/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 10 Use \(f(t)=10 e^{-t}\). For wh... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 10

Use \(f(t)=10 e^{-t}\). For what value of \(t\) will \(f(t)=2 ?\)

Short Answer

Expert verified
The value of \(t\) for which \(f(t)=2\) is \(t = - \ln(0.2)\)

Step by step solution

01

Set the Function Equal to 2

The first thing to do is to set the function \( f(t) = 10e^{-t}\) equal to 2. This gives us the equation \( 10e^{-t} = 2 \).
02

Isolate the Exponential Term

Next, we isolate the exponential term on one side of the equation. This is done by dividing both sides of the equation by 10 to get \( e^{-t} = 0.2 \) .
03

Apply Natural Logarithm

We then apply the natural logarithm to both sides of the equation. The natural logarithm of \( e^{-t} \) is just \(-t\). So, by applying the natural logarithm to both sides, we get the equation \( -t = \ln(0.2) \) .
04

Solve for t

The final step is to solve for \(t\). Because the equation is \(-t = \ln(0.2)\), we multiply both side of the equation by -1 to find \( t = -\ln(0.2)\) .

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Key Concepts

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

Natural Logarithm
The natural logarithm, denoted as \(\ln(x)\), is a mathematical function that is the inverse of the exponential function \(e^x\), where \(e\) is Euler's number, approximately equal to 2.71828. This means that if \(y = e^x\), then \(x\) is the natural logarithm of \(y\), or \(x = \ln(y)\).

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Most popular questions from this chapter

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$f(x)=\frac{1}{x}$$

Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$\log |x-2|+\log |x|=1.2$$

The following table gives the temperature, in degrees Celsius, of a cup of hot water sitting in a room with constant temperature. The data was collected over a period of 30 minutes. (Source: www.phys. unt.edu, Dr. James A. Roberts)$$\begin{array}{|c|c|} \hline\text { Time } & \text { Temperature } \\\\(\mathrm{min}) & (\text { degrees Celsius }) \\ \hline0 & 95 \\\1 & 90.4 \\\5 & 84.6 \\\10 & 73 \\\15 & 64.7 \\\20 & 59 \\\25 & 54.5 \\\29 & 51.4\\\\\hline\end{array}$$ (a) Make a scatter plot of the data and find the exponential function of the form \(f(t)=C a^{2}\) that best fits the data. Let \(t\) be the number of minutes the water has been cooling. (b) Using your modicl, what is the projected temperature of the water after 1 hour?

The following table gives the total amount spent by all candidates in each presidential election, beginning in \(1988 .\) Each amount listed is in millions. (Source: Federal Election Commission) $$\begin{array}{|c|c|} \hline\text { Year } & \text { Price } \\\\\hline1988 & 495 \\\1988 & 550 \\\1992 & 560 \\\1996 & 649.5 \\\2000 & 1,016.5 \\\2004 & 1,016.5 \\\ \hline\end{array}$$ (a) Make a scatter plot of the data and find the exponential function of the form \(P(t)=C a^{2}\) that best fits the data. Let \(t\) be the number of years since 1988 (b) Using your model, what is the projected total amount all candidates will spend during the 2012 presidential election?

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$g(x)=2 x^{5}-6$$
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