Problem 38 - Algebra and Trigonometry

Question

Value of a Lot The value of a building lot on Galiano Island is jointly proportional to its area and the quantity of water produced by a well on the property. A 200 ft by 300 ft lot has a well producing 10 gallons of water per minute, and is valued at \(\$ 48,000\) . What is the value of a 400 ft by 400 ft lot if the well on the lot produces 4 gallons of water per minute?

Step-by-Step Solution

Answer

The value of the 400 ft by 400 ft lot is $51,200.

Step 1: Understand Joint Proportionality

The value of the lot is jointly proportional to the area and the water production. This means the value can be expressed as \( V = k \times A \times W \), where \( V \) is the value of the lot, \( A \) is the area, \( W \) is the water production, and \( k \) is the constant of proportionality.

Step 2: Determine the Constant of Proportionality

First, calculate the area of the 200 ft by 300 ft lot: \( A_1 = 200 \times 300 = 60,000 \) square feet. The value for this lot is \( 48,000 \), and the well produces \( W_1 = 10 \) gallons per minute.\[V_1 = k \times A_1 \times W_1\]\[48,000 = k \times 60,000 \times 10\]\[k = \frac{48,000}{600,000} = \frac{48}{600}\] Simplify this to find \( k = \frac{1}{12.5} \).

Step 3: Calculate the Area of the New Lot

Calculate the area of the 400 ft by 400 ft lot: \( A_2 = 400 \times 400 = 160,000 \) square feet.

Step 4: Calculate the Value of the New Lot

Now use the constant \( k = \frac{1}{12.5} \), the new area \( A_2 = 160,000 \), and the water production \( W_2 = 4 \) gallons per minute to find the value of the new lot:\[V_2 = k \times A_2 \times W_2\]\[V_2 = \frac{1}{12.5} \times 160,000 \times 4\] \[V_2 = \frac{640,000}{12.5}\] Calculate to find \( V_2 = 51,200 \).

Key Concepts

Constant of Proportionality

In mathematics, when two variables are said to be proportional, they change at a constant rate with respect to one another. In our exercise, the value of the building lot is jointly proportional to both its area and the water production of its well. This means that if one increases, the value increases as well. This relationship can be summarized by the equation \( V = k \times A \times W \), where \( V \) stands for value, \( A \) is the area, and \( W \) is the water production. The key element here is \( k \), the constant of proportionality, which links these quantities.

Understanding the constant \( k \) is crucial because it quantifies how much influence each contributing factor (area and water production) has on the value. In the given problem, once we know \( k \) for one scenario, we can apply it to other scenarios by simply adjusting the area and water production to find new property values. It's like a universal code that translates these inputs into a specific property value.

Here's a simplified way to think about it:

• For larger areas (\( A \)), the value tends to be higher if all else is equal.
• For higher water production (\( W \)), the value also tends to increase.
• \( k \) determines the sensitivity of the property's value to these inputs.

Area Calculation

Calculating the area of a property is a straightforward task essential for determining its value when dealing with joint proportionality. The area is simply the length multiplied by the width of the lot. In this exercise, two lots are considered: a 200 ft by 300 ft lot and a 400 ft by 400 ft lot.

For the 200 ft by 300 ft lot, the area \( A_1 \) is \( 200 \times 300 = 60,000 \) square feet. This calculation gives us a two-dimensional measure of how much ground the property covers.

For the 400 ft by 400 ft lot, the area \( A_2 \) is \( 400 \times 400 = 160,000 \) square feet. In terms of proportionality, a larger area like 160,000 square feet indicates the property is likely more valuable, all other factors being equal.

• Ensures an accurate basis for later calculations.
• Integral to finding the ultimate property value.

By knowing the area, you lay the foundational brick for the remaining calculations, including the value, according to the relationship \( V = k \times A \times W \).

Value Calculation

After understanding the constant of proportionality and calculating the area, the final step in this exercise is to determine the value of the property using these factors. In the context of our problem, once \( k \) is known, and you have calculated \( A \) and \( W \), you can find the value \( V \) easily using the formula \( V = k \times A \times W \).

Let's examine the specifics. The constant of proportionality \( k \) is calculated from the first lot, as \( k = \frac{1}{12.5} \). With this constant, and for the second property with an area \( A_2 = 160,000 \) square feet and water production \( W_2 = 4 \) gallons per minute, you plug these values into the formula:

\[ V_2 = \frac{1}{12.5} \times 160,000 \times 4 \]

This calculation gives you \[ V_2 = \frac{640,000}{12.5} \] which simplifies to \( V_2 = 51,200 \).

This method confirms that a complex interplay between area, water production, and proportionality affects property value. The final value calculated shows how sensitive property value is to changes in these inputs, guided by the established proportionality constant \( k \). Understanding this process helps in practical scenarios where property assessment relies on multiple factors beyond just size. Knowing how to compute this ensures more accurate valuations.