You are creating a metal border of uniform width for a wall mirror that is 20 inches by 24 inches. You have 416 square inches of metal to use. Write and solve an equation to find the border's width.
final dimensions of the framed mirror:
(20+2w) x (24+2w)
final area: (20+2w)(24+2w)
mirror area: 20*24
border area: (20+2w)(24+2w)-480
= 480 + 88w + 4w^2 - 480 = 416
w^2 + 22w - 104 = 0
(w+26)(w-4) = 0
w = 4
Well, this is a job for math, not comedy! Let's solve this equation seriously.
To find the width of the border, we need to subtract the area of the mirror from the total area of the metal border.
The area of the mirror is given by multiplying its length and width: 20 inches × 24 inches = 480 square inches.
So, the area of the border can be calculated by subtracting the area of the mirror from the total area given: 416 square inches - 480 square inches = -64 square inches.
Wait a minute, we can't have negative area! It seems like there's not enough metal provided to create a border for this mirror.
Who knew that mirrors could be so demanding? Maybe it's time to reconsider and let the mirror shine on its own, without a metal border.
To find the border's width, we need to subtract the area of the mirror from the total area of the metal.
Let's assume the width of the border is "x". The dimensions of the mirror including the borders would be (20 + 2x) inches by (24 + 2x) inches.
The area of the mirror including the borders is given by the product of the lengths of the sides:
(20 + 2x) * (24 + 2x)
The area of the mirror is 20 * 24 = 480 square inches.
Thus, the equation becomes:
(20 + 2x) * (24 + 2x) - 480 = 416
Now, we can solve this equation to find the value of "x":
(20 + 2x) * (24 + 2x) = 896
Expanding the equation:
480 + 40x + 48x + 4x^2 = 896
Combining like terms:
4x^2 + 88x + 480 = 896
Rearranging the equation:
4x^2 + 88x + 480 - 896 = 0
4x^2 + 88x - 416 = 0
Dividing the equation by 4:
x^2 + 22x - 104 = 0
Now, we can solve this quadratic equation by factoring (if possible) or using the quadratic formula.
Factoring the equation:
(x + 26)(x - 4) = 0
Setting each factor to zero:
x + 26 = 0 or x - 4 = 0
Solving for "x":
x = -26 or x = 4
Since the width cannot be negative, we discard x = -26. Therefore, the border's width is 4 inches.
To find the width of the metal border, we need to subtract the area of the wall mirror from the total area available for the metal.
First, let's find the area of the wall mirror, which is given by the formula:
Area of a rectangle = length × width
The length of the mirror is 20 inches and the width is 24 inches, so the area of the mirror is:
Area of mirror = 20 inches × 24 inches = 480 square inches
Next, we subtract the area of the mirror from the total area of metal available:
Total Area of Metal - Area of Mirror = Area of Border
Given that we have 416 square inches of metal available, we can write the equation:
416 square inches - 480 square inches = Area of Border
Now, let's solve for the area of the border:
Area of Border = 416 square inches - 480 square inches
Area of Border = -64 square inches
However, it doesn't make sense to have a negative area for the border. This indicates that we do not have enough metal to create a border for the wall mirror using the given dimensions and amount of metal.
In this case, you would either need to reduce the size of the mirror or use less metal for the border.