Problem 8 - Applied Calculus

Question

A product costs \(\$ 80\) today. How much will the product cost in \(t\) days if the price is reduced by (a) \(\quad \$ 4\) a day (b) \(5 \%\) a day

Step-by-Step Solution

Answer

(a) \(80 - 4t\), (b) \(80 \times (0.95)^t\)

Step 1: Identify Initial Cost

Start with the initial cost of the product, which is given as \( \$ 80 \). This is the cost of the product on day 0.

Step 2: Calculate Reduced Price per Day (Part a)

For part (a), the problem states that the price is reduced by \( \$ 4 \) a day. To find the cost after \( t \) days, reduce the initial price by \( 4 \times t \). The formula becomes:\[\text{Price after } t ext{ days} = 80 - 4t\]

Step 3: Calculate Reduced Price per Day (Part b)

For part (b), the price is reduced by \(5\%\) a day. This means each day, the product retains \(95\%\) of its previous day's price. To express it mathematically:\[\text{Price after } t ext{ days} = 80 \times (0.95)^t\]

Step 4: Formulate the General Formula

For part (a), the general formula is \( 80 - 4t \). For part (b), it is \( 80 \times (0.95)^t \). These equations will give you the price at any given day \( t \).

Key Concepts

Mathematical Modeling

Mathematical modeling is a powerful tool that helps us to understand and predict real-world phenomena through mathematical formulas and calculations. In the context of this exercise, we are using mathematical models to forecast how the price of a product changes over time based on given conditions.
There are different types of mathematical models, depending on how changes occur:

• For linear changes, we use linear equations where something increases or decreases steadily over time.
• For percentage changes, we apply exponential equations that consider the compounding effect of percentages.

This exercise presents two real-life scenarios in which we use both linear and exponential models to predict price changes. By using mathematical modeling, we can create a clear picture of how the product's price will evolve, making it easier to make informed decisions.

Exponential Decay

Exponential decay refers to a process where a quantity decreases at a consistent percentage rate over time. This concept is essential in part (b) of the exercise, where the product's price decreases by 5% per day.
Exponential decay can be identified by its unique compounding property, where tomorrow’s value depends on today’s value. In formulas, it is often represented as\[P(t) = P_0 \times (1 - r)^t\]where \(P(t)\) is the price after \(t\) days, \(P_0\) is the initial price, and \(r\) is the rate of decay as a decimal.
In our case, the price of the product after \(t\) days is represented by:\[80 \times (0.95)^t\]Here, 0.95 represents the remaining 95% of the price each day, modeling how a consistent percentage decrease affects the price over time.

Linear Decrease

Linear decrease happens when a quantity reduces by a fixed amount at regular intervals. This is illustrated in part (a) of the exercise, where the price decreases by \(\\(4\) daily.
A linear decrease can be described by a simple linear equation:\[P(t) = P_0 - d \times t\]where \(P(t)\) is the price after \(t\) days, \(P_0\) is the initial price, and \(d\) is the daily decrease amount.
For our problem, we use:\[80 - 4t\]This shows how the price is reduced by exactly \(\\)4\) every day. Such a linear model is straightforward as the decrease is uniform, making predictions about future prices clear and simple.