A picture frame has a square picture in the center of the frame, the top and bottom borders are each 2 inches wide and the left and right borders are each 4 inches wide. What is the value of x if the total area of the picture frame is 65.25 inches
No idea.
What is x supposed to represent?
The two x's inside the center frame
To find the value of x, we first need to determine the dimensions of the square picture in the center of the frame.
Let's denote the side length of the square picture as s. Since the top and bottom borders are each 2 inches wide, the height of the picture frame is s + 2 + 2 = s + 4. Similarly, since the left and right borders are each 4 inches wide, the width of the picture frame is s + 4 + 4 = s + 8.
The area of the picture frame is given by the product of its height and width: (s + 4) * (s + 8).
According to the problem, the total area of the picture frame is 65.25 square inches. Therefore, we can set up an equation:
(s + 4) * (s + 8) = 65.25
To solve this equation, we can expand and rearrange it:
s^2 + 12s + 32 = 65.25
s^2 + 12s + 32 - 65.25 = 0
s^2 + 12s - 33.25 = 0
Now we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. I will use the quadratic formula in this explanation.
The quadratic formula states:
s = (-b ± √(b^2 - 4ac))/(2a)
In this case, a = 1, b = 12, and c = -33.25.
s = (-(12) ± √((12)^2 - 4(1)(-33.25)))/(2(1))
s = (-12 ± √(144 + 133))/2
s = (-12 ± √(277))/2
Now we have two possible solutions for s. Let's calculate them separately:
s₁ = (-12 + √(277))/2
s₂ = (-12 - √(277))/2
These are the two possible side lengths for the square picture in the center of the frame. To find the value of x, we need to subtract the top and bottom border widths (2 inches) from either solution:
x = s₁ - 2
x = s₂ - 2
Now you can substitute the respective values of s₁ and s₂ into the equations above to find the values of x.
just subtract the picture area from the total, and that is the area of the frame:
(x+2*2)(x+2*4)-x^2 = 65.25