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Chapter 5: Problem 69

PUBLIC TRANSPORTATION It is estimated that \(x\) weeks from now, the number of commuters using a new subway line will be increasing at the rate of \(18 x^{2}+500\) per week. Currently, 8,000 commuters use the subway. How many will be using it 5 weeks from now?

Short Answer

Expert verified
11250 commuters.

Step by step solution

01

- Understand the Rate of Change

The rate of change of the number of commuters is given by the function \(18x^{2} + 500\). This represents the derivative of the number of commuters with respect to time \(x\) in weeks.
02

- Set Up the Integral

To find the total number of commuters after 5 weeks, integrate the rate of change function from 0 to 5 weeks. Integrate \(18x^{2} + 500\).
03

- Integrate the Function

Compute the integral: \[ \int_{0}^{5} (18x^{2} + 500) \, dx \] First, separate the integral into two parts: \[ \int_{0}^{5} 18x^{2} \, dx + \int_{0}^{5} 500 \, dx \]
04

- Solve Each Integral

Integrate each term separately:\[ \int 18x^{2} \, dx = 18 \int x^{2} \, dx = 18 \left( \frac{1}{3} x^{3} \right) = 6x^{3} \] \[ \int 500 \, dx = 500x \]
05

- Calculate Definite Integrals

Calculate the definite integrals by evaluating from 0 to 5:\[ \left[ 6x^{3} + 500x \right]_{0}^{5} \] This gives:\[ \left( 6 \times 5^{3} + 500 \times 5 \right) - \left( 6 \times 0^{3} + 500 \times 0 \right) \] \[ 6 \times 125 + 2500 - 0 \] \[ 750 + 2500 = 3250 \]
06

- Add Initial Commuters

Currently, there are 8000 commuters. Add the 3250 from step 5 to find the total number of commuters:\[ 8000 + 3250 = 11250 \]

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Key Concepts

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

Rate of Change
The rate of change is a key concept in calculus. It measures how a quantity, such as the number of subway commuters, changes over time. In this problem, the rate of change is given by the function \(18x^{2} + 500\). This function means that each week, the number of additional commuters using the subway increases based on the formula.

Here are some important points about the rate of change:
  • It is the derivative of a function. In this problem, \(18x^{2} + 500\) is the derivative of the number of commuters.
  • The variable \(x\) represents time in weeks. When \(x\) increases, the rate changes accordingly.
  • Understanding the rate of change helps us predict future values by integrating it over a specific interval.
By integrating the rate of change function from week 0 to week 5, we can find the total number of new commuters added over that time.
Definite Integral
A definite integral helps us find the accumulated value of a function over a certain interval. In this problem, we use the definite integral of the rate of change function to determine the total number of new commuters over 5 weeks.

Here's a breakdown of the steps:
  • Set up the integral from the start time (0 weeks) to the end time (5 weeks): \( \int_{0}^{5} (18x^{2} + 500) \ dx \).
  • Separate the integral into two parts for easier computation: \( \int_{0}^{5} 18x^{2} \ dx + \int_{0}^{5} 500 \ dx \).
  • Calculate the integral of each part: \( \int 18x^{2} \ dx = 6x^{3} \) and \( \int 500 \ dx = 500x \).
  • Evaluate each integral from 0 to 5: \( \left[ 6x^{3} + 500x \right]_{0}^{5} \). This results in \( \left( 6 \times 5^{3} + 500 \times 5 \right) \), or 3250.
Once we've computed the definite integral, we add this value to the initial number of commuters to get the total number of commuters after 5 weeks.
Initial Value Problem
An initial value problem involves finding a function that satisfies a differential equation and meets an initial condition. In this problem, the differential equation is given by the rate of change function \(18x^{2} + 500\), and the initial condition is that currently, there are 8000 commuters.

Steps to solve an initial value problem:
  • Identify the rate of change function, which is the derivative of the original function we need to find.
  • Integrate the rate of change function over the given interval. We did this to find the new commuters added over 5 weeks.
  • Use the initial condition to determine the specific value of the function at the starting point. In this case, it is 8000 commuters at week 0.
  • Add the accumulated change (determined by the definite integral) to the initial value to find the total number of commuters.
By following these steps, we find that 8000 commuters grow by 3250 new commuters over 5 weeks, resulting in a total of 11250 commuters.

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