Problem 38 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

After \(x\) weeks, the number of people using a new rapid transit system was approximately \(N(x)=6 x^{3}+500 x+8,000\). a. At what rate was the use of the system changing with respect to time after 8 weeks? b. By how much did the use of the system change during the eighth week?

Step-by-Step Solution

Answer

a. 1,652 users per week. b. 1,514 users.

Step 1: Understanding the problem

We need to find two things: (a) the rate of change of the number of users with respect to time after 8 weeks, and (b) the change in the number of users from the end of the 7th week to the end of the 8th week.

Step 2: Finding the rate of change

To determine how the usage of the system is changing over time, we need to find the derivative of the function with respect to time, i.e., calculate \(N'(x)\). The given function is \(N(x) = 6x^3 + 500x + 8,000\).

Step 3: Compute the derivative

Calculate the derivative \(N'(x)\): \[ N'(x) = \frac{d}{dx}(6x^3 + 500x + 8,000) \]. \[ N'(x) = 18x^2 + 500 \].

Step 4: Evaluate the derivative at 8 weeks

Substitute \(x = 8\) into the derivative to find the rate of change after 8 weeks: \[ N'(8) = 18(8^2) + 500 \]. \[ N'(8) = 18(64) + 500 \]. \[ N'(8) = 1,652 \]. So, the rate of change at 8 weeks is 1,652 users per week.

Step 5: Determine change in users during the eighth week

To find the change in the number of users during the eighth week, calculate \(N(8) - N(7)\). First, compute \(N(8)\) and \(N(7)\): \[ N(8) = 6(8^3) + 500(8) + 8,000 \]. \[ N(8) = 6(512) + 4,000 + 8,000 \]. \[ N(8) = 3,072 + 4,000 + 8,000 \]. \[ N(8) = 15,072 \].

Step 6: Compute the number of users at 7 weeks

Now calculate \(N(7)\): \[ N(7) = 6(7^3) + 500(7) + 8,000 \]. \[ N(7) = 6(343) + 3,500 + 8,000 \]. \[ N(7) = 2,058 + 3,500 + 8,000 \]. \[ N(7) = 13,558 \].

Step 7: Calculate the difference

Now, subtract the number of users at 7 weeks from the number of users at 8 weeks: \[ N(8) - N(7) = 15,072 - 13,558 \]. \[ N(8) - N(7) = 1,514 \]. Thus, the number of users increased by 1,514 during the eighth week.

Key Concepts

Rate of Change

In the context of calculus, the rate of change helps us understand how a quantity changes over a given interval of time. For this problem, we are interested in the rate at which the number of users of a rapid transit system is increasing after a certain number of weeks. The rate of change can be found by computing the derivative of the function that describes the number of users with respect to time. This derivative tells us how quickly the number of users is changing at any given week. Always remember, a positive rate of change means the number is increasing, while a negative rate means it’s decreasing.

Derivative Computation

To compute the rate of change, we start with the given function for the number of users, which is \[N(x) = 6x^3 + 500x + 8000\]. The derivative of this function, denoted as \(N'(x)\), will tell us how the number of users changes over time, or more precisely, over weeks. Calculating the derivative involves applying basic rules of differentiation:

• The derivative of \(x^n\) with respect to \(x\) is \(nx^{n-1}\).
• The derivative of a constant is zero.

Applying these rules, the derivative of the given function is calculated as: \[N'(x) = 18x^2 + 500\]. Now, by substituting \(x = 8\), we can find the rate of change after 8 weeks: \[N'(8) = 18(8^2) + 500 = 1652\]. This means that after 8 weeks, the number of users is increasing at a rate of 1652 users per week.

Function Evaluation

Function evaluation is a critical step in many problems involving calculus. It involves substituting a specific value into the function. To understand the change in the number of users during a specific week, we need to evaluate the function at different points. In this problem, to find the number of users during the eighth week, we evaluate the function at weeks 7 and 8:

• First, we calculate \(N(8)\):\[N(8) = 6(8^3) + 500(8) + 8000 = 15072\].
• Then, we calculate \(N(7)\):\[N(7) = 6(7^3) + 500(7) + 8000 = 13558\].

Now to find the change in users during the eighth week, we subtract the number of users at week 7 from week 8: \[N(8) - N(7) = 15072 - 13558 = 1514\]. Hence, the system had an increase of 1514 users during the eighth week.