Problem 46 - Applied Calculus

Question

With time, \(t,\) in years since the start of \(1980,\) textbook prices have increased at \(6.7 \%\) per year while inflation has been \(3.3 \%\) per year. \(^{68}\) Assume both rates are continuous growth rates. (a) Find a formula for \(B(t),\) the price of a textbook in year \(t\) if it \(\operatorname{cost} \$ B_{0}\) in 1980 (b) Find a formula for \(P(t),\) the price of an item in year \(t\) if it cost \(\$ P_{0}\) in 1980 and its price rose according to inflation. (c) A textbook cost \(\$ 50\) in \(1980 .\) When is its price predicted to be double the price that would have resulted from inflation alone?

Step-by-Step Solution

Answer

The textbook price doubles the inflation price in about 20.39 years since 1980.

Step 1: Define Continuous Growth Formula

The formula for continuous growth can be represented as \( A(t) = A_0 e^{rt} \), where \( A_0 \) is the initial amount, \( r \) is the growth rate, and \( t \) is the time in years.

Step 2: Formula for Textbook Price Over Time

According to the problem, textbook prices grow at a rate of \(6.7\%\) per year. If \( B_0 \) is the initial cost, the formula for the price \( B(t) \) at time \( t \) is given by \( B(t) = B_0 e^{0.067t} \).

Step 3: Formula for Inflation Price Over Time

Given an inflation rate of \(3.3\%\) per year, for an item costing \( P_0 \) initially, the formula for its inflated price \( P(t) \) at time \( t \) is \( P(t) = P_0 e^{0.033t} \).

Step 4: Calculate Textbook Price to Double Inflation Price

We need to find \( t \) such that the textbook price \( B(t) = 2P(t) \). Start by setting \( 50e^{0.067t} = 2 \times 50e^{0.033t} \). Simplifying gives \( e^{0.067t} = 2e^{0.033t} \) and hence \( e^{0.067t - 0.033t} = 2 \).

Step 5: Solve for Time \( t \)

Continue solving \( e^{0.034t} = 2 \) by taking the natural logarithm on both sides: \( 0.034t = \ln(2) \). Therefore, \( t = \frac{\ln(2)}{0.034} \). Calculate this value to get \( t \approx 20.39 \).

Key Concepts

Understanding Exponential Growth

Exponential growth is a key concept in mathematics that describes how quantities increase over time. It occurs when the growth rate of a value is proportional to its current size, leading to faster and faster growth as time passes.
To visualize exponential growth, consider the formula \( A(t) = A_0 e^{rt} \), where

• \( A_0 \): initial value or starting amount
• \( r \): continuous growth rate (as a decimal)
• \( t \): time elapsed

This model effectively describes many real-world situations, such as population growth, radioactive decay, and of course, pricing in economics where we're considering textbook prices.
If a textbook's price increases at a continuous rate of 6.7%, it means each year's price is roughly 6.7% more than the previous year. Importantly, because exponential growth compounds, it results in significantly larger increases over longer periods. This makes understanding the exponential growth model crucial for predicting future values, such as how much a textbook might cost after several years of steady increases.

Exploring the Inflation Rate

The inflation rate is a measure of how the prices of goods and services rise over time, reducing purchasing power. Measured annually, inflation reflects how much more one might need to spend to buy the same item one year later.
For instance, in our textbook problem, inflation is set at 3.3% per year. This means everything that cost a certain amount in the base year will cost, on average, 3.3% more each subsequent year. In the continuous growth context, the formula becomes \( P(t) = P_0 e^{0.033t} \), where

• \( P_0 \): initial price
• \( t \): time in years

This continuous rate approach treats inflation as steadily compounding, rather than occurring at specific intervals. Understanding inflation in this way helps clarify how ongoing small increases can accumulate into substantial price hikes. Recognizing inflation's impact allows us to adjust budgets and expectations about future purchasing power.

Understanding Textbook Pricing Dynamics

Textbook pricing, much like other goods, is affected significantly by rates of growth and inflation. The distinction with textbook pricing is often the larger growth rate compared to general inflation, leading to higher increases over time.
Consider a scenario where a textbook priced at $50 in 1980 grows annually by 6.7%, using the exponential formula \( B(t) = B_0 e^{0.067t} \). As calculated, the price will double sooner than items growing at standard inflation rates. This disparity is essential for students, educators, and publishers to understand, as it affects affordability and purchasing decisions.
The key takeaway is that a higher growth rate than inflation means textbooks could become significantly more expensive than general goods, emphasizing a need for strategies to manage or mitigate these costs. Understanding the intricacies of textbook pricing through continuous growth helps stakeholders plan effectively and seek solutions to avoid prohibitive costs for learners.