A pond is enclosed by a wooden deck that is 3 feet wide. The fence surrounding the deck is 100 feet long. If the pond is rectangular and the length of the pond is to be three times its width, what are its dimensions?
width of pond --- x
length of pond --- 3x
width of whole thing = x+6 , (3 feet on each end)
length = 3x + 6
2(x+6) + 2(3x+6) = 100
2x + 12 + 6x + 12 = 100
8x = 76
x = 9.5 ft
the pond is 9.5 ft wide and 28.5 ft long
Let's assume that the width of the pond is "w" feet.
According to the given information, the length of the pond will be three times its width, so the length will be 3w feet.
To calculate the dimensions of the pond, we need to consider the wooden deck as well. The deck surrounds the pond on all sides and has a width of 3 feet.
So, the overall width of the pond (including the deck) will be the width of the pond (w) plus twice the width of the deck (2 * 3 feet = 6 feet). Therefore, the overall width will be w + 6 feet.
Similarly, the overall length of the pond (including the deck) will be the length of the pond (3w) plus twice the width of the deck (2 * 3 feet = 6 feet). Therefore, the overall length will be 3w + 6 feet.
We are given that the fence surrounding the deck is 100 feet long.
The total length of the fence is equal to the perimeter of the deck plus the length of the deck.
Perimeter of a rectangle = 2 * (length + width)
So, the total length of the deck's perimeter will be 2 * (overall length + overall width).
We can now set up the equation:
100 feet = 2 * (3w + 6 feet + w + 6 feet)
Now, let's solve this equation step-by-step to find the value of "w" (width of the pond).
Step 1: Distribute the 2 to the terms inside the parentheses:
100 feet = 6w + 12 feet + 2w + 12 feet
Step 2: Combine like terms on each side:
100 feet = 8w + 24 feet
Step 3: Move the constant term to the other side:
100 feet - 24 feet = 8w
Step 4: Simplify:
76 feet = 8w
Step 5: Divide both sides by 8:
w = 76 feet / 8
Step 6: Simplify:
w = 9.5 feet
Therefore, the width of the pond is 9.5 feet.
To find the length of the pond, we can substitute this value back into the equation:
Length = 3w = 3 * 9.5 feet = 28.5 feet
So, the dimensions of the pond are 9.5 feet (width) and 28.5 feet (length).
To find the dimensions of the pond, we can follow these steps:
Step 1: Let's assume the width of the pond as "W". Since the length of the pond is three times its width, we can say that the length of the pond is "3W".
Step 2: We know that the wooden deck is 3 feet wide on each side. Therefore, the total width of the pond, including the deck, would be "W + 2(3 feet)".
Step 3: Similarly, the total length of the pond, including the deck, would be "3W + 2(3 feet)".
Step 4: The perimeter of the fence surrounding the deck is given as 100 feet. The fence goes around the entire perimeter of the pond, including the deck. So, it would be equal to the sum of the widths and lengths of the pond, including the deck.
Therefore, the equation would be:
Perimeter of the fence = Total width of the pond + Total length of the pond
Substituting the values, we have:
100 feet = (W + 2(3 feet)) + (3W + 2(3 feet))
Step 5: Simplify and solve the equation to find the value of "W", which will represent the width of the pond. Once we have the width, we can calculate the length of the pond.
Let's calculate it:
100 feet = W + 6 feet + 3W + 6 feet
100 feet = 4W + 12 feet
Step 6: Subtract 12 feet from both sides:
88 feet = 4W
Step 7: Divide both sides by 4:
22 feet = W
Now that we have the width of the pond as 22 feet, we can find the length of the pond by multiplying it by 3:
Length of the pond = 3W = 3 * 22 feet = 66 feet
Therefore, the dimensions of the pond are 22 feet (width) and 66 feet (length).