, and when we take a derivative of T that gives us 1, we'll have 3 multiply by 1, so we'll have plus 3 as our next term. And for the final term, we're taking a derivative with respect to T of 15. And here we're taking a derivative of a constant, and anytime we take the derivative of a constant, we get 0. So we'll have plus 0 for the last term. Now that we have an expression for the rate at which our area is changing per year, We need to evaluate this in the year of 2025. So that means that T, It's going to be equal to 2025 minus our initial year that we started counting T, which is 2000. And this is going to be equal to 25. So, I'm gonna need to calculate S of 25. And that is going to be equal to -1 divided by 20. Multiplied by 25. Plus 3. And this is going to be equal to. 1.75, and this has units of kilometers squared per year. Since our area was given in kilometer squared. And our time was in years. So now that we have the rate at which our area is expanding, we can use it to calculate our population growth rate. So our population growth rate, which we'll label as P, is going to be equal to our population density, which we're told was hold steady at 800 people per square kilometer. And so, we'll have P is equal to 800. Multiplied by S of 25. So this is going to be equal to 800. Multiplied by 1.75. And when we evaluate that expression, this is going to be equal to 1400, and this has units of people per year. And we get that unit by looking at the units of our population density, which has units of people per square kilometer, and the rate of our area expansion, or S. Has units of kilometers squared per year. So the square kilometers, cancel out, I'm left with just units of people per year. So, this is the answer that I was actually looking for, and when I compare our answer to our answer choices, the correct answer choice turns out to be answer choice C. So I hope that this explanation has been useful, and I'll see you in the next video.

Key Concepts

Function Analysis

Population Density

Rate of Change

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