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Chapter 9: Problem 17

Write an equation to represent the value of \(K\) in terms of \(n\) $$ \begin{array}{|c|c|c|c|c|c|}\hline n & {0} & {1} & {2} & {3} & {4} \\ \hline K & {1} & {0.4} & {0.16} & {0.064} & {0.0256} \\ \hline\end{array} $$

Short Answer

Expert verified
The equation is \(K = (0.4)^n\).

Step by step solution

01

Identify the Pattern

Observe the values of \(K\) as \(n\) increases. Notice how each value of \(K\) can be seen as a percentage of the previous value.
02

Calculate the Ratio

Determine the ratio between consecutive \(K\) values. For example, \(0.4 / 1 = 0.4\), \(0.16 / 0.4 = 0.4\), \(0.064 / 0.16 = 0.4\), \(0.0256 / 0.064 = 0.4\). The consistent ratio is \(0.4\).
03

Establish the Formula

Given that the initial value \(K_0 = 1\) and a consistent ratio of \(0.4\), use an exponential decay formula: \(K_n = K_0 \times r^n\) where \(r\) is the ratio. Substitute \(K_0 = 1\) and \(r = 0.4\) to get \(K_n = 1 \times (0.4)^n\).
04

Write the Final Equation

The final equation for \(K\) in terms of \(n\) is \(K = (0.4)^n\).

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Key Concepts

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

Algebra Patterns
In algebra, recognizing patterns is a crucial skill. Here, we observe the pattern in how the value of K changes as n increases. The pattern is that each K value is a consistent percentage of the previous K value. By identifying this pattern, you can predict future values or formulate equations. Spotting these recurring relationships simplifies the process of finding solutions to algebraic problems. Pay close attention to sequences and how elements change from one to another, as these changes often hold the key to forming the algebraic relationship you need.
Ratio Calculation
Calculating ratios helps determine how quantities compare with each other. In this problem, we need to find the ratio between consecutive K values. By dividing one value of K by the previous one, we see that each ratio is 0.4. For example:
  • 0.4 / 1 = 0.4
  • 0.16 / 0.4 = 0.4
  • 0.064 / 0.16 = 0.4
  • 0.0256 / 0.064 = 0.4
This consistent ratio shows us how the numbers decrease, and is fundamental for developing the equation representing K in terms of n.
Function Representation
Representing mathematical relationships with functions makes it easier to understand and analyze them. In the given problem, we can describe the relationship between K and n using a function. The identified ratio suggests an exponential decay model, where the base of the decay is 0.4. Using the initial value (when n=0) which is 1, we represent the function as:
\( K_n = 1 \times (0.4)^n \)
This compact form of the function helps in simplifying calculations and predicting values for any given n.
Mathematical Equations
Mathematical equations are tools to express relationships between variables. In this problem, we've established that the value of K changes exponentially with respect to n. With the initial value at \(K_0 = 1\) and a constant ratio of 0.4, we use the formula:
\( K_n = K_0 \times r^n \)
Substituting known values, the equation becomes:
\( K = (0.4)^n \)
This equation precisely conveys how K relates to n and allows for easy computation of K for any value of n. Using this final equation helps in quickly finding the needed solutions without recalculating the pattern every time.

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Most popular questions from this chapter

Automobile license plates in one state consist of three different letters followed by three different digits. The state does not use vowels or the letter \(Y\) , which prevents slang words from accidentally appearing on license plates. a. Each letter can appear only once on a given license plate. How many different sets of three letters are possible? b. Each digit can appear only once on a given license plate. How many different sets of three digits are possible? c. Altogether, how many different license plates with three letters followed by three digits are possible for this state?

Simplify each expression. $$ \frac{m-3}{m(2 m-6)} $$

Use the quadratic formula to solve each equation. $$ -6 a^{2}+3 a=-4 $$

Challenge In Investigation \(2,\) you always assumed that two teams have the same chances of winning a single game. For this exercise, assume that Team \(\mathrm{A}\) has a 60\(\%\) chance of defeating Team \(\mathrm{B}\) in every game they play against each other. a. Suppose there is a one-game tournament between the teams and the winner of the game wins the tournament. What is the probability that Team \(\mathrm{A}\) will win? That Team \(\mathrm{B}\) will win? b. Use a tree diagram to show all the possibilities for the tournament. For example, in the first game, there are two branches: A wins or \(\mathrm{B}\) wins. (Hint: If \(\mathrm{A}\) wins the first two games, is a third game played?) c. Suppose the teams played \(1,000\) tournaments. In how many tournaments would you expect Team \(\mathrm{A}\) to win the first game? In how many of those tournaments would you expect Team \(\mathrm{A}\) to also win the second game? d. For each combination in your tree diagram, use similar reasoning to find the number of tournaments out of \(1,000\) you would expect to go that way. For example, one combination should be ABB; in how many tournaments out of \(1,000\) would you expect the winner to be \(\mathrm{A},\) then \(\mathrm{B}\) , and then \(\mathrm{B}\) ? (Hint. Check your answers by adding them; they should total to \(1,000 . )\) e. Find the total number of tournaments out of \(1,000\) in which each team wins the tournament. What is the probability that Team \(\mathrm{A}\) wins a tournament? f. Which tournament, one-game or best-two-out-of-three, is better for Team B?

Petra wants to make a withdrawal from an automated teller machine, but she can鈥檛 remember her personal identification number. She knows that it includes the digits 2, 3, 5, and 7, but she can鈥檛 recall their order. She decides to try all the possible orders until she finds the right one. a. How many orders are possible? b. Petra remembers that the first digit is an odd number. Now how many orders are possible? c. Petra then remembers that the first digit is 5. How many orders are possible now?
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