Problem 52 - Calculus: Early Transcendentals

Question

The rates at which rain fell, in inches per hour, in two different locations \( t \) hours after the start of a storm are given by \( f(t) = 0.73t^3 - 2t^2 + t + 0.6 \) and \( g(t) = 0.17t^2 - 0.5t + 1.1 \). Compute the area between the graphs for \( 0 \le t \le 2 \) and interpret your result in this context.

Step-by-Step Solution

Answer

Compute the integral of the absolute function difference from 0 to 2, representing comparative rainfall volume.

Step 1: Setup the Problem

To find the area between the graphs of the two functions, we need to compute the definite integral of the absolute difference of the functions over the given interval. Specifically, we need the integral of \(|f(t) - g(t)|\) from \(t = 0\) to \(t = 2\).

Step 2: Determine the Functions Difference

Calculate the difference between the two functions: \[f(t) - g(t) = (0.73t^3 - 2t^2 + t + 0.6) - (0.17t^2 - 0.5t + 1.1) \]Simplify the expression:\[0.73t^3 - 2t^2 + t + 0.6 - 0.17t^2 + 0.5t - 1.1 = 0.73t^3 - 2.17t^2 + 1.5t - 0.5 \]

Step 3: Find Intersection Points

To ensure correctness in calculating the area, check for any interception points of the function differences within the interval \([0, 2]\). The equation to solve is:\[0.73t^3 - 2.17t^2 + 1.5t - 0.5 = 0\]Solve this equation to find any critical points, but as an initial approach, we assume no intersections within the given interval if not easily solvable, or rely on numerical methods.

Step 4: Integrate the Absolute Difference

Integrate the absolute difference of the functions from \(t = 0\) to \(t = 2\): \[\int_{0}^{2} \left|0.73t^3 - 2.17t^2 + 1.5t - 0.5\right| \, dt\]If intersections exist in the interval, split the integral at those points, else perform as a single integral.

Step 5: Interpret the Result

The result from the integration step, after solving the integral, gives the total rainfall volume difference in inches per unit area over the interval from \(t = 0\) to \(t = 2\) hours between the two locations. This measurement indicates how much more or less rain location one received compared to location two over the specified time.

Key Concepts

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers. In the given exercise, we have two polynomial functions: one of degree three, representing the rate at which rain fell at one location, and another of degree two for the second location.

• The first function, \( f(t) = 0.73t^3 - 2t^2 + t + 0.6 \), includes terms with powers ranging from zero to three, making it a cubic polynomial.
• The second function, \( g(t) = 0.17t^2 - 0.5t + 1.1 \), includes terms with powers up to two, making it a quadratic polynomial.

Polynomial functions like these are fundamental in mathematics because they are continuous and smooth, and can model a wide range of situations. They can represent various real-world phenomena including fluid rates, economic models, and population growth. These models allow us to calculate not just the function values, but also the area under or between them over specific intervals.

Area Between Curves

Calculating the area between curves is an essential task in integral calculus. It often helps in understanding the difference between two varying quantities over a certain period. In this exercise, determining the area between the graphs of the rain rate functions over the interval \([0, 2]\) helps us understand the total difference in rainfall between the two locations.

• The goal is to compute \( \int_{0}^{2} |f(t) - g(t)| \, dt \), where \(|f(t) - g(t)|\) represents the absolute difference between the rates at which the rain fell at the two locations.
• This calculation involves integrating, which is essentially summing up tiny slices of area between the curves over the interval.
• If the graphs of the two functions cross each other, we must split the integral at the point of intersection to accurately measure the total area in segments.

By evaluating this definite integral, we ascertain the total measurable difference, which reflects how the intensity of the rain compared at both sites during the storm period. The area provides insights into weather patterns and water distribution between the locations.

Rate of Change

The rate of change is a crucial mathematical concept, often represented by the derivative in calculus, but in our context, it’s expressed through the functions \( f(t) \) and \( g(t) \). These functions describe how much rain falls per hour at each location over time.

• The function \( f(t) \) changes at a non-linear rate because it is a cubic polynomial, implying varying speeds of change in the rainfall rate, perhaps becoming more intense or slowly subsiding as the storm progresses.
• Conversely, \( g(t) \) is a quadratic function, meaning its rate of change is linear but varies due to the squared term, showing possibly a simpler pattern that accelerates or decelerates consistently.

Understanding these rates of change allows meteorologists to make predictions and assessments about weather conditions. It helps in seeing how rapidly shifts occur during storms, critical for planning and response strategies in weather-related phenomena.