Problem 44 - 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 \(\mathrm{ft}\) by 300 \(\mathrm{ft}\) lot has a well producing 10 gallons of water per minute, and is valued at \(\$ 48,000\) . What is the value of a 400 \(\mathrm{ft}\) by 400 \(\mathrm{ft}\) lot if the well on the lot produces 4 gallons of water per minute?

Step-by-Step Solution

Answer

The value of the second lot is \( \$51,200 \).

Step 1: Understand Joint Proportionality

The value of the lot is jointly proportional to both its area and the rate at which its well produces water. This means we can express the relationship as \( V = k \times A \times W \), where \( V \) is the value of the lot, \( A \) is the area, \( W \) is the water production rate, and \( k \) is the proportionality constant.

Step 2: Determine the Proportionality Constant

Using the information for the first lot, which has dimensions 200 ft by 300 ft and a well producing 10 gallons/min, we can calculate its area as \( A = 200 \times 300 = 60,000 \ \text{sq ft} \). Substituting into the formula: \( 48000 = k \times 60000 \times 10 \). Solve for \( k \):\[k = \frac{48000}{60000 \times 10} = \frac{48}{600} = 0.08\]

Step 3: Calculate the Area of the Second Lot

The second lot has dimensions 400 ft by 400 ft. Calculate its area: \( A = 400 \times 400 = 160,000 \ \text{sq ft} \).

Step 4: Calculate the Value of the Second Lot

Using the value of \( k \) obtained in Step 2 and the area and water production rate of the second lot, substitute into the joint proportionality formula:\[V = 0.08 \times 160,000 \times 4\]Calculate to find the value of the second lot:\[V = 0.08 \times 160,000 \times 4 = 51,200\]Thus, the value of the second lot is \( \$51,200 \).

Key Concepts

Understanding the Proportionality Constant

Joint proportionality might sound complicated, but it is quite straightforward when broken down. In this case, the value of a lot depends on two factors: its area and the water production rate. This relationship is expressed mathematically using a proportionality constant, denoted as \( k \). The general formula for joint proportionality is \( V = k \times A \times W \), where \( V \) is the value of the lot, \( A \) is the area, and \( W \) is the water production rate.

The proportionality constant \( k \) is a crucial number that helps weigh the contributions of both area and water production rate on the lot's value. It's calculated using known data, like in our scenario with the first lot, to solve for \( k \). Once determined, \( k \) remains consistent across different scenarios, allowing us to predict values for lots with different sizes and water outputs. This highlights how powerful a proportionality constant can be in linking seemingly unrelated variables into a coherent equation.

In summary, understanding and finding this "glue" constant makes it simpler to estimate property values logically on Galiano Island and beyond.

A Closer Look at Area Calculation

Calculating area is fundamental in various fields, from real estate to agriculture. For rectangular plots of land like those on Galiano Island, the area \( A \) is determined simply by multiplying the length by the width. So, for the first lot, the dimensions were 200 feet by 300 feet:
*\[ A = 200 \times 300 = 60,000 \ ext{sq ft} \]*

This calculation was straightforward, but understanding its place in the joint proportionality formula is important. The area serves as one of the components that collectively influence the property's value. Hence, knowing how to accurately calculate this aspect is crucial for precise estimations.

For the second lot, with dimensions of 400 feet by 400 feet, the calculation expands to:
*\[ A = 400 \times 400 = 160,000 \ ext{sq ft} \]*

With this result, it's clear that larger areas yield higher property values, provided the water production rate remains a contributing factor. Recognizing the significance of area in property valuation can help in making informed real estate decisions.

Exploring Water Production Rate

The water production rate of a well on a property can significantly affect the value of that property. In real estate, especially for lots on an island, water availability can make or break a deal. The water production rate, \( W \), indicates how much water the well produces over a given period. In our examples, it is given in gallons per minute.

For the first lot, the well produces 10 gallons per minute, contributing to its overall value. Meanwhile, the second lot's well offers a lesser rate of 4 gallons per minute. This aspect emphasizes the fluctuating value associated with water resources in land valuation.

Whenever we deploy the joint proportionality formula, \( W \) directly multiplies the area \( A \) and the proportionality constant \( k \) to determine value \( V \). Thus, understanding the water production rate's role can help stakeholders assess property value realistically. Remember, while more water often brings more value, it's the proportionate increase that is key.