Problem 11
Question
Measurements of the width of a pond are taken every 20 yards along its length. The measurements are: 0 yards, 60 yards, 50 yards, 70 yards, 50 yards, and 30 yards. Approximate the surface area of the pond using the Trapezoidal rule.
Step-by-Step Solution
Verified Answer
The approximate surface area of the pond using the Trapezoidal Rule is 2600 square yards.
1Step 1: Understanding the Trapezoidal Rule
The Trapezoidal Rule is a numerical integration method to estimate the area under the curve. The Rule is stated as follows: \[Area \approx \frac{1}{2}h(y_0 + 2y_1 + 2y_2 + \ldots + 2y_{n-1} + y_n)\] where h is the width of the intervals (in this scenario, it is 20 yards, the distance between the measurements) and \(y_0, y_1, ..., y_n\) are the heights of the trapezoids (in this scenario, they are the measurements of the pond's width).
2Step 2: Plugging the values into the formula
The given measurements of the pond's width are 0, 60, 50, 70, 50, 30 yards respectively. Plugging these values and h = 20 yards into the trapezoidal rule formula, we get \[Area \approx \frac{1}{2}*20*(0 + 2*60 + 2*50 + 2*70 + 2*50 + 30)\].
3Step 3: Calculate the Surface Area
Calculate the expression to get the surface area. \[Area \approx \frac{1}{2}*20*(260) = 2600\] square yards. This is the approximated surface area of the pond using the trapezoidal rule.
Key Concepts
Numerical IntegrationSurface Area ApproximationCalculus
Numerical Integration
Numerical integration is a method used when finding the exact integral of a function is difficult or impossible to achieve analytically. In simple terms, it involves estimating the area under a curve by calculating approximate values using numerical methods.
It is especially useful when dealing with irregular shapes or situations where we need to use discrete data points, like measurements from a physical object. Imagine trying to find the area of a plot of land with no regular geometric shape. Without a formula, numerical integration methods can help us find approximate values.
It is especially useful when dealing with irregular shapes or situations where we need to use discrete data points, like measurements from a physical object. Imagine trying to find the area of a plot of land with no regular geometric shape. Without a formula, numerical integration methods can help us find approximate values.
- Numerical integration techniques, such as the Trapezoidal Rule, are practical in real-world applications when data is collected at intervals.
- The technique breaks down a continuous function into discrete sections that are easier to estimate.
- For instance, in our pond example, measurements are taken at specific points, and the Trapezoidal Rule helps us estimate the entire pond's surface area.
Surface Area Approximation
Approximating surface area is a common problem in mathematics, which often requires special techniques like numerical integration or geometric approximations.
For a pond with complex boundaries, we can't simply use length times width as the borders vary. Instead, we create an approximation model.
Using the Trapezoidal Rule, the pond's curved outline is treated as a series of linear sections, or trapezoids, whose areas can be easily calculated and summed.
For a pond with complex boundaries, we can't simply use length times width as the borders vary. Instead, we create an approximation model.
Using the Trapezoidal Rule, the pond's curved outline is treated as a series of linear sections, or trapezoids, whose areas can be easily calculated and summed.
- In our pond example, we assumed straight lines between each pair of successive width measurements.
- The sum of the areas of these trapezoids gave us an estimated total area.
- Such estimation methods are often used in engineering, environmental science, and other fields where precise but quick calculations are required.
Calculus
Calculus is an essential branch of mathematics that deals with the concept of change and motion. It helps in understanding phenomena involving rates of change by using derivatives and integrals.
In the context of our problem, calculus isn't directly solving it, but it provides the foundation for methods like the Trapezoidal Rule.
With integrals being one of the fundamental concepts in calculus, they help quantify areas under curves or the total accumulation of quantities. However, when exact integration isn't feasible, we turn to numerical methods like those defined in calculus principles.
In the context of our problem, calculus isn't directly solving it, but it provides the foundation for methods like the Trapezoidal Rule.
With integrals being one of the fundamental concepts in calculus, they help quantify areas under curves or the total accumulation of quantities. However, when exact integration isn't feasible, we turn to numerical methods like those defined in calculus principles.
- The Trapezoidal Rule is derived from integral concepts, approximating the integral by dividing the area into simple shapes.
- This is a great example of how calculus principles are adapted for practical use, giving students a bridge between theoretical and applied mathematics.
- It's crucial for students to understand these foundations to appreciate how theoretical concepts are utilized in everyday applications.
Other exercises in this chapter
Problem 7
Give upper and lower bounds for \(\int_{0}^{2} \frac{10}{2+x^{5}} d x\) such that the upper and lower bounds differ by less than \(0.01\).
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Approximate each of the following integrals with error less than \(1 / 100\). (Note: If you look at all the questions and think about a strategy in advance, you
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