Problem 44 - Precalculus Mathematics for Calculus

Question

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 \(400 \mathrm{ft}\) 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 the Joint Proportionality

The statement that the value of the lot is jointly proportional to its area and the water produced by the well means that the value can be represented as \( V = k \cdot A \cdot Q \), where \( V \) is the value of the lot, \( A \) is the area, \( Q \) is the quantity of water produced by the well, and \( k \) is the constant of proportionality.

Step 2: Calculate the Area

The area \( A \) of the first lot is given by the dimensions 200 ft by 300 ft. Thus, \( A = 200 \times 300 = 60,000 \ \text{sq ft} \). For the second lot, the dimensions are 400 ft by 400 ft, so \( A = 400 \times 400 = 160,000 \ \text{sq ft} \).

Step 3: Set up the Equation for the First Lot

Using the relationship \( V = k \cdot A \cdot Q \), substitute the known values for the first lot: \( V = 48000 \), \( A = 60000 \), \( Q = 10 \). Thus, \( 48000 = k \cdot 60000 \cdot 10 \).

Step 4: Solve for the Constant k

Simplify the equation from Step 3 to find \( k \): \( 48000 = 600000k \). Thus, \( k = \frac{48000}{600000} = 0.08 \).

Step 5: Calculate the Value of the Second Lot

Using the same equation \( V = k \cdot A \cdot Q \) for the second lot, substitute \( k = 0.08 \), \( A = 160000 \), and \( Q = 4 \): \( V = 0.08 \times 160000 \times 4 \).

Step 6: Compute the Final Value

Compute the expression from Step 5: \( V = 0.08 \times 160000 \times 4 = 51200 \). Therefore, the value of the second lot is \( \$ 51,200 \).

Key Concepts

Area calculation

When determining the value of a building lot based on joint proportionality, calculating the area is a crucial step. The area of a property is typically determined by multiplying its length by its width. For instance, in the original problem, the area of the first lot was calculated using its dimensions: 200 ft by 300 ft. The formula becomes:

• Area = Length × Width = 200 ft × 300 ft = 60,000 sq ft

Similarly, the area of the second lot was found by:

• Area = Length × Width = 400 ft × 400 ft = 160,000 sq ft

Calculating the area is necessary because it directly impacts the value when joint proportionality is at play. Make sure to always confirm the dimensions to ensure you're working with accurate figures. This will lead to an accurate estimation of the property value when combining it with other proportional factors.

Proportionality constant

In joint proportionality, the relationship between variables is represented through a constant known as the "proportionality constant." Understanding how to determine this constant is key to solving problems involving joint proportional relationships.
Initially, the equation used is:

• \( V = k \cdot A \cdot Q \)

Here, \( V \) is the value, \( A \) the area, \( Q \) the quantity of water produced, and \( k \) is the proportionality constant. By substituting the known values from the specific example of the first lot (value: $48,000, area: 60,000 sq ft, water produced: 10 gallons per minute), we can solve for \( k \):

• \( 48000 = k \cdot 60000 \cdot 10 \)
• \( k = \frac{48000}{600000} = 0.08 \)

The calculated constant \( k \) not only characterizes the specific relationship between the variables for one scenario, but is used again to determine the value of the second lot. This ensures consistency in understanding how much more or less value an additional factor (such as an increase in area or water production) contributes to the total worth.

Mathematical modeling

Mathematical modeling uses mathematical expressions to represent real-world situations and solve practical problems. In this exercise, joint proportionality is the model applied to find the value of different properties. The general form used here is:

• \( V = k \cdot A \cdot Q \)

This model helps us understand how the property value changes with varying area and water yield. By establishing \( k \) from the first scenario, we apply it as a constant in evaluating the second property. Mathematical modeling simplifies complex situations by identifying key relationships and expressing them through equations.
It uses known data points to predict unknowns. This is powerful in planning and decision-making, especially in real estate, as it allows for quick assessment of potential investments. With this model, you can simulate many what-if scenarios and understand their impact, making it a versatile tool across different domains.