An artist intends to paint a 98-square-foot mural on the large wall show below. Find the dimensions of the mural if the artist leaves a border of uniform width around it.
No wall shown. you cannot copy and paste here.
To find the dimensions of the mural, we need to determine the area of the wall and subtract the area of the border.
Let's assume the uniform width of the border is represented by 'w'.
The total dimensions of the wall including the border will be:
Length: L + 2w
Height: H + 2w
Therefore, the area of the wall including the border is given by:
(L + 2w) * (H + 2w)
Now, we know that the area of the mural is 98 square feet.
So, the area of the wall without the border is:
L * H
To find the dimensions of the mural, we need to subtract the area of the border from the total area of the wall:
(L + 2w) * (H + 2w) - L * H = 98
Simplifying the equation gives us:
LH + 2wL + 2wH + 4w^2 - LH = 98
Simplifying further:
2wL + 2wH + 4w^2 = 98
Factoring out 2w:
2w(L + H) + 4w^2 = 98
Dividing both sides by 2 to simplify:
w(L + H) + 2w^2 = 49
Moving terms around:
w(L + H) = 49 - 2w^2
Now, let's assume 'x' as the width of the mural (excluding the border). So, the length of the mural will be 'x' as well.
From the equation above, we have:
w(x + x) = 49 - 2w^2
Simplifying further:
2wx = 49 - 2w^2
Dividing both sides by 2w:
x = (49 - 2w^2) / (2w)
Now, we need to find the appropriate value of 'w' that will give us the desired 98 square foot area.
Let's try different values of 'w' and calculate the corresponding dimensions (x and x). We can start with small increments or decrements of 'w' until we get close to the desired area.
For example, let's assume w = 1.
Using the equation we derived earlier:
x = (49 - 2(1)^2) / (2 * 1)
x = (49 - 2) / 2
x = 47 / 2
x = 23.5
So, the dimensions of the mural with a border of width 1 foot will be approximately 23.5 feet by 23.5 feet.
You can repeat this process with different values of 'w' until you find the dimensions that give you the desired 98 square foot area.