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Chapter 2: Problem 40

It is projected that \(t\) months from now, the population of a certain town will be \(P(t)=3 t+5 t^{3 / 2}+6,000 .\) At what percentage rate will the population be changing with respect to time 4 months from now?

Short Answer

Expert verified
The population is changing at approximately 0.2974% per month at the 4-month mark.

Step by step solution

01

- Differentiate the population function

The population function is given by \[P(t) = 3t + 5t^{3/2} + 6,000.\]To find the rate of change of the population with respect to time, differentiate the function with respect to \(t\).
02

- Apply the power rule

Using the power rule for differentiation \( \frac{d}{dt}[t^n] = nt^{n-1}, \)we differentiate each term individually:\[P'(t) = \frac{d}{dt}[3t] + \frac{d}{dt}[5t^{3/2}] + \frac{d}{dt}[6,000].\]
03

- Differentiate each term

Differentiate each term using the power rule:1. \( \frac{d}{dt}[3t] = 3 2. \)\( \frac{d}{dt}[5t^{3/2}] = 5 \cdot \frac{3}{2}t^{1/2} = \frac{15}{2}t^{1/2} \)3. \( \frac{d}{dt}[6,000] = 0 \)So,\[P'(t) = 3 + \frac{15}{2}t^{1/2}.\]
04

- Evaluate the derivative at \(t=4\)

Substitute \(t = 4\) into the derivative to find the rate of change at \(t = 4\): \[P'(4) = 3 + \frac{15}{2}\cdot 4^{1/2}\]Since \(4^{1/2} = 2\): \[P'(4) = 3 + \frac{15}{2}\cdot 2 = 3 + 15 = 18.\]
05

- Convert rate of change to a percentage

To express the rate of change as a percentage, consider the rate of change per month. Here, \(P'(4)=18\) people per month. The percentage rate is given as \[\left(\frac{P'(4)}{P(4)}\right) \cdot 100\%\]First, find \(P(4)\): \[P(4) = 3\cdot 4 + 5\cdot 4^{3/2} + 6,000\]\[= 12 + 5 \cdot 8 + 6,000\]\[= 12 + 40 + 6,000 = 6,052.\]Now, calculate the percentage: \[\left(\frac{18}{6,052}\right) \cdot 100\% \approx 0.2974\%.\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Differentiation
Differentiation is a fundamental concept in calculus. It is the process of finding the derivative of a function. The derivative represents how a function changes as its input changes, which is known as the rate of change. In the context of population growth, differentiation helps us determine how the population size changes over time.
For the function given in the exercise, we need to differentiate the population function \( P(t) = 3t + 5t^{3/2} + 6,000 \) with respect to time \( t \). This process allows us to find the rate at which the population is changing at any given moment.
Differentiation involves applying certain rules, such as the power rule, which we will discuss later. The result of differentiation is a new function, called the derivative, which gives us the rate of change for the original function at different points.
Rate of Change
The rate of change measures how one quantity changes in relation to another. In the context of population growth, it tells us how fast the population is increasing or decreasing over time. This is extremely important for understanding trends and making predictions.
In the exercise, we found the rate of change of the population with respect to time by calculating the derivative of the population function \( P(t) = 3t + 5t^{3/2} + 6,000 \). After differentiating, we get \( P'(t) = 3 + \frac{15}{2}t^{1/2} \).
To determine the rate of change at a specific time, say 4 months from now, we substitute \( t = 4 \) into the derivative function. This gives us \( P'(4) = 3 + 15 = 18 \).
This means the population is growing by 18 people per month four months from now.
Power Rule
The power rule is a quick way to find the derivative of functions where the variable is raised to a power. The rule states: \[ \frac{d}{dt}[t^n] = nt^{n-1} \]. This is highly useful for differentiating polynomial functions.
In the exercise, we applied the power rule to differentiate each term of the population function. Let's break down the steps:
  • For \( 3t \), we get \( 3 \).
  • For \( 5t^{3/2} \), we get \( 5 \cdot \frac{3}{2} t^{(3/2)-1} = \frac{15}{2} t^{1/2} \).
  • For the constant term 6,000, its derivative is 0 since constants do not change.
Combining these, we get \( P'(t) = 3 + \frac{15}{2}t^{1/2} \). The power rule simplifies the process and helps us quickly find the rate of change for each component of the function.

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Most popular questions from this chapter

It is estimated that the monthly cost of producing \(x\) units of a particular commodity is \(C(x)=0.06 x+3 x^{1 / 2}+20\) hundred dollars. Suppose production is decreasing at the rate of 11 units per month when the monthly production is 2,500 units. At what rate is the cost changing at this level of production?

Find all the points \((x, y)\) on the graph of the function \(y=4 x^{2}\) with the property that the tangent to the graph at \((x, y)\) passes through the point \((2,0)\).

It is projected that \(t\) months from now, the average price per unit for goods in a certain sector of the economy will be \(P\) dollars, where $$ P(t)=-t^{3}+7 t^{2}+200 t+300 $$ a. At what rate will the price per unit be increasing with respect to time 5 months from now? b. At what rate will the rate of price increase be changing with respect to time 5 months from now? c. Use calculus to estimate the change in the rate of price increase during the first half of the sixth month. d. Compute the actual change in the rate of price increase during the first half of the sixth month.

The formula \(D=36 m^{-1.14}\) is sometimes used to determine the ideal population density \(D\) (individuals per square kilometer) for a large animal of mass \(m\) kilograms (kg). a. What is the ideal population density for humans, assuming that a typical human weighs about \(70 \mathrm{~kg}\) ? b. The area of the United States is about \(9.2\) million square kilometers. What would the population of the United States have to be for the population density to be ideal? c. Consider an island of area \(3,000 \mathrm{~km}^{2}\). Two hundred animals of mass \(m=30 \mathrm{~kg}\) are brought to the island, and \(t\) years later, the population is given by $$ P(t)=0.43 t^{2}+13.37 t+200 $$ How long does it take for the ideal population density to be reached? At what rate is the population changing when the ideal density is attained?

Find the derivative of the given function. $$ f(x)=6 x^{4}-7 x^{3}+2 x+\sqrt{2} $$
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