a 12 cm by 16 cm picture is mounted with border of uniform width on a rectangular frame. if the total area of the border is 288 cm2, what is the length of the side of the frame?
24
let l be the border width. Then the total area of the new picuture with border is..
Areatotal=areapicutre+areaborder.
areaborder=(12+2L)(16+2L)-12x16
288=24L+32L+4L^2
so solve for L, and the length of the fram is 16+2L
Let's assume that the width of the border is x cm.
The dimensions of the picture, including the border, will be increased by 2x cm in each direction.
So, the dimensions of the picture will be (12+2x) cm by (16+2x) cm.
The area of the picture including the border is given by the product of these dimensions:
(12+2x) cm * (16+2x) cm
The area of just the picture is:
12 cm * 16 cm = 192 cm^2
The area of just the border is the difference between the total area of the picture including the border and the area of just the picture:
(12+2x) cm * (16+2x) cm - 192 cm^2 = 288 cm^2
Expanding the equation:
(12+2x)(16+2x) - 192 = 288
192 + 48x + 32x + 4x^2 - 192 = 288
80x + 4x^2 = 288
Rearranging:
4x^2 + 80x - 288 = 0
Factoring out a common factor of 4:
4(x^2 + 20x - 72) = 0
Dividing both sides by 4:
x^2 + 20x - 72 = 0
Now we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. In this case, let's factor:
(x + 24)(x - 3) = 0
So, either x + 24 = 0 or x - 3 = 0
If x + 24 = 0, then x = -24, but since we are dealing with lengths, which cannot be negative, this solution is not valid.
If x - 3 = 0, then x = 3.
Therefore, the width of the border is 3 cm.
The length of the side of the frame will be the dimension of the picture including the border, minus the dimension of just the picture:
(12+2x) cm - 12 cm = 3 cm
Simplifying:
(12+2(3)) cm - 12 cm = 3 cm
(12+6) cm - 12 cm = 3 cm
18 cm - 12 cm = 3 cm
6 cm = 3 cm
Therefore, the length of the side of the frame is 6 cm.
To find the length of the side of the frame, we can start by calculating the area of the picture itself and then deducting it from the total area of the border.
1. Calculate the area of the picture:
The picture has dimensions of 12 cm by 16 cm, so the area of the picture is given by: Area_p = length_p × width_p = 12 cm × 16 cm = 192 cm²
2. Deduct the area of the picture from the total area of the border:
The total area of the border is given as 288 cm². Since the border has a uniform width around the picture, we can calculate the width of one side of the border by dividing the total area by the length or the width of the picture (as they are the same in this case):
Border_width = Total_area / length_p = 288 cm² / 12 cm = 24 cm
3. Determine the dimensions of the frame:
Since the frame is mounted around the picture with a uniform border width, the dimensions of the frame will be equal to the dimensions of the picture increased by twice the border width for each side.
Length_frame = length_p + 2 × Border_width = 12 cm + 2 × 24 cm = 60 cm
Therefore, the length of one side of the frame is 60 cm.