/*! 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 4 In \(3-10,\) find the value of \... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 4

In \(3-10,\) find the value of \(x\) to the nearest hundredth. $$ x=e^{1.5} $$

Short Answer

Expert verified
The value of \( x \) is approximately 4.48.

Step by step solution

01

Understanding the Problem

Our task is to find the value of \( x \) given as \( x = e^{1.5} \). \( e \) is a mathematical constant approximately equal to 2.71828, and the expression involves raising \( e \) to the power of 1.5.
02

Evaluate the Exponential Expression

To find \( x = e^{1.5} \), calculate \( e \) raised to the power of 1.5. This can be done using a scientific calculator or evaluation software.
03

Calculation Process

Using a calculator, input \( e^{1.5} \) to get approximately 4.48169.
04

Rounding to the Nearest Hundredth

Rounding 4.48169 to the nearest hundredth means looking at the third decimal place (1 in this case). Since this is less than 5, we round down to get 4.48.

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Key Concepts

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

Understanding Euler's Number
Euler's Number, often denoted by the letter \( e \), is a fundamental mathematical constant. It is similar in importance to the number \( \pi \) in the realm of natural phenomena mathematics.
Unlike physical constants, Euler's Number is about growth in processes like compounding interest, population growth, and radioactive decay. Its value is approximately \( 2.71828 \).
  • \( e \) arises naturally in problems involving exponential growth or decay.
  • It is the base of natural logarithms, meaning the logarithm of \( e \) itself is 1.
When you see an expression like \( e^{1.5} \), understanding Euler's number helps comprehend how such growth or multiplication happens exponentially. Thus, every increase on the exponent implies significant growth of the whole number.
Rounding Numbers Made Easy
Rounding numbers is a crucial skill in math that helps simplify figures without significantly losing accuracy. It's about deciding how many digits are appropriate for a given situation.
The process involves looking at the number right after the place you wish to round to. For instance, rounding to the nearest hundredth means focusing on the thousandths digit.
  • If the thousandths digit is 5 or more, you round up the hundredths digit by one.
  • If it's less than 5, you keep the hundredths digit as it is.
In our example, we rounded 4.48169 to 4.48 because the third digit (1) is less than 5. Always take care when rounding, as every decimal place holds value depending on context.
Using a Scientific Calculator Effectively
A scientific calculator is a handy tool when dealing with complex computations, including exponential expressions. Here are steps to get the best of it when calculating powers of \( e \):
First, ensure your calculator has an \("e^x"\) function or similar. Most scientific calculators do because of Euler's Number's importance in advanced math calculations.
  • Type \( 1.5 \) into the calculator.
  • Press the \("e^x"\) or equivalent function.
  • Your calculator should display the result of the exponential operation.
Remember, practice will improve your speed and confidence. Knowing how to use your calculator effectively means you do not just rely on what it tells you, but also understand why it shows a specific number.

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Most popular questions from this chapter

a. When Kyle was born, his grandparents invested \(\$ 5,000\) in a college fund that paid 4\(\%\) per year, compounded yearly. What was the value of this investment when Kyle was ready for college at age 18\(?\) (Note that \(r=0.04 . )\) b. If Kyle's grandparents had invested the \(\$ 5,000\) in a fund that paid 4\(\%\) compounded continuously, what would have been the value of the fund after 18 years?

In \(58-73\) , write each power as a radical expression in simplest form. The variables are positive numbers. $$ \frac{8^{\frac{1}{4}} a^{\frac{5}{6}} b^{\frac{3}{6}}}{\left(27 c^{4}\right)^{\frac{1}{6}}} $$

In \(1986,\) the worst nuclear power plant accident in history occurred in the Chernobyl Nuclear Power Plant located in the Ukraine. On April \(26,\) one of the reactors exploded, releasing large amounts of radioactive isotopes into the atmosphere. The amount of plutonium present after \(t\) years can be modeled by the function: $$ y=P e^{-0.0000288 t} $$ where \(P\) represents the amount of plutonium that is released. a. Graph this function over the interval \(0 \leq t \leq 100,000\) and \(P=10\) grams. b. If 10 grams of the isotope plutonium- 239 were released into the air, to the nearest hundredth, how many grams will be left after 10 years? After 100 years? c. Using the graph, approximate how long it will take for the 10 grams of plutonium- 239 to decay to 1 gram.

In \(38-57,\) write each radical expression as a power with positive exponents and express the answer in simplest form. The variables are positive numbers. $$ \sqrt{25 a} $$

In \(58-73\) , write each power as a radical expression in simplest form. The variables are positive numbers. $$ 5^{\frac{1}{2}} $$
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