Problem 3 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Traditionally, when a college football team seems certain to receive a bid to play in the post-season Orange Bowl, fans begin to throw oranges onto the field. Suppose that at one point during a game, the number of oranges per square yard between the goal line and the 30 -yard line is given by \(\rho(x)=\frac{30-x}{3}\) oranges per square yard, where \(x\) is the number of yards from the goal line. If the field is 160 feet ( \(160 / 3\) yards) wide, how many oranges lie between the goal line and the 30 -yard line?

Step-by-Step Solution

Answer

The total number of oranges between the goal line and the 30-yard line is 16,000.

Step 1: Set up the Integral

The total number of oranges between the goal line and the 30-yard line can be obtained by integrating the 'density' function \(\rho(x)\) over the area of the field in this region. This is done by multiplying \(\rho(x)\) by the width (160 feet = 160/3 yards) and a small length \(dx\) at position \(x\), then integrating from 0 to 30 yards. This gives \[\int_{0}^{30} \frac{30-x}{3} * \frac{160}{3} \,dx \] which represents the total number of oranges.

Step 2: Evaluate the Integral

Evaluate the integral: The integral can be simplified as \(\frac{160}{9} \int_{0}^{30} (30 - x) \,dx\). Expanding the integrand and integrating term by term gives \(\frac{160}{9} [30x - \frac{1}{2}x^2]_0^{30}\).

Step 3: Find the Total Number of Oranges

The total number of oranges is given by evaluating the expression obtained in step 2 at \(x=30\) and subtracting the value at \(x=0\) (which is zero), to get \(\frac{160}{9} [30*30 - \frac{1}{2}*30^2] = 16,000\) oranges.

Key Concepts

Density Function

In mathematics, a density function describes how a quantity is distributed over a particular area or volume. In our example from football, the density function \(\rho(x) = \frac{30-x}{3}\) represents how the number of oranges varies over the field, depending on the distance \(x\) from the goal line. Density functions often reflect real-life scenarios where density isn't uniform, such as the number of trees in a forest or pollutants in the air. The aim is to quantify 'how much' is present at any given point or interval.

• Understanding \(\rho(x)\): Here, as \(x\) increases from 0 to 30, \(\rho(x)\) decreases, indicating that fewer oranges are thrown as we move away from the goal line.
• Dimensions: The function is given in terms of yards, so it's crucial to keep units consistent when calculating areas or volumes.

This concept helps us form the foundation for calculating totals, particularly when combined with integration.

Definite Integral

A definite integral calculates the total accumulation of a function over an interval. It effectively sums up an infinite number of infinitesimally small quantities, providing a precise total.In our problem, the integral of the density function \(\rho(x)\) from 0 to 30 yards helps determine the total number of oranges between those limits. Specifically, it measures how the orange density accumulates across the field's length.The expression for the definite integral is:\[\int_{0}^{30} \left( \frac{30-x}{3} \right) \times \frac{160}{3} \,dx\]This function is the product of the density at each point and the width of the field, summing the result over 30 yards.

• Breaking it down: We integrate from 0 to 30, considering the density function and field width at each point simply because density is multiplied by area to find a total count.
• Simplification: Inside the integral, multiply through the constants and integrate step by step to simplify the expression first before computing.

Application of Integration

Integration is a powerful tool beyond mathematics, providing answers to real-world problems by determining total quantities from density functions. In the orange-pitching scenario, integration reveals how to sum up an entire area's scattered objects effectively. This reflects its utility in multiple fields:

• Physics: Calculating center of mass or moment of inertia using mass density functions.
• Economics: Determining total profit or cost over time with rate functions.
• Environmental Science: Calculating pollutant spread in an ecosystem based on concentration density functions.

For our example, integration allows us to calculate the total number of oranges by evaluating how their density varies with distance. By setting up the integral correctly and performing the calculations step-by-step, we discovered the impressive number of 16,000 oranges, illustrating integration's capability to address complex, practical situations.