Problem 19 - The Calculus with Analytic Geometry

Question

A one-story building having a rectangular floor space of \(13,200 \mathrm{ft}^{2}\) is to be constructed where a 22 - \(\mathrm{ft}\) easement is required in the front and back and a \(15-\mathrm{ft}\) easement is required on each side. Find the dimensions of the lot having the least area on which this building can be located.

Step-by-Step Solution

Answer

The lot dimensions are approximately 183.13 ft long and 124.88 ft wide.

Step 1 - Write down the problem

We need to determine the dimensions of a lot with the least area that can accommodate a building with a floor space of 13,200 ft², considering easements of 22 ft in the front and back and 15 ft on each side.

Step 2 - Express the dimensions of the building

Let's denote the length and width of the building as L and W respectively. Since the area of the building is 13,200 ft², we have the equation: \[ L \times W = 13,200 \text{ ft}^2 \]

Step 3 - Incorporate the easements

Due to the easements, the total length of the lot, including the building and the front and back easements, will be: \[ L + 2(22) = L + 44 \text{ ft} \]Likewise, the total width of the lot, including the building and the side easements, will be: \[ W + 2(15) = W + 30 \text{ ft} \]

Step 4 - Formulate the area of the lot

The area of the lot is given by multiplying the adjusted length and width: \[ (L + 44)(W + 30) \]

Step 5 - Substitute the building area formula

Since we know that \( L \times W = 13,200 \), we express W in terms of L: \[ W = \frac{13,200}{L} \]

Step 6 - Substitute W into the lot area formula and simplify

Substituting \( W = \frac{13,200}{L} \) into the lot area formula: \[ A = (L + 44) \left( \frac{13,200}{L} + 30 \right) \]Simplify this to: \[ A = 13,200 + 30L + \frac{580,800}{L} + 1,320 \]Combine like terms: \[ A = 14,520 + 30L + \frac{580,800}{L} \]

Step 7 - Find the minimum area by taking the derivative

To find the minimum area, take the derivative of A with respect to L and set it to zero. \[ \frac{dA}{dL} = 30 - \frac{580,800}{L^2} \]Set the derivative to zero:\[ 30 - \frac{580,800}{L^2} = 0 \]

Step 8 - Solve the derivative equation for L

Solve for L:\[ 30L^2 = 580,800 \]\[ L^2 = 19,360 \]\[ L = \sqrt{19,360} \approx 139.13 \text{ ft} \]

Step 9 - Find W using the building area formula

Now, find W using \( W = \frac{13,200}{L} \):\[ W = \frac{13,200}{139.13} \approx 94.88 \text{ ft} \]

Step 10 - Calculate the lot dimensions

The total lot dimensions are:\[ (139.13 + 44) \text{ ft} \approx 183.13 \text{ ft} \text{ long} \]\[ (94.88 + 30) \text{ ft} \approx 124.88 \text{ ft} \text{ wide} \]

Key Concepts

Understanding Area Minimization

Area minimization is a common problem in calculus where you aim to find the smallest possible area that meets given requirements. Here, we want to find the lot dimensions that can house a rectangular building with a specific floor area while considering given easements. By reducing the area, we save costs and resources. Key to solving this is expressing the area as a function, incorporating the building dimensions and easements. Minimizing this function involves calculus, particularly derivatives.

Role of Derivatives in Optimization

Derivatives help us find the points where the function is at its maximum or minimum. These points are called critical points. To find the minimum area for the lot, we first express the total area as a function of one variable. Then we take its derivative and set it to zero to locate the critical points. For example, in this problem, after simplifying, we get the area function: . Taking the derivative and setting it to zero isolates any values for which the function's slope is zero. Next, we solve for these values to find dimensions that minimize the area.

Considering Easement Constraints

Easements are restrictions placed on a property that limit what can be built, and they often span across specific distances. In this problem, the front and back easements are 22 feet each, while the side easements are 15 feet each. This means the total lot size includes both the building dimensions and the easements. Acknowledge the front/back and side easements when formulating the total lot dimensions: . These constraints must be incorporated into your area function and any calculations.

Rectangular Floor Space and Lot Dimensions

In our scenario, the building's rectangular floor space is fixed at 13,200 square feet. We represent this as: . From this equation, we can express either dimension in terms of the other. This relationship is crucial when expressing the lot area function. Ultimately, solving for the optimal dimensions requires substituting (i.e., the width as a function of the length) back into the lot area function and finding its minimum. Here, the final lot dimensions, including easements, are determined by adding the easement lengths to the building length and width, respectively.