A rectangular swimming pool is 6 meters longer than the width of the pool. A pool deck (sidewalk) that surrounds the pool is 4 meters wide (all the way around the pool). The combined area of the pool deck and pool together (outer rectangle) has an area of 576 square meters more than the area of the pool (inner rectangle) alone. Find the dimensions of the pool.
c'mon, guy, just write the words as algebra. If the pool has width w, then its length is w+6. Now work with the total area:
(w+8)(w+6+8) = w(w+6) + 576
Now solve for w and then w+6.
Let's assume the width of the pool is x meters.
According to the problem, the length of the pool is 6 meters longer than the width, so the length of the pool is x + 6 meters.
The area of the pool alone (inner rectangle) is given by the formula: Area = length * width.
So, the area of the pool is x * (x + 6).
The pool deck surrounds the pool, and its width is 4 meters on each side, so the total additional width and length of the outer rectangle will be 2 * 4 = 8 meters.
The length of the outer rectangle will be the length of the pool plus the additional length from the deck, so it will be (x + 8) meters.
The width of the outer rectangle will be the width of the pool plus the additional width from the deck, so it will be (x + 4) meters.
The area of the pool deck and pool together (outer rectangle) is given by the formula: Area = length * width.
So, the area of the outer rectangle is (x + 8)(x + 4).
According to the problem, the combined area of the outer rectangle is 576 square meters more than the area of the pool alone.
Therefore, we can set up the equation:
(x + 8)(x + 4) = x(x + 6) + 576
Expanding the equation:
x^2 + 12x + 32 = x^2 + 6x + 576
Simplifying the equation by canceling out the common terms:
12x - 6x = 576 - 32
6x = 544
Dividing both sides by 6:
x = 544/6
x = 90.67
Since we can't have a fractional value for the width, we can round down to the nearest whole number.
Therefore, the width of the pool is approximately 90 meters.
Now, we can find the length of the pool:
Length = width + 6
Length = 90 + 6
Length = 96 meters
So, the dimensions of the pool are approximately 90 meters by 96 meters.
To find the dimensions of the pool, let's start by assigning variables to the unknowns in the problem:
Let the width of the pool be "x" meters.
The length of the pool will then be "x + 6" meters.
The area of the pool will be x * (x + 6) square meters.
Next, we need to determine the dimensions of the entire combined area of the pool deck and pool (outer rectangle). The pool deck has a uniform width of 4 meters all the way around the pool, so we can calculate the dimensions based on the width and length of the pool:
The width of the entire combined area will be "x + 4 + 4 = x + 8" meters.
The length of the entire combined area will be "x + 6 + 4 + 4 = x + 14" meters.
The area of the combined area will be (x + 8) * (x + 14) square meters.
According to the problem, the combined area of the pool deck and pool is 576 square meters more than the area of the pool alone. So, we can set up the equation:
(x + 8) * (x + 14) = x * (x + 6) + 576
Expanding and simplifying the equation:
x^2 + 22x + 112 = x^2 + 6x + 576
Rearranging and simplifying further:
x^2 - x^2 + 22x - 6x = 576 - 112
16x = 464
Dividing by 16:
x = 29
Therefore, the width of the pool is 29 meters, and the length of the pool is 29 + 6 = 35 meters.
So, the dimensions of the pool are 29 meters by 35 meters.