A framed picture has length 50 cm and width 40 cm . The picture itself has area 1344 cm^2. How far is it from the edge of the picture to the edge of the frame if thid didtance is uniform around the picture ?
if the width of the frame is w, the dimensions of the picture are 50-2w and 40-2w (subtract w on each side). So, since we know the area of the picture,
(50-2w)(40-2w) = 1344
w = 4
Check:
picture area is 32x42 = 1344
1325
To add on to Steve's answer ( No guess and check )
(50-2x)(40-2x) = 1344
4x^2 -180x +2000 = 1344
4x^2 -180x +656=0
Then we will use the Quadratic Formula
180 ± √32,400 - 10496 / 8
-180 ± 148 / 8
4 or 41. We can not use 41 as it will be two big. If u substitute 4 instead of x in the original equation.
50 - 2(4) * 40 - 2(4)
50 - 8 * 40 * 8
So the width of the frame is 8 Centameters
Let m be the width of the frame. Then:
(50-2m)(40-2m)=1344
2000-180m+4m²=1344
4m²-180m+656=0
m²-45m+164=0
(m-41)(m-4)=0
m=41 or 4
The frame is 4 inches wide ....................
To find the distance from the edge of the picture to the edge of the frame, we need to subtract the area of the picture from the area of the frame.
Given:
Length of the picture = 50 cm
Width of the picture = 40 cm
Area of the picture = 1344 cm²
Let's calculate the area of the frame:
Area of the frame = (Length of the frame × Width of the frame) - (Length of the picture × Width of the picture)
Since the distance around the picture is uniform, the length and width of the frame are equal to each other.
Let's assume the width of the frame as x cm.
Length of the frame = Length of the picture + 2 × Width of the frame
Width of the frame = x
Area of the frame = (50 + 2x) × (40 + 2x) - 50 × 40
Now, we'll plug in the given values to calculate the area of the frame:
1344 = (50 + 2x) × (40 + 2x) - 50 × 40
Expanding the equation:
1344 = (2000 + 140x + 4x²) - 2000
Simplifying the equation:
1344 = 140x + 4x²
Rearranging the equation into a quadratic form:
4x² + 140x - 1344 = 0
To solve this quadratic equation, we can either factorize it or use the quadratic formula. In this case, let's use the quadratic formula:
x = [-b ± sqrt(b² - 4ac)] / 2a
For our equation, a = 4, b = 140, and c = -1344.
x = [-140 ± sqrt(140² - 4(4)(-1344))] / 2(4)
Now, let's calculate the value of x using the quadratic formula:
x = [-140 ± sqrt(19600 + 21504)] / 8
x = [-140 ± sqrt(41104)] / 8
x = [-140 ± 202.753] / 8
Simplifying further:
x₁ = (-140 + 202.753) / 8 ≈ 16.969
x₂ = (-140 - 202.753) / 8 ≈ -47.969
Since we are dealing with measurements, the width of the frame cannot be negative. Therefore, we can disregard the negative value of x.
Finally, the distance from the edge of the picture to the edge of the frame is approximately 16.969 cm.