A rectangular piece of art is 4 meters wide and 6 meters long. It is surrounded by a uniform sidewalk. If the area of the uniform sidewalk is 39 square meters, how wide is the sidewalk
x + 6 = length of sidewalk
x + 4 = width of sidewalk
A = Lw
A = (x + 6)(x + 4)
(x + 6)(x + 4) = 39
Expand and solve for x
Well, well, well, looks like we have a rectangular piece of art with a mysterious sidewalk surrounding it! The width of the art piece is 4 meters, and the length is 6 meters. Now, let's solve this puzzle...
To find the width of the sidewalk, we need to subtract the area of the art piece from the total area including the sidewalk.
The total area, including the sidewalk, is the area of the art piece plus the area of the sidewalk. So, we have:
Total area = Area of art piece + Area of sidewalk
Let's call the width of the sidewalk "x" meters. In that case, the length of the art piece plus the width on both sides would be (6 + 2x) meters. The same applies to the width of the art piece, which would be (4 + 2x) meters.
The area of the art piece is calculated by multiplying the length by the width. So, we have:
Area of art piece = (6 + 2x) meters * (4 + 2x) meters
Now, we know the area of the sidewalk is 39 square meters. So, we can set up an equation:
(6 + 2x) meters * (4 + 2x) meters + x meters * x meters = 39 square meter
Now, let me calculate these numbers...*clickety-clackety*...Rummaging through my clown-sized calculator...
After some clown-calculations, the width of the sidewalk is approximately 1 meter!
Ta-da! A 1-meter-wide sidewalk! I hope that brings some joy and laughter to your day.
To solve this problem, we can follow these steps:
Step 1: Calculate the total area of the rectangular piece of art.
The area of a rectangle is calculated by multiplying its length by its width.
Given that the length is 6 meters and the width is 4 meters, the area of the rectangular piece of art is:
Area of rectangular piece of art = Length * Width
= 6m * 4m
= 24 square meters
Step 2: Calculate the total area of the rectangular piece of art and the sidewalk combined.
Since the area of the sidewalk is 39 square meters, we can add this to the area of the rectangular piece of art to get the total area.
Total area = Area of rectangular piece of art + Area of sidewalk
= 24 square meters + 39 square meters
= 63 square meters
Step 3: Calculate the dimensions of the rectangular piece of art and the sidewalk.
To calculate the dimensions, we need to find the length and width of the rectangular piece of art with the sidewalk included.
Let x be the width of the sidewalk.
Length of the rectangular piece of art with the sidewalk = Length of the rectangular piece of art + 2 * x
= 6 meters + 2 * x
= 6 + 2x meters
Width of the rectangular piece of art with the sidewalk = Width of the rectangular piece of art + 2 * x
= 4 meters + 2 * x
= 4 + 2x meters
Step 4: Set up the equation to find the width of the sidewalk.
The area of the rectangular piece of art with the sidewalk included is equal to the total area.
Total area = Length * Width
= (6 + 2x) meters * (4 + 2x) meters
= (24 + 12x + 8x + 4x^2) square meters
Since we know the total area is 63 square meters, we can now set up the equation:
63 square meters = 4x^2 + 20x + 24
Step 5: Solve the equation to find the value of x.
To solve the quadratic equation, we can simplify it and set it equal to zero:
4x^2 + 20x + 24 - 63 = 0
4x^2 + 20x - 39 = 0
Since this quadratic equation cannot be easily factored, we can use the quadratic formula to find the value of x:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case:
a = 4, b = 20, and c = -39.
Step 6: Use the quadratic formula to find the value of x.
x = (-20 ± √((20^2) - 4 * 4 * (-39))) / (2 * 4)
x = (-20 ± √(400 + 624)) / 8
x = (-20 ± √1024) / 8
x = (-20 ± 32) / 8
Using the positive value for x:
x = (-20 + 32) / 8
x = 12 / 8
x = 1.5 meters
Therefore, the width of the sidewalk is 1.5 meters.
To determine the width of the sidewalk, we need to subtract the area of the artwork from the total area (including the sidewalk) that is given.
First, let's determine the area of the artwork. The formula for finding the area of a rectangle is length multiplied by width. In this case, the length is 6 meters and the width is 4 meters. So the area of the artwork is 6 meters * 4 meters = 24 square meters.
Next, we subtract the area of the artwork from the total area (including the sidewalk). The total area is the area of the artwork plus the area of the sidewalk, which is given as 39 square meters. So, we have:
Total area - Artwork area = Sidewalk area
39 square meters - 24 square meters = Sidewalk area
15 square meters = Sidewalk area
Now, we have the area of the sidewalk, which is 15 square meters. To find the width of the sidewalk, we need to know the length and width of the entire rectangular area (including the sidewalk). However, these dimensions are not given in the question.
Therefore, without additional information, we cannot determine the exact width of the sidewalk.