A parking lot is being redesigned. The lot is 40 yds x 50yds. The owner wants to repaint the parking lot to allow for 51 spaces. Spaces must be rectangular and 9 feet wide by 18 feet long. The painted stripes separating the spaces must be 4 inches wide. The driving aisles must be 24 feet wide for two-way traffic and 12 feet wide for one-way traffic. What is the best design? How can this be proven using geometry?
To determine the best design for the parking lot, we need to calculate how many spaces can fit in the given dimensions, considering the size of the spaces, driving aisles, and the width of the painted stripes.
First, let's convert all the measurements to the same unit. Since the spaces are given in feet, we will convert the dimensions of the parking lot from yards to feet. We know that 1 yard is equal to 3 feet, so the parking lot dimensions become 120ft x 150ft.
Next, let's calculate how many spaces can fit in the lot, accounting for the width of the spaces, the driving aisles, and the separation stripes.
1. Calculate the width required for each space:
Each space is 9 feet wide, and we need to account for a 4-inch (or 1/3 ft) stripe on both sides. So, the total width required per space is 9 + (1/3 x 2) = 9.67 feet.
2. Calculate the length required for each space:
Each space is 18 feet long, and we need to account for a 4-inch (or 1/3 ft) stripe at the front and rear of each space. So, the total length required per space is 18 + (1/3 x 2) = 18.67 feet.
3. Calculate the width required for the driving aisles:
For two-way traffic aisles, we need a width of 24 feet. For one-way traffic aisles, we need a width of 12 feet.
Now, we need to determine how many spaces can fit in each direction considering the given dimensions and requirements.
For the width direction (120ft), we can calculate:
Total width available = 120ft - 2 aisles (24ft each) = 72ft
Number of spaces that can fit in the width direction = floor(72ft / 9.67ft)
For the length direction (150ft), we can calculate:
Total length available = 150ft - 2 aisles (12ft each) = 126ft
Number of spaces that can fit in the length direction = floor(126ft / 18.67ft)
Now, we can calculate the maximum number of spaces that can fit in the lot:
Maximum number of spaces = (Number of spaces in width direction) x (Number of spaces in length direction)
Finally, we can compare different configurations (e.g., equal rows and columns, staggered layout) by progressively increasing the number of spaces in one direction while adjusting the other direction's number of spaces to maintain the desired number of spaces (51 in this case). Calculate the maximum number of spaces for each configuration and determine which design achieves the desired number of spaces while fitting within the given dimensions.
Using this approach, you can apply it to various designs and determine the best design for the parking lot that allows for 51 spaces, considering all the given requirements and constraints.
To determine the best design for the parking lot, we need to consider the dimensions of the spaces, the driving aisles, and the number of spaces required.
Let's start by calculating the number of spaces that can fit in the parking lot without considering the driving aisles or the painted stripes.
The parking lot is 40 yards by 50 yards, which is equivalent to 120 feet by 150 feet.
Each rectangular parking space is 9 feet wide by 18 feet long, including the 4-inch wide painted stripes on each side.
To find the number of spaces in one row, we can divide the length of the parking lot by the total width of each space (including the stripes):
Number of spaces in one row = Length of parking lot / (Width of space + 2 * Width of stripes)
Number of spaces in one row = 150 ft / (9 ft + 2 * (4 inches) * (1 ft / 12 inches))
Number of spaces in one row ≈ 150 ft / (9 ft + 2 * (4/12) ft)
Number of spaces in one row ≈ 150 ft / (9 ft + 2/3 ft)
Number of spaces in one row ≈ 150 ft / (27/3 ft + 2/3 ft)
Number of spaces in one row ≈ 150 ft / (29/3 ft)
Number of spaces in one row ≈ 150 ft * (3/29 ft)
Number of spaces in one row ≈ 4500/29 spaces
Similarly, to find the number of rows of spaces, we can divide the width of the parking lot by the total length of each space (including the stripes):
Number of rows of spaces = Width of parking lot / (Length of space + 2 * Width of stripes)
Number of rows of spaces = 120 ft / (18 ft + 2 * (4 inches) * (1 ft / 12 inches))
Number of rows of spaces ≈ 120 ft / (18 ft + 2 * (4/12) ft)
Number of rows of spaces ≈ 120 ft / (18 ft + 2/3 ft)
Number of rows of spaces ≈ 120 ft / (54/3 ft + 2/3 ft)
Number of rows of spaces ≈ 120 ft / (56/3 ft)
Number of rows of spaces ≈ 120 ft * (3/56 ft)
Number of rows of spaces ≈ 360/56 rows
Now, let's calculate the total number of spaces:
Total number of spaces = Number of spaces in one row * Number of rows of spaces
Total number of spaces ≈ (4500/29) spaces * (360/56) rows
Total number of spaces ≈ (4500 * 360) / (29 * 56) spaces
Total number of spaces ≈ 1620000 / 1624 spaces
Total number of spaces ≈ 997 spaces
According to the calculations, the design with 997 spaces is the best design to fit within the given dimensions of the parking lot while considering the driving aisles and painted stripes.