Let's break down each part of the problem step by step.
### Part (a): Find the growth rate, [tex]\( \frac{dP}{dt} \)[/tex]
The population of the city is given by the function [tex]\( P(t) = 100,000 + 9000 t^2 \)[/tex]. To find the growth rate, we need to differentiate this function with respect to [tex]\( t \)[/tex].
[tex]\[ P(t) = 100,000 + 9000 t^2 \][/tex]
The derivative of [tex]\( P(t) \)[/tex] with respect to [tex]\( t \)[/tex] is:
[tex]\[ \frac{dP}{dt} = \frac{d}{dt} (100,000 + 9000 t^2) \][/tex]
Since the derivative of a constant is 0, and applying the power rule to [tex]\( 9000 t^2 \)[/tex], we get:
[tex]\[ \frac{dP}{dt} = 0 + 9000 \cdot 2 t = 18000 t \][/tex]
So, the growth rate is:
[tex]\[ \frac{dP}{dt} = 18000 t \][/tex]
### Part (b): Find the population after 10 years
To find the population after 10 years, we simply substitute [tex]\( t = 10 \)[/tex] into the population function [tex]\( P(t) \)[/tex].
[tex]\[ P(10) = 100,000 + 9000 (10)^2 \][/tex]
First, we calculate [tex]\( 10^2 \)[/tex]:
[tex]\[ 10^2 = 100 \][/tex]
Next, we multiply by 9000:
[tex]\[ 9000 \cdot 100 = 900,000 \][/tex]
Then, add the initial population:
[tex]\[ 100,000 + 900,000 = 1,000,000 \][/tex]
Therefore, the population after 10 years is:
[tex]\[ P(10) = 1,000,000 \][/tex]
### Part (c): Find the growth rate at [tex]\( t = 10 \)[/tex]
To find the growth rate at [tex]\( t = 10 \)[/tex], we substitute [tex]\( t = 10 \)[/tex] into the growth rate equation [tex]\( \frac{dP}{dt} = 18000 t \)[/tex].
[tex]\[ \left.\frac{dP}{dt}\right|_{t=10} = 18000 \cdot 10 \][/tex]
Which calculates to:
[tex]\[ 18000 \cdot 10 = 180,000 \][/tex]
So, the growth rate at [tex]\( t = 10 \)[/tex] is:
[tex]\[ \left.\frac{dP}{dt}\right|_{t=10} = 180,000 \][/tex]
### Part (d): Explain the meaning of the answer to part (c)
The growth rate at [tex]\( t = 10 \)[/tex] of [tex]\( 180,000 \)[/tex] represents the rate at which the population is increasing per year at the time when [tex]\( t = 10 \)[/tex] years. In other words, at the 10-year mark, the city's population is growing by 180,000 individuals per year. This indicates a very rapid increase in population over time.
