A picture measuring 5cm by 8cm is mounted on a frame so as to leave a uniform margin of width X cm all round .if the area of the margin is 30cm² ,find x
The margin has a rectangle at the top and bottom its length = 5 cm
Its area = 5 ∙ x
That means margin has two rectangles of area 5 x
The margin also has a rectangle on the left and right sides its length = 8 - x - x = 8 - 2 x
Its area = ( 8 - 2 x ) ∙ x
That means margin has two rectangles of area ( 8 - 2 x ) ∙ x
Total area of margin is:
2 ∙ 5 x + 2 ∙ ( 8 - 2 x ) ∙ x = 30 cm²
10 x + 2 ∙ ( 8 x - 2 x² ) = 30
10 x + 16 x - 4 x² = 30
- 4 x² + 26 x = 30
Now you must solve this equation.
Multiply both sides by - 1
4 x² - 26 x = - 30
Add 30 to both sides
4 x² - 26 x + 30 = 0
The solutions are:
x = 3 / 2 = 1.5 cm
and
x = 5 cm
The margin cannot be 5 cm, because in this case the length of the rectangle at the top and bottom of the picture would be:
5 - 2 x = 5 - 2 ∙ 5 = 5 - 10 = - 5 cm
This is impossible because the length of the rectangle cannot be negative.
So the solution is:
x = 1.5 cm
(5+2x)(8+2x) - 5*8 = 30
now solve for x
Yes
To find the width of the margin, we need to determine the dimensions of the framed picture.
Let's start by calculating the area of the framed picture. The original picture measures 5cm by 8cm, so the area of the picture is:
Area of the picture = Length × Width = 5cm × 8cm = 40cm²
Now, let's consider the framed picture. We know that there is a uniform margin of width X cm all around the picture. This means that both the length and width of the framed picture will be increased by 2X cm.
The length of the framed picture is:
Length of framed picture = Length of picture + 2 × Width of margin
Length of framed picture = 5cm + 2X cm
The width of the framed picture is:
Width of framed picture = Width of picture + 2 × Width of margin
Width of framed picture = 8cm + 2X cm
To calculate the area of the framed picture, we multiply the length by the width:
Area of framed picture = (Length of framed picture) × (Width of framed picture)
Area of framed picture = (5cm + 2X cm) × (8cm + 2X cm)
The area of the framed picture consists of the area of the picture and the area of the margin. Therefore, we can write the following equation:
Area of framed picture = Area of picture + Area of margin
Substituting the values we know:
(5cm + 2X cm) × (8cm + 2X cm) = 40cm² + 30cm²
Expanding the equation:
(40 + 16X + 4X^2) = 70
Rearranging and simplifying the equation:
4X^2 + 16X + 40 - 70 = 0
4X^2 + 16X - 30 = 0
Now we have a quadratic equation. We can solve it using the quadratic formula:
X = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 4, b = 16, and c = -30. Plugging in these values:
X = (-16 ± √(16^2 - 4 × 4 × -30)) / (2 × 4)
X = (-16 ± √(256 + 480)) / 8
X = (-16 ± √736) / 8
X = (-16 ± 27.11) / 8
Now, we can solve for the positive value of X:
X = (27.11 - 16) / 8
X = 11.11 / 8
Therefore, the width of the margin, X, is approximately 1.39 cm.