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Real Estate

To estimate the area of a vacant lot with a 100 ft frontage in a development, we use the following approach:

1. Introduce a coordinate system where the x-axis coincides with the edge of the straight road forming the lower boundary of the property.
2. Consider the upper boundary of the property as the graph of a continuous function \( f \) over the interval [0, 100].
3. The problem is equivalent to finding the area under the graph of \( f \) on [0, 100].

To estimate the area using a Riemann sum:

1. Divide the interval [0, 100] into five equal subintervals, each of length 20 ft.
2. Measure the distance from the midpoint of each subinterval to the upper boundary of the property using surveyor's equipment.
3. These measurements provide the values of \( f(x) \) at \( x = 10, 30, 50, 70, \) and \( 90 \).

Question: What is the approximate area of the lot?

Answer :

To estimate the area of the lot using a Riemann sum, we'll use the midpoint rule.

[tex]The midpoint rule states that for a partition of the interval \([a, b]\) into \(n\) subintervals of equal width \(\Delta x = \frac{b - a}{n}\), the Riemann sum is approximated by:[/tex]

[tex]\[ \sum_{i=1}^{n} f(x_i^*) \Delta x \]where \( x_i^* \) is the midpoint of the \(i\)-th subinterval.[/tex]

In this case, the interval is [tex]\([0, 100]\) and we divide it into five equal subintervals, so each subinterval has a width of \( \Delta x = \frac{100}{5} = 20 \) ft.[/tex]

The midpoints of the subintervals are [tex]\( x_i^* = 10, 30, 50, 70, 90 \) ft.[/tex]

[tex]Given the values of \( f(x) \) at these midpoints: \( f(10), f(30), f(50), f(70), \) and \( f(90) \), we'll use these values to calculate the Riemann sum.[/tex]

The approximate area of the lot is:

[tex]\[ \text{Area} \approx f(10) \times 20 + f(30) \times 20 + f(50) \times 20 + f(70) \times 20 + f(90) \times 20 \]\[ \text{Area} \approx (f(10) + f(30) + f(50) + f(70) + f(90)) \times 20 \][/tex]

Now, substitute the given values of [tex]\( f(x) \)[/tex] into this formula and calculate the result.

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