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Chapter 8: Problem 14

If revenue flows into a company at a rate of \(f(t)=9000 \sqrt{1+2 t}\), where \(t\) is measured in years and \(f(t)\) is measured in dollars per year, find the total revenue obtained in the first four years.

Short Answer

Expert verified
The total revenue for the first four years is $78,000.

Step by step solution

01

Identify the Problem

The task is to find the total revenue obtained over a period of four years, given the rate of revenue as a function of time: \(f(t) = 9000 \sqrt{1+2t}\).
02

Set Up the Integral

To find the total revenue over the period of four years \((0 \leq t \leq 4)\), we need to integrate the function \(f(t)\). The integral to solve is: \[ R = \int_{0}^{4} 9000 \sqrt{1+2t} \, dt \]
03

Calculate the Integral

To solve the integral, use substitution. Let \( u = 1 + 2t \), then \( du = 2 \, dt \) or \( dt = \frac{1}{2} du \). Change the limits of integration: when \( t = 0, u = 1 \), and when \( t = 4, u = 9 \). Substitute and integrate:\[ R = \int_{1}^{9} 9000 \sqrt{u} \cdot \frac{1}{2} \, du \]\[ R = 4500 \int_{1}^{9} u^{1/2} \, du \]
04

Perform the Integration

Integrate \( u^{1/2} \): \[ \int u^{1/2} \, du = \frac{2}{3} u^{3/2} + C \]Apply this to the integral:\[ R = 4500 \cdot \left[ \frac{2}{3} u^{3/2} \right]_{1}^{9} \]
05

Evaluate the Definite Integral

Calculate the definite integral by substituting the limits:\[ R = 4500 \cdot \left( \frac{2}{3} (9)^{3/2} - \frac{2}{3} (1)^{3/2} \right) \]\[ R = 4500 \cdot \left( \frac{2}{3} \cdot 27 - \frac{2}{3} \cdot 1 \right) \]\[ R = 4500 \cdot \left( 18 - \frac{2}{3} \right) \]
06

Simplify and Compute

Simplify the expression:\[ R = 4500 \cdot \left( 18 - 0.6667 \right) \]\[ R = 4500 \cdot 17.3333 \]Calculate the final result:\[ R = 78000 \]

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Key Concepts

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

Revenue Function
A Revenue Function describes the rate at which revenue flows into a company over time. In this scenario, the function is given by \[ f(t)=9000 \sqrt{1+2t} \] where \( t \) represents time in years, and \( f(t) \) represents revenue in dollars per year.
  • The square root component \( \sqrt{1+2t} \) indicates that revenue changes over time. Depending on the value of \( t \), the revenue rate increases.
  • By integrating this function over a specific time interval, you can find the total revenue collected during that period.
It's a powerful tool for businesses to understand how their income progresses.
Integration by Substitution
Integration by substitution is a technique used to simplify the process of integrating a function. In this exercise, it was applied to the integral: \[ \int_{0}^{4} 9000 \sqrt{1+2t} \, dt \] Here's how substitution worked in this context:
  • Choose substitution: Set \( u = 1 + 2t \). It simplifies the equation within the integral.
  • Calculate the differential: If \( u = 1 + 2t \), then \( du = 2 \, dt \), leading to \( dt = \frac{1}{2} \, du \).
  • Change the integration limits: When \( t = 0 \), \( u = 1 \). When \( t = 4 \), \( u = 9 \).
Integrating in terms of \( u \) thus results in a more manageable integral to solve.
Limits of Integration
When dealing with definite integrals, it's crucial to determine the limits of integration correctly. In this exercise, the limits are defined as the time period from year 0 to year 4:
  • Lower Limit: The starting point of integration, \( t = 0 \).
  • Upper Limit: The ending point of integration, \( t = 4 \).
After substitution, these limits are transformed. Initially, \( t = 0 \) corresponds to \( u = 1 \) and \( t = 4 \) corresponds to \( u = 9 \).This transformation is essential because it changes the variable of integration to \( u \), making the integral easier to evaluate.
Evaluating Integrals
After setting up the integral and applying substitution, the next step is to evaluate it:The integral to compute is: \[ 4500 \int_{1}^{9} u^{1/2} \, du \]By using the power rule of integration: \[ \int u^{n} \, du = \frac{u^{n+1}}{n+1} + C \]Apply it to \( u^{1/2} \): \[ \int u^{1/2} \, du = \frac{2}{3} u^{3/2} + C \]Then, plug in the limits of integration: \[ 4500 \cdot \left[ \frac{2}{3} u^{3/2} \right]_{1}^{9} \]Evaluate and simplify:
  • Plug in the upper limit \( u = 9 \).
  • Subtract the result of the lower limit \( u = 1 \).
The result gives the total revenue over the four-year period.

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Most popular questions from this chapter

In a purely competitive market, the price of a good is naturally driven to the value where the quantity demanded by consumers matches the quantity made by producers, and the market is said to be in equilibrium. These values are the coordinates of the point of intersection of the supply and demand curves. $$ \begin{array}{l}{\text { (a) Given the demand curve } p=50-\frac{1}{20} x \text { and the supply }} \\ {\text { curve } p=20+\frac{1}{10} x \text { for a good, at what quantity and }} \\ {\text { price is the market for the good in equilibrium? }} \\ {\text { (b) Find the consumer surplus and the producer surplus }} \\ {\text { when the market is in equilibrium. Illustrate by sketch- }} \\ {\text { ing the supply and demand curves and identifying the }} \\ {\text { surpluses as areas. }}\end{array} $$

Prove that the centroid of any triangle is located at the point of intersection of the medians. [Hints: Place the axes so that the vertices are \((a, 0),(0, b),\) and \((c, 0)\). Recall that a median is a line segment from a vertex to the midpoint of the opposite side. Recall also that the medians intersect at a point two-thirds of the way from each vertex (along the median) to the opposite side. \(]\)

Use the Theorem of Pappus to ind the volume of the given solid. $$ \begin{array}{l}{\text { The solid obtained by rotating the triangle with vertices }} \\ {(2,3),(2,5), \text { and }(5,4) \text { about the } x \text { -axis }}\end{array} $$

(a) Set up an integral for the area of the surface obtained by rotating the curve about (i) the \(x\) -axis and (ii) the \(y\) -axis. (b) Use the numerical integration capability of a calculator to evaluate the surface areas correct to four decimal places. $$ x=\ln (2 y+1), 0 \leq y \leq 1 $$

Find the centroid of the region bounded by the given curves. $$ y=2-x^{2}, \quad y=x $$
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