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Chapter 3: Problem 37

In Exercises 33 - 38, evaluate the function at the indicated value of \( x \). Round your result to three decimal places. Function \( h(x) = 5000e^{0.06x} \) Value \( x = 6 \)

Short Answer

Expert verified
After calculating the expression, round your result to three decimal places.

Step by step solution

01

Substitute the given value

Firstly, replace \( x \) in the equation \( h(x) = 5000e^{0.06x} \) with the given value \( x = 6 \). Thus, the equation becomes: \( h(6) = 5000e^{0.06*6} \).
02

Calculate the exponent

Calculate the exponent, \(0.06 * 6 = 0.36\). Hence, the expression now reads \( h(6) = 5000e^{0.36} \).
03

Calculate the value of the function

Now, calculate the value of the function \( h(6) = 5000e^{0.36} \). Remember, \( e \) here represents Euler's Number (approx. \( 2.71828 \)), and 5000 is being multiplied by \( e^{0.36} \).

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Key Concepts

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

Exponential Growth
Exponential growth is a process that increases quantity over time. It is characterized by the presence of a variable exponent, as seen in the function h(x) = 5000e^{0.06x}. The exponential function, crucially, grows at a rate proportional to its current value, meaning the larger the value, the quicker it grows.

In the context of real-world applications, this kind of growth is seen in populations, finance, such as compound interest, and in natural processes, like radioactive decay. The defining feature of exponential growth is that the variable x is in the exponent, which leads to a rapid increase of the function's output as x increases.
Euler's Number
Euler's Number, denoted as e, is a mathematical constant approximately equal to 2.71828. It is the base of the natural logarithm and is fundamental to the exponential growth described by the equation h(x) = 5000e^{0.06x}.

The number e arises naturally in mathematics when dealing with continuously growing processes and is an irrational number, which means it has an infinite number of decimal places without repeating. Its properties make it a vital component in calculus, especially when dealing with differential equations and compound interest calculations.
Function Evaluation
Function evaluation is the process of determining the output of a function for a particular input. In the given function, h(x) is evaluated at x = 6. This requires substituting 6 into the function in place of x and calculating the result.

Evaluating h(6) involves following a series of steps to process the exponentiation and multiplication. By understanding the syntax of the function and correctly substituting the values, one can find the value of h(x) at any given point, which in this context represents the growth of a quantity, like money investment or population, after 6 time units.
Rounding Decimals
Rounding decimals is a technique used to reduce the number of decimal places in a number to make it easier to work with or understand. When rounding to three decimal places, one looks to the fourth decimal place.
  • If the fourth decimal place is 5 or greater, the third decimal is increased by one.
  • If it is less than 5, the third decimal remains the same.

This approach simplifies the result while maintaining a close approximation to the original number. It is a widely used method in various fields requiring precision, such as science, engineering, and finance, but where an exact figure is not mandatory, or too many decimal places would complicate the calculations.

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Most popular questions from this chapter

The values \( y \) (in billions of dollars) of U.S. currency in circulation in the years \( 2000 \) through \( 2007 \) can be modeled by \( y = -451 + 444 \ln t \), \( 10 \le t \le 17 \), where \( t \) represents the year, with \( t = 10 \) corresponding to \( 2000 \).During which year did the value of U.S. currency in circulation exceed \( \$690 \) billion?(Source: Board of Governors of the Federal Reserve System)

In Exercises 121 - 128, solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. \( e^{-2x} - 2xe^{-2x} = 0 \)

Carbon \( 14 \) dating assumes that the carbon dioxide on Earth today has the same radioactive content as it did centuries ago. If this is true,the amount of \( ^{14} C \) absorbed by a tree that grew several centuries ago should be the same as the amount of \( ^{14} C \) absorbed by a tree growing today. A piece of ancient charcoal contains only \( 15\% \) as much radioactive carbon as a piece of modern charcoal. How long ago was the tree burned to make the ancient charcoal if the half-life of \( ^{14} C \) is \( 5715 \) years?

In Exercises 69 - 74, use the acidity model given by \( pH = -\log \left[H^+\right] \), where acidity \( (pH) \) is a measure of the hydrogen ion concentration \( \left[H^+\right] \) (measured in moles of hydrogen per liter) of a solution. find the \( pH \) if \( \left[H^+\right] = 2.3 \times 10^{-5} \).

In Exercises 25 and 26, determine the time necessary for \( \$1000 \) to double if it is invested at interest rate \( r \) compounded (a) annually, (b) monthly, (c) daily, and (d) continuously. \( r = 6.5\% \)
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