To find the dimensions that produce the maximum floor area for a one-story house that is rectangular in shape with a perimeter of 153 feet, follow these steps:
1. Understanding the perimeter relation:
The perimeter [tex]\( P \)[/tex] of a rectangle is given by the formula:
[tex]\[
P = 2 \times (\text{length} + \text{width})
\][/tex]
Given the perimeter is 153 feet:
[tex]\[
2 \times (\text{length} + \text{width}) = 153
\][/tex]
2. Solving for the sum of length and width:
Divide both sides by 2 to isolate the sum of length and width:
[tex]\[
\text{length} + \text{width} = \frac{153}{2} = 76.5
\][/tex]
3. Considering the quadratic function for area:
To maximize the area ([tex]\( A \)[/tex]), which is given by [tex]\( A = \text{length} \times \text{width} \)[/tex], observe that length + width = 76.5 means:
[tex]\[
\text{Area} = \text{width} \times (76.5 - \text{width})
\][/tex]
4. Finding the dimensions that maximize area:
The function [tex]\( \text{width} \times (76.5 - \text{width}) \)[/tex] is a quadratic function, which reaches its maximum value when the width is half of 76.5 (the value at which a parabola opens downwards has its vertex). Therefore:
[tex]\[
\text{width} = \frac{76.5}{2} = 38.25 \ \text{feet}
\][/tex]
Given that width and length are equal in this scenario:
[tex]\[
\text{length} = 76.5 - 38.25 = 38.25 \ \text{feet}
\][/tex]
5. Calculating the maximum area:
[tex]\[
\text{Area} = 38.25 \times 38.25 = 1463.0625 \ \text{square feet}
\][/tex]
The dimensions that yield the maximum floor area are 38.25 feet by 38.25 feet, as the maximum area is achieved when the rectangle is a square.
Therefore, the correct answer is:
OD. 38.25 ft x 38.25 ft
