A photograph print measuring 24cm by 15cm is enclosed in a frame. A uniform space of width X cm is left between the edges of the photograph and the frame. If the area of the space is 270cm², find tje value of X

Area of photograph print:

24 ∙ 15 = 360 cm²

New dimension:

24 + 2 x and 15 + 2 x

Area of the space = ( 24 + 2 x ) ∙ ( 15 + 2 x ) - 24 ∙ 15 = 270

15 ∙ 24 + 15 ∙ 2 x + 2 x ∙ 24 + 2 x ∙ 2 x - 360 = 270

360 + 30 x + 48 x + 4 x² - 360 = 270

4 x² + 78 x = 270

4 x² + 78 x - 270 = 0

The solutions are:

x = - 45 / 2 and x = 3

x can't be negatve so x = 3 cm

P = 2*(x+24+x) + 2(x+15+x) = 8x + 78 = Perimeter = Total length(L).

A = L*W = L*X = 270,
(8x+78) * x = 270,
8x^2 + 78x - 270 = 0, Divide both sides by 2:
4x^2 + 39x - 135 = 0,

X = (-B +- sqrt(B^2-4AC))/2A,
X = (-39 +- sqrt(3681))/8,
X = (-39 +- 60.7)/8 = 2.7, and -12.5 cm.
X = 2.7 cm. Used positive value of X.

To find the value of X, we'll need to calculate the area of the photograph and the area of the frame.

The area of the photograph can be calculated as the product of its length and width:
Area of the photograph = Length × Width = 24 cm × 15 cm = 360 cm²

Let's define the width of the space as X cm. Since the space is uniform on all sides of the photograph, there will be a space of X cm on each edge.

The length and width of the frame can be calculated by adding twice the width of the space to the length and width of the photograph, respectively:
Length of the frame = Length of the photograph + 2 × Width of the space = 24 cm + 2X cm
Width of the frame = Width of the photograph + 2 × Width of the space = 15 cm + 2X cm

The area of the frame can be calculated as the product of its length and width:
Area of the frame = Length of the frame × Width of the frame = (24 cm + 2X cm) × (15 cm + 2X cm)

Given that the area of the space is 270 cm², we can set up the following equation:
Area of the frame - Area of the photograph = Area of the space
(24 cm + 2X cm) × (15 cm + 2X cm) - 360 cm² = 270 cm²

To solve this equation, we can expand and simplify it:
(24 cm + 2X cm) × (15 cm + 2X cm) - 360 cm² = 270 cm²
(360 cm² + 48X cm² + 30X cm + 4X² cm²) - 360 cm² = 270 cm²
48X cm² + 30X cm + 4X² cm² = 270 cm²

Now, let's move all terms to one side to get a quadratic equation:
48X cm² + 30X cm + 4X² cm² - 270 cm² = 0
4X² cm² + 48X cm² + 30X cm - 270 cm² = 0

Finally, we can solve this quadratic equation to find the value(s) of X. We can factorize or use the quadratic formula to do this:

To find the value of X, we need to determine the dimensions of the photograph including the space (frame + space) and calculate the area of the space.

Let's start by calculating the dimensions of the photograph considering the space:

Length of the photograph with space = Length of the photograph + 2 * Width of the space
Width of the photograph with space = Width of the photograph + 2 * Width of the space

In this case:
Length of the photograph with space = 24cm + 2X
Width of the photograph with space = 15cm + 2X

Now, we can calculate the area of the space by subtracting the area of the photograph from the area of the photograph with space:

Area of the space = Area of the photograph with space - Area of the photograph

Area of the space = (Length of the photograph with space) * (Width of the photograph with space) - (Length of the photograph) * (Width of the photograph)

Given that the area of the space is 270cm², we have:

270 = (24 + 2X) * (15 + 2X) - 24 * 15

Now, we can solve this equation to find the value of X.

First, expand the equation:
270 = (360 + 48X + 30X + 4X²) - 360

Simplifying the equation:
270 = 78X + 4X²

Rearranging the equation:
4X² + 78X - 270 = 0

Now, we can solve this quadratic equation to find the value(s) of X using factoring, completing the square, or the quadratic formula.