Problem 84 - College Algebra

Question

The percentage of U.S. households with cable television can be modeled by $$f(x)=18.32+15.94 \ln x$$ where \(x\) represents the number of years after 1979 and \(f(x)\) represents the percentage of U.S. households with cable television. What percentage of U.S. households had cable television in \(1990 ?\)

Step-by-Step Solution

Answer

The percentage of U.S households that had cable television in 1990 is approximately 50.79%.

Step 1: Determine the value of \(x\)

The variable \(x\) is the number of years after 1979. For the year 1990, \(x = 1990 - 1979 = 11\).

Step 2: Plug the year into the function

Substitute the value of \(x\) into the function, that is, compute \(f(x) = f(11) = 18.32+15.94 \ln 11\).

Step 3: Evaluate the expression

Using the properties of logarithms, perform the calculation. \(f(11) = 18.32 + 15.94 \times \ln 11\), which approximately equals 50.79.

Key Concepts

Logarithmic Functions

Logarithmic functions are an essential part of understanding how certain types of growth or decay work. They are the inverses of exponential functions.
In simpler terms, while exponential functions represent rapid growth, logarithmic functions describe a way to measure how quickly something happens against that growth.
When you see a logarithmic function like:

• \( f(x) = 18.32 + 15.94 \ln x \)

This tells us there's a relationship where the change is proportional to a logarithm of some variable. Here,

• \( \ln x \) is the natural logarithm, which uses the mathematical constant \( e \approx 2.718 \)

Instead of linear growth (like adding a set number each time), with logarithmic functions, the growth rate decreases as the input value increases.
That's why they're frequently used in modeling situations where growth tapers off, like the spread of technology.

Mathematical Modeling

Mathematical modeling is like making a mini-version of a real problem using math. This model helps us predict or understand complex systems better.
The function

• \( f(x) = 18.32 + 15.94 \ln x \)

makes use of a logarithmic function to describe how the percentage of households with cable TV increases over time.
To build such a model, we use data to find patterns, choose a suitable mathematical formula, and then plug values into this formula to get predictions.
In our example, we plug in the year from our original reference year (1979) to find out how much the adoption of cable changed.The strength of modeling is its predictive power.

• By understanding past behavior, we can make informed guesses about the future.
• Technological adoption is often non-linear, capturing these nuances with a model gives clearer insights.

This effort helps stakeholders make decisions, be it in business or public policy.

Percentage Calculations

Percentage calculations represent part of a whole as a fraction of 100.
They are exceedingly common in daily life, from discounts in shops to calculating interest rates.In our scenario, the calculation determines how many households out of a theoretical total 100 have cable TV.
Here's a quick guide to using our formula for percentage calculations:

• Identify your variables: We have \( x \), representing years since 1979.
• Use the formula: Plug the year into \( f(x) \).
• Example: For 1990, \( x = 11 \).
• Result: Calculate \( 18.32 + 15.94 \ln 11 \) which gives us approximately 50.79%

Thus, if you were looking to find out how many in a neighborhood had cable TV back in 1990, this calculation suggests about 50.79% of U.S. households had access.