A rectangular wimming pool is 40 ft wide and 45 ft long. It is surrounded on all four sides by a wooden deck of fixed width. The total area enclosed within the perimeter of the deck is 2354 sq ft. What is the width of the deck?
So the area of the deck is
Area=(40+2w)w +45(2w)
solve for w.
area of pool and deck
= (40+2x)(45+2x)
= 2354
1800 + 170x + 4x^2 = 2354
2x^2 + 85x - 277 = 0
by formula
x = 3.04 or a negative
the deck is 3.04 ft wide
check: length = 45+2(3.04) = 51.08
width = 40 + 2(3.04) = 46.08
area = 51.08 x 46.08 = appr 2353.8
hmmm. unfortunately, we only have the total area of pool+deck is 2354.
deck area = total area - pool area
deck area = 2354 - 40*45 = 554
now you can solve bob's equation for the deck width, with the small correction that there are two short sides:
Area=(40+2w)(2w) +45(2w)
Xxzx
To find the width of the deck, we need to subtract the area of the pool from the total area enclosed within the perimeter of the deck.
We know the dimensions of the pool:
Width = 40 ft
Length = 45 ft
The area of the pool is given by:
Area of pool = Width × Length
Substituting the given values, we find:
Area of pool = 40 ft × 45 ft = 1800 sq ft
Now, let's assume the width of the deck as 'x' feet. Since the deck surrounds the pool on all four sides, the overall dimensions of the deck, including the pool, will be:
Width of deck = Width of pool + 2 × Width of deck
Length of deck = Length of pool + 2 × Width of deck
The area enclosed within the perimeter of the deck is 2354 sq ft, which is the total deck area minus the pool area:
Area of deck = Total area - Area of pool
Substituting the values, we get:
2354 sq ft = (Width of pool + 2 × Width of deck) × (Length of pool + 2 × Width of deck) - Area of pool
2354 sq ft = (40 ft + 2x) × (45 ft + 2x) - 1800 sq ft
Expanding this expression:
2354 sq ft = (1800 sq ft + 90 ft x + 80 ft x + 4x^2) - 1800 sq ft
Simplifying the expression:
2354 sq ft = 170 ft x + 4x^2
Now, we have a quadratic equation:
4x^2 + 170x - 2354 = 0
We can solve this quadratic equation to find the width of the deck, 'x', using the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
Substituting the values from our equation:
a = 4, b = 170, c = -2354
Let's solve for x:
x = (-170 ± sqrt(170^2 - 4 × 4 × -2354)) / (2 × 4)
Simplifying this expression will give us the two possible values for the width of the deck.