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Chapter 8: Problem 12

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 in the first four years is approximately $78,000.

Step by step solution

01

Understand the Problem

We want to calculate the total revenue obtained by the company over a period of four years when the rate of revenue flow is given by the function \(f(t)=9000 \sqrt{1+2t}\). To find this total revenue over a specified time interval, we need to find the definite integral of \(f(t)\) from \(t=0\) to \(t=4\).
02

Write the Definite Integral

The total revenue over the first 4 years is given by the integral \[ R = \int_{0}^{4} 9000 \sqrt{1+2t} \, dt \]. This will calculate the total accumulation of revenue from \(t=0\) to \(t=4\).
03

Simplify the Integrand

In the integrand \(9000 \sqrt{1+2t} \), the inner expression can be simplified by substitution. Let \( u = 1 + 2t \), so that \( du = 2 dt \), or \( dt = \frac{1}{2} du \). When \(t=0\), \(u=1\); when \(t=4\), \(u=9\).
04

Substitute and Solve the Integral

Substituting \(u\) in the integral gives us \[ R = \int_{1}^{9} 9000 \sqrt{u} \frac{1}{2} \, du \ = 4500 \int_{1}^{9} u^{1/2} \, du \]. The antiderivative of \(u^{1/2}\) is \(\frac{2}{3}u^{3/2}\).
05

Evaluate the Antiderivative

Apply the limits to the antiderivative: \[ R = 4500 \times \frac{2}{3} \left[ u^{3/2} \right]_{1}^{9} = 4500 \times \frac{2}{3} \left[ 9^{3/2} - 1^{3/2} \right] \].
06

Calculate the Result

First, calculate \(9^{3/2} = (3^2)^{3/2} = 3^3 = 27\). Substitute the values into the expression: \[ R = 4500 \times \frac{2}{3} (27 - 1) = 4500 \times \frac{2}{3} \times 26 \]. This simplifies to: \[ R = 4500 \times \frac{52}{3} = 4500 \times 17.33 \approx 78000 \].
07

Conclude the Findings

Therefore, by evaluating the definite integral over the specified interval, we find that the total revenue obtained over the first four years is approximately \(78,000\) dollars.

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Key Concepts

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

Definite Integral
A definite integral helps us calculate the total amount accumulated over a specific interval. In this context, it sums up the revenue the company earns over four years. To compute this, we use the function provided: \( f(t)=9000 \sqrt{1+2t} \). This tells us how fast revenue is coming in at any given time \( t \).
In our problem, we want to sum up all the revenue from \( t = 0 \) to \( t = 4 \), which requires integrating \( f(t) \) over this interval. Think of definite integrals as a way to "add up" all the tiny slices of revenue over time.
  • The limits of integration \( 0 \) to \( 4 \) determine the start and end of this accumulation.
  • The result of a definite integral is a specific numerical value, representing the total accumulation.
The outcome provides the total revenue collected during the specified time period.
Revenue Calculation
Revenue calculation using integrals is a powerful method for understanding how income accumulates over time. This function, \( f(t) = 9000 \sqrt{1+2t} \), describes how fast money comes in every year. By evaluating the definite integral of this function, we can estimate the total income during a period.
When a company knows its revenue rate function, they can predict future finances.
  • It helps in planning budgets and financial strategies.
  • Businesses can determine how efforts or events affect revenue streams.
Calculating the integral in our exercise shows that in four years, the company accumulates approximately \$78,000, guiding future corporate decisions.
Antiderivative
To solve the integral, we need to find the antiderivative, a function whose derivative gives us back the original function \( f(t) \). This step involves reversing differentiation and is crucial for computing definite integrals.
In our example, the integrand is \( u^{1/2} \) after substitution. We find its antiderivative as \( \frac{2}{3}u^{3/2} \).
  • The antiderivative mimics 'undoing' the derivative process.
  • It captures how accumulated quantities change over time.
Antiderivatives thus play a key role in converting the rate of revenue into total revenue, enlightening how accumulation manifests.
Substitution Method
The substitution method is a technique used to simplify integrals by changing variables, making integration easier. Here, we substitute \( u = 1 + 2t \) to replace the more complex parts of the function.
Substitution helps by transforming difficult integrals into more manageable forms. Converting our limits of integration when \( t = 0 \) to \( u = 1 \) and when \( t = 4 \) to \( u = 9 \) simplifies the problem.
  • Substitution eliminates clutter and complexity in the integrand.
  • It often leads to straightforward expressions for integration.
By making these changes, we facilitated easier computation, ensuring the total revenue was found efficiently and accurately.

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Most popular questions from this chapter

\(3-6\) Set up, but do not evaluate, an integral for the length of the curve. $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$

The hydrogen atom is composed of one proton in the nucleus and one electron, which moves about the nucleus. In the quan- tum theory of atomic structure, it is assumed that the electron does not move in a well-defined orbit. Instead, it occupies a state known as an orbital, which may be thought of as a "cloud of negative charge surrounding the nucleus. At the state of lowest energy, called the ground state, or Is-orbital, the shape of this cloud is assumed to be a sphere centered at the nucleus. This sphere is described in terms of the probability density function $$p(r)=\frac{4}{a_{0}^{3}} r^{2} e^{-2 r / a_{0}} \quad r \geqslant 0$$ where \(a_{0}\) is the Bohr radius\(\left(a_{0} \approx 5.59 \times 10^{-11} \mathrm{m}\right) .\) The integral $$P(r)=\int_{0}^{r} \frac{4}{a_{0}^{3}} s^{2} e^{-2 s / a_{0}} d s$$ gives the probability that the electron will be found within the sphere of radius \(r\) meters centered at the nucleus. (a) Verify that \(p(r)\) is a probability density function. (b) Find lim_{r\rightarrow\infty} \(p(r) .\) For what value of \(r\) does \(p(r)\) have its maximum value? (c) Graph the density function. (d) Find the probability that the electron will be within the sphere of radius 4\(a_{0}\) centered at the nucleus. (e) Calculate the mean distance of the electron from the nucleus in the ground state of the hydrogen atom.

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(a) A type of lightbulb is labeled as having an average lifetime of 1000 hours. It's reasonable to model the probability of failure of these bulbs by an exponential density function with mean \(\mu=1000 .\) Use this model to find the probability that a bulb (i) fails within the first 200 hours, (ii) burns for more than 800 hours. (b) What is the median lifetime of these lightbulbs?

A demand curve is given by \(p=450 /(x+8) .\) Find the consumer surplus when the selling price is \(\$ 10 .\)
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