A rectangular parking space is marked out by painting three of its sides. The remaining unpainted side has a length of 9 feet, and the sum of the lengths of the three painted sides is 37 feet. What is the area of this rectangular parking space in square feet?

Introduction:
This is a rectangle perimeter puzzle where only partial information about the sides is given. Three sides are painted and their total length is known, while the fourth unpainted side's length is also given. From these details, we must reconstruct the dimensions of the rectangle and then find its area. The problem checks understanding of properties of rectangles (opposite sides equal) and ability to form and solve linear equations from word statements.

Given Data / Assumptions:

• The shape of the parking space is a rectangle. • One unpainted side has length 9 feet. • The sum of the lengths of the three painted sides is 37 feet. • Opposite sides of a rectangle are equal in length. • We are asked to find the area in square feet.

Concept / Approach:
Let the rectangle have sides of length L and B. Since one side is unpainted and has length 9 feet, that side must be either L or B. The opposite side, which has the same length, is part of the painted sides. The three painted sides therefore consist of one side of length 9 feet and the other two sides of the rectangle. Using the total length of painted sides, we can solve for the other side, and then compute the area L * B.

Step-by-Step Solution:
Step 1: Let us assume the unpainted side is of length 9 feet. Its opposite side is also 9 feet long and is painted. Step 2: Let the other dimension of the rectangle be x feet. Step 3: The painted sides are: one side of 9 feet (opposite the unpainted side) and the two sides of length x feet each. Step 4: Sum of painted sides is therefore 9 + x + x = 9 + 2x. Step 5: We are given that 9 + 2x = 37. Step 6: Solve for x: 2x = 37 - 9 = 28, so x = 14. Step 7: The dimensions of the rectangle are 9 feet and 14 feet. Step 8: Compute the area: area = length * breadth = 9 * 14 = 126 square feet.

Verification / Alternative check:
Check the perimeter and painted sides for consistency. The full perimeter of the rectangle is 2 * (9 + 14) = 2 * 23 = 46 feet. One side of 9 feet is unpainted, so the three painted sides must be 14 + 14 + 9 = 37 feet, which matches the given condition. The area computed from these dimensions is 126 sq ft, confirming that our reasoning and calculations are correct.

Why Other Options Are Wrong:
Option 120 sq ft: This would require dimensions such as 10 ft by 12 ft or some other pair that does not satisfy the painted sides condition. Option 130 sq ft: There is no integer pair of sides near 9 and 14 that would yield this exact area while fitting the painted side sum. Option 135 sq ft: This would correspond to something like 9 ft by 15 ft, whose perimeter and painted side totals do not match the problem statement. Option 144 sq ft: This might correspond to 12 ft by 12 ft, which would make it a square and would change both the perimeter and the sum of painted sides.

Common Pitfalls:
Some students mistakenly assume that the three painted sides form three equal sides or that the unpainted side must be the longer side. Others incorrectly treat 37 as the full perimeter instead of the sum of three sides. Understanding that opposite sides of a rectangle are equal and clearly identifying which sides are counted in the 37 feet is essential to avoid algebraic errors.

Final Answer:
The area of the parking space is 126 square feet.