Question

Revenue The revenue (in dollars per year) for a new product is modeled by \(R=10,000\left[1-\frac{1}{\left(1+0.1 t^{2}\right)^{1 / 2}}\right]\) where \(t\) is the time in years. Estimate the total revenue from sales of the product over its first 2 years on the market.

Short answer

The total revenue from sales of the product over its first 2 years on the market is given by the integral \(T = \int_{0}^{2} R dt = \int_{0}^{2} 10,000\left[1-\frac{1}{\left(1+0.1 t^{2}\right)^{1 / 2}}\right] dt\). This integral is best computed using numerical methods or software.

Step-by-step solution

Understand and Write Down the Given Information

The revenue function is given by \(R=10,000\left[1-\frac{1}{\left(1+0.1 t^{2}\right)^{1 / 2}}\right]\), where \(t\) is the time in years.

Set Up the Integral for Total Revenue

The total revenue over the first two years, \(T\), can be found by integrating the revenue function from 0 to 2. The integral thus looks like: \(T = \int_{0}^{2} R dt = \int_{0}^{2} 10,000\left[1-\frac{1}{\left(1+0.1 t^{2}\right)^{1 / 2}}\right] dt \).

Compute the Integral

Computing this integral might be quite complicated directly. The integral can be evaluated using numerical methods like Simpson's Rule, Trapezoidal Rule, or numerical computation software.

Interpret the Result

The result of the integral gives the total revenue from the sales of the product over its first two years on the market.

Key concepts

Integral Calculus

In integral calculus, we focus on the concept of integration, which is essentially the process of finding the whole by summing its parts. This is particularly useful when we need to find total values, like revenue over time, from a rate of change — in this case, the revenue function given by \[R=10,000\left[1-\frac{1}{\left(1+0.1 t^{2}\right)^{1 / 2}}\right]\]Where \(t\) represents the time in years. The integral of the revenue function over a specific interval computes the total revenue over that period.
For our scenario, we calculate the integral of \(R\) from \(t = 0\) to \(t = 2\) to find the total revenue for the product's first two years. This definite integral gives us a single number which represents the full accumulation of revenue. The nature of integration makes it the perfect tool for converting rates (in dollars per year) into totals (total revenue).
Without integration, our understanding of total revenue over variable periods would remain quite limited.

Numerical Methods

Numerical methods offer powerful techniques to evaluate mathematical problems that might not have straightforward solutions. When dealing with complex integrals like our revenue function \[\int_{0}^{2} 10,000 \left[1-\frac{1}{\left(1+0.1 t^{2}\right)^{1 / 2}}\right] dt\]It might not be easy to find a solution analytically by hand. Here is where numerical methods come into play.
### Trapezoidal and Simpson’s Rule Two common numerical integration methods are the Trapezoidal Rule and Simpson’s Rule:

• Trapezoidal Rule: Approximates the region under the graph of the function as a series of trapezoids and sums their areas. This method is straightforward and works well for approximating definite integrals.
• Simpson's Rule: Uses parabolic arcs instead of line segments to approximate each segment of the curve. This increases accuracy with fewer subintervals than the trapezoidal rule.

These methods simplify integration into manageable computations, suitable for calculators or computational tools, providing useful results where symbolic methods fail.

Product Revenue Estimation

Product revenue estimation is a critical component of financial planning and analysis for businesses. Accurately predicting the revenue a new product can generate enables firms to make informed decisions on investment and strategy.
The revenue model in our problem is a concise equation that expresses revenue as a function of time, making it a dynamic tool for foresight:\[R=10,000\left[1-\frac{1}{\left(1+0.1 t^{2}\right)^{1 / 2}}\right]\]This model considers the change in revenue as the product matures in the market, giving insight into potential growth patterns.
Understanding how to estimate total revenue over a given period, such as the first two years, requires integrating the revenue function. This not only aids in financial forecasting but also helps gauge the viability and potential success of introducing new products in competitive markets.
Such models empower businesses to allocate resources efficiently, maximize profitability, and uphold strategic objectives.