A pool measuring 20 meters by 22 meters is surrounded by a path of uniform width, as shown in the figure. If the area of the pool and the path combine is 1088 square meters, what is the width of the path?
Ah, the old pool and path dilemma! Let's dive right into it, shall we?
Given that the pool and path combined area is 1088 square meters, we can start by subtracting the area of just the pool from that total. The pool measures 20 meters by 22 meters, which gives us an area of 20 * 22 = 440 square meters.
So, if we deduct 440 square meters from the total area of 1088 square meters, we're left with the area of just the path. Are you following along?
Now comes the fun part – math! By subtracting the pool area from the total, we find that the path area is 1088 - 440 = 648 square meters.
To determine the width of the path, we need to know the length of the inner rectangle formed by the pool. Since the pool measures 20 meters by 22 meters, the inner rectangle's dimensions will be 20 - 2x and 22 - 2x, where x represents the width of the path.
The area of this inner rectangle will be the length multiplied by the width, so we have (20 - 2x)(22 - 2x) = 648.
Now, let's put a smile on that math face and solve this equation. Distributing and simplifying, we get 440 - 84x + 4x^2 = 648. Rearranging, we have 4x^2 - 84x + 208 = 0.
Using factoring or the quadratic formula, we find that x ≈ 6.33 meters or x ≈ 8.17 meters. Since negative widths don't make sense for our path, we can safely say that the width of the path is approximately 6.33 meters.
And there you have it! The path surrounding the pool has a width of about 6.33 meters. Now you can relax by the pool and start working on your math-themed synchronized swimming routine.
To find the width of the path, we need to subtract the area of the pool from the total combined area of the pool and the path.
Step 1: Calculate the area of the pool.
The area of the pool is equal to the length multiplied by the width: 20 meters * 22 meters = 440 square meters.
Step 2: Subtract the area of the pool from the total combined area.
Total combined area - area of the pool = area of the path
1088 square meters - 440 square meters = 648 square meters
Step 3: Determine the width of the path.
The width of the path is equal to the area of the path divided by the length or width of the pool (since both sides of the pool have the same width).
Width of the path = Total area of the path / Length or Width of the pool
Width of the path = 648 square meters / 20 meters = 32.4 meters
So, the width of the path is 32.4 meters.
(20 + 2 w)(22 + 2 w) = 1088 ... (10 + w)(11 + w) = 272
w^2 + 21 w - 162 = 0
factor to find w
width of path ---- x metres
area of outer rectangle = (22+2x)(20+2x)
(22+2x)(20+2x) = 1088
expand, simplify and solve the resulting equation.
You will get two integer answers, make sure to reject the negative root.
width of path ---- x metres
area of outer rectangle = (22+2x)(20+2x)
(22+2x)(20+2x) = 1088
the width of the path is 3
If the path has width w, then since the path is along both sides and both ends, the total area is
(20+2w)(22+2w) = 1088
Now just solve for w.
or, just ask, which two factors of 1088 differ by 2? Since they are so close together, they will both be close to √1088 ≈ 33. So, I'd try 32 and 34.