Problem 116 - A Graphical Approach to Precalculus with Limits

Question

Life Span of Robins \(\quad\) Use the equation $$y=\frac{2-\log (100-x)}{0.42}$$ from Example 11 for Exercises 115 and 116. Estimate the number of years elapsed for \(75 \%\) of the robins to die.

Step-by-Step Solution

Answer

It takes approximately 1.43 years for 75\% of the robins to die.

Step 1: Understand the Problem

We need to estimate the number of years (denoted as \(y\)) it takes for 75% of the robins to die. This means we set \(x = 75\) in the equation provided.

Step 2: Substitute the Percentage

Substitute \(x = 75\) into the equation \(y = \frac{2 - \log (100 - x)}{0.42}\). This yields: \[ y = \frac{2 - \log(100 - 75)}{0.42} \].

Step 3: Simplify the Expression Inside the Logarithm

Calculate \(100 - 75\), which gives us \(25\). Therefore, the equation becomes: \[ y = \frac{2 - \log(25)}{0.42} \].

Step 4: Calculate the Logarithm

Find \( \log(25) \). Assuming the base of the logarithm is 10, \( \log(25) \approx 1.39794\).

Step 5: Substitute the Logarithm Value

Substitute the value of \( \log(25) \) back into the expression: \[ y = \frac{2 - 1.39794}{0.42} \].

Step 6: Perform Arithmetic Operations

Subtract 1.39794 from 2 to get 0.60206. Then, divide 0.60206 by 0.42 to find \(y\): \[ y \approx \frac{0.60206}{0.42} \].

Step 7: Final Calculation

Complete the division: \[ y \approx 1.43348 \].

Step 8: Round the Answer

Round the answer to a reasonable precision, such as hundredths: \[ y \approx 1.43 \].

Key Concepts

Logarithmic Functions

Logarithmic functions are used to reverse the process of exponentiation, which means finding the power to which a number (the base) must be raised to produce a given number. In mathematical terms, if you have \( a^b = c \), the logarithm form is \( \log_a(c) = b \). Logarithms are extremely useful in solving equations where the variable is an exponent.

In the exercise about robins' life span, the logarithmic function \( \log(100-x) \) helps us understand how long it might take for a certain percentage of robins to die. Here, the base of the logarithm is 10, a common choice, known as a common logarithm.

When you're computing logarithms manually, you might use a calculator or a logarithm table, especially for non-integer values like \( \log(25) \), which is approximately 1.39794. Understanding the log function is crucial because it simplifies multiplicative processes into additive ones. This property is especially useful when dealing with data that changes exponentially.

Solving Equations

Solving equations involves finding the value of a variable that makes the equation true. In this exercise, we have the equation \( y = \frac{2-\log(100-x)}{0.42} \). Our goal is to find \( y \) when \( x = 75 \). This process combines substitution, arithmetic, and knowledge of logarithmic functions.

Steps to solve it:

• First, substitute \( x = 75 \) into the equation.
• Simplify inside the parenthesis: calculate \( 100 - 75 = 25 \).
• Compute the logarithm \( \log(25) \).
• Substitute this logarithm value back into the equation.
• Simplify the numerator by subtracting the logarithmic value from 2.
• Finally, divide by the denominator, 0.42, to solve for \( y \).

This method demonstrates the systematic approach needed to solve complex equations, combining multiple calculation steps systematically to find the correct solution.

Percentage Calculations

Percentage calculations are crucial in many mathematical contexts, giving us a way to express parts of a whole in relation to 100. For this problem, 75% indicates the proportion of robins we expect to die.

Here's how we handled it:

• Identify what percentage needs to be calculated (in this case, the percentage of robins that have died, 75%).
• Translate this percent into the context of the problem: \( x = 75 \) for the calculation.
• Using percentages helps us understand parts of a group in a universal way, which is especially valuable in population studies and statistical analyses.

Understanding how and when to apply percentage calculations enables efficient problem solving and ensures accuracy when translating real-world situations into mathematical ones. In our exercise, the percentage told us the value to use in the logarithmic equation and allowed us to track changes in the robin population over time.