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Chapter 1: Problem 16

For Activities 11 through \(18,\) calculate the output value that corresponds to each of the given input values of the function. (Round answers to three decimal places when appropriate.) $$ R(p)=26\left(0.78^{p}\right) ; p=2.1, p=-1 $$

Short Answer

Expert verified
For \( p=2.1, R(p) \approx 15.184 \), and for \( p=-1, R(p) \approx 33.332 \).

Step by step solution

01

Understand the Function

The given function is \( R(p) = 26 \cdot (0.78^p) \). This is an exponential function where 26 is the coefficient and 0.78 is the base of the exponent \( p \).
02

Substitute Input p=2.1

Substitute \( p = 2.1 \) into the function: \[ R(2.1) = 26 \times (0.78^{2.1}) \]
03

Calculate Exponent for p=2.1

Calculate \( 0.78^{2.1} \) using a calculator:\( 0.78^{2.1} \approx 0.584 \, (rounded \ to \ three \ decimal \ places) \)
04

Multiply by Coefficient for p=2.1

Now multiply the result by 26:\[ R(2.1) = 26 \times 0.584 \approx 15.184 \] So, \( R(2.1) = 15.184 \) rounded to three decimal places.
05

Substitute Input p=-1

Substitute \( p = -1 \) into the function: \[ R(-1) = 26 \times (0.78^{-1}) \]
06

Calculate Exponent for p=-1

Calculate \( 0.78^{-1} \) using a calculator:\( 0.78^{-1} \approx 1.282 \) (because \( a^{-1} = \frac{1}{a} \))
07

Multiply by Coefficient for p=-1

Now multiply the result by 26:\[ R(-1) = 26 \times 1.282 \approx 33.332 \] So, \( R(-1) = 33.332 \) rounded to three decimal places.

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Key Concepts

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

Coefficient
In mathematics, a coefficient is a crucial number found in many equations and expressions, particularly in exponential functions like the one given, \( R(p) = 26 \cdot (0.78^p) \). The coefficient in this equation is the number 26. It serves a vital role: it's the constant factor that scales the entire function. Essentially, the coefficient multiplies the exponential part of the function, affecting the final outcome dramatically.
  • In our example, regardless of the value of \( p \), every resulting output is multiplied by 26.
  • A larger coefficient would amplify the function's outputs, whereas a smaller coefficient would reduce them.
  • Changing the coefficient uniformly scales the outputs of most graphs of functions upward or downward.
Understanding how a coefficient impacts a function's result helps in predicting and adjusting calculations efficiently.
Exponent
An exponent is a mathematical notation indicating the number of times a number, known as the base, is multiplied by itself. In the function \( R(p) = 26 \cdot (0.78^p) \), \( p \) acts as the exponent. Understanding the role of the exponent can help grasp how functions behave under various inputs.
  • If \( p \) is positive, the base \( 0.78 \) is multiplied by itself \( p \) times. For example, \( 0.78^{2.1} \) requires exponential calculation.
  • If \( p \) is negative, as with \( p = -1 \), it inverts the base to \( \frac{1}{0.78} \).
  • The exponent regulates the rate at which a function's value grows or decays.
The knowledge of exponents is key in solving exponential functions effectively, as it defines the function's complexity and direction.
Function Evaluation
Evaluating a function means determining the output when given an input value. In the context of exponential functions, the process involves several steps to attain the final result. For the function \( R(p) = 26 \cdot (0.78^p) \), we plug different values of \( p \) to find the corresponding outcomes. Let's break this down:
  • First, substitute the value of \( p \) into the function.
  • Next, compute the exponent, calculating \( 0.78 \) raised to the power of \( p \).
  • Finally, multiply the exponential result by the coefficient, in this case, 26.
  • Round the result to the desired precision, such as three decimal places here.
For example, substituting \( p = 2.1 \) results in \( R(2.1) \approx 15.184 \), and with \( p = -1 \), we calculate \( R(-1) \approx 33.332 \). Such evaluation steps are essential, ensuring accuracy in solving and understanding exponential functions.

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Most popular questions from this chapter

A fishing club on the Restigouche River in Canada kept detailed records on the numbers of fish caught by its members between 1880 and \(1930 .\) Average catch tends to be cyclic each decade with the average minimum catch of 0.9 salmon per day occurring in years ending with 0 and average maximum catch of 1.7 salmon per day occurring in years ending with 5 (Source: E. R. Dewey and E. F. Dakin, Cycles: The Science of Prediction. New York: Holt, 1947 ) a. Calculate the period and horizontal shift of the salmon catch cycle. Use these values to calculate the parameters \(b\) and \(c\) for a model of the form \(f(x)=a \sin (b x+c)+d\) b. Calculate the amplitude and average value of the salmon catch cycle. c. Use the constants from parts \(a\) and \(b\) to construct a sine model for the average daily salmon catch.

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