Problem 95 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

During a thunderstorm, rain was falling at the rate of $$ \frac{8}{(t+4)^{2}} \quad(0 \leq t \leq 2) $$ inches/hour. a. Find an expression giving the total amount of rainfall after \(t\) hr. Hint: The total amount of rainfall at \(t=0\) is zero. b. How much rain had fallen after \(1 \mathrm{hr}\) ? After \(2 \mathrm{hr}\) ?

Step-by-Step Solution

Answer

a. The total amount of rainfall after \(t\) hours is given by the expression: \(A(t) = \frac{2t}{(t + 4)}\) b. After 1 hour, \(\frac{2}{5}\) inches of rain had fallen and after 2 hours, \(\frac{2}{3}\) inches of rain had fallen.

Step 1: Write down and analyze the rain rate given

The rain rate is given as: \(\frac{8}{(t+4)^{2}}\) inches/hour, and is valid for the time interval \(0 \leq t \leq 2\).

Step 2: Integrate the rain rate to find the expression for the total rainfall

The total amount of rainfall can be found by integrating rain rate over time. We will integrate the rate function with respect to time from 0 to t: \(A(t) = \int_0^{t} \frac{8}{(t+4)^{2}} dt\)

Step 3: Perform the Integration

Using the substitution method for integration, let \(u = t + 4\), so, \(du = dt\): \(A(t) = \int_0^{t} \frac{8}{u^{2}} du\) Now, integrate the function with respect to \(u\): \(A(t) = -8(\frac{1}{u}) \Big|^{t}_{0}\) But, since we have substituted, we have to replace \(u\) back with \(t + 4\): \(A(t) = -8(\frac{1}{t + 4}) \Big|^{t}_{0}\)

Step 4: Evaluate the Definite Integral

Now, apply the upper and lower limits: \(A(t) = -8(\frac{1}{t + 4} - \frac{1}{4})\) We can simplify further: \(A(t) = -8(\frac{4 - (t + 4)}{4(t + 4)})\) \(A(t) = -8(\frac{-t}{4(t + 4)})\) \(A(t) = \frac{2t}{(t + 4)}\) Now, we have the expression for the total amount of rainfall after \(t\) hours: \(A(t) = \frac{2t}{(t + 4)}\) #b. Calculate the amount of rain fallen after 1 hr and 2 hr#

Step 5: Find the amount of rain fallen after 1 hr

For the time \(t=1\), substitute it into the expression we derived for the total amount of rainfall: \(A(1) = \frac{2(1)}{(1 + 4)}\) \(A(1) = \frac{2}{5}\) So, after 1 hour, \(\frac{2}{5}\) inches of rain had fallen.

Step 6: Find the amount of rain fallen after 2 hr

For the time \(t=2\), substitute it into the expression for the total amount of rainfall: \(A(2) = \frac{2(2)}{(2 + 4)}\) \(A(2) = \frac{4}{6}\) \(A(2) = \frac{2}{3}\) So, after 2 hours, \(\frac{2}{3}\) inches of rain had fallen.

Key Concepts

Definite Integral

Understanding the concept of a definite integral is fundamental for learning how to quantify accumulations over an interval, much like calculating the total rainfall during a storm. In our rain rate example, the rainfall rate is represented as a function where the variable is time.

To find the total amount of rainfall over a specific period, we use a definite integral, which sums up all the tiny bits of rainfall that occurred at each moment within the time interval. This is conceptually similar to adding up slices to find the total area under a curve.

The definite integral has both an upper and a lower limit, which correspond to the start and end times of the interval we're interested in. Integrating the rate function from 0 to some time t calculates the total rainfall from the start of the storm up to that moment. Thus, the definite integral serves as a powerful tool to translate rates of change into accumulated totals in various real-world scenarios, such as meteorology, economics, and physics.

Integration Techniques

Applying integration techniques correctly is essential for solving calculus problems involving definite integrals. In our rainfall problem, the substitution method is employed due to the nature of the function involved.

This technique simplifies the original integral by introducing a substitute variable, making the function easier to integrate. In this case, we set the substitute variable u equal to the expression within the denominator, t + 4. The process involves finding 'du' and then performing the integral with respect to the new variable u.

Once integrated, we replace the variable u with the original terms to express the answer in terms of t. It is critical to also adjust the limits of integration if you change the variable within the integral. Our problem stays simple because we revert to the original variable t before applying the limits. Besides substitution, other techniques include integration by parts, partial fractions, and trigonometric substitutions, which are selected based on the function form.

Applied Mathematics

Applied mathematics is all about solving real-world problems using mathematical methods, and our exercise illustrates this perfectly. By calculating the amount of rainfall during a given time period, we provide a practical application of calculus concepts, showing the significance of mathematical modeling in understanding natural phenomena.

Often, applied mathematics involves formulating relationships using functions, as seen with the rate of rainfall, and determining total quantities over time using definite integrals. These models can guide decision-making in resource management, environmental predictions, and engineering designs.

The real-life implications of applied mathematics highlight the importance of knowing not just how to perform mathematical operations, but also understanding their relevance and impact. In careers like meteorology, civil engineering, and environmental science, these calculations help in predicting weather patterns, designing effective drainage systems, and understanding fluid dynamics, demonstrating that math is more than just numbers—it's a way to interact with and solve tangible challenges in our world.