A circular pool with a diameter of 32 feet is surrounded by a wood deck of uniform width.
If the area of the deck is 68π square feet, what is x, the width of the deck?
x = r - 16
pi (r^2 - 16^2) = 68 pi
so
r^2 - 16^2 = 68
r^2 = 324
r = 18
x = 18-16 = 2 feet
2 feet
To find the width of the deck, we need to subtract the area of the pool from the total area.
Step 1: Finding the radius of the circular pool
The diameter of the pool is 32 feet, so the radius is half of that, which is 32/2 = 16 feet.
Step 2: Calculating the area of the circular pool
The formula for the area of a circle is A = πr^2, where A is the area and r is the radius.
Substituting the given values, we have A = π(16)^2 = 256π square feet.
Step 3: Subtracting the area of the pool from the total area
The area of the deck is given as 68π square feet.
So, the width of the deck is the difference between the total area and the area of the pool, divided by the perimeter of the pool.
The perimeter of a circle is given by the formula P = 2πr, where P is the perimeter and r is the radius.
Substituting the given values, we have P = 2π(16) = 32π feet.
The width of the deck is given by the formula:
Width of the deck = (Total area - Area of the pool) / Perimeter of the pool
Width of the deck = (68π - 256π) / 32π
Width of the deck = (68 - 256) / 32
Width of the deck = -188 / 32
Width of the deck = -5.875
However, it doesn't make sense to have a negative width, so it seems there is an error in the given information or calculation.
Please double-check the given information or calculations to ensure accuracy.
To find the width of the deck, we can subtract the area of the pool from the area of the deck.
The area of the pool can be found using the formula for the area of a circle: A = π * r^2, where A is the area and r is the radius. Since the diameter of the pool is 32 feet, the radius is half of that, which is 16 feet. So the area of the pool is π * 16^2 = 256π square feet.
Now, we can subtract the area of the pool from the area of the deck to find the area of the deck alone:
Area of the deck = Area of the deck + pool - Area of the pool
68π = Area of the deck + 256π
To isolate the "Area of the deck", we subtract 256π from both sides:
68π - 256π = Area of the deck
Simplifying the equation gives:
-188π = Area of the deck
Now, we know the area of the deck, but we need to find the width (x) of the deck.
Since the deck is surrounding the circular pool, the deck has a rectangular shape. The formula for the area of a rectangle is A = l * w, where A is the area, l is the length, and w is the width.
In this case, since the pool is circular, the length of the deck is the same as the diameter of the pool, which is 32 feet.
Substituting the values into the formula, we have:
-188π = 32 * x
To isolate "x", we divide both sides by 32:
(-188π) / 32 = x
Simplifying the equation gives:
x ≈ -5.875π
Therefore, the approximate width of the deck (x) is -5.875π feet.