a painting 16cms wide and 9 cms long is surrounded by a frame of uniform width x. If the area of the frame is 84 square centimeters, find the uniform width of the frame?
(16+2x)(9+2x) - 16*9 = 84
To find the uniform width of the frame, we need to subtract the area of the painting from the total area of the frame and then divide it by the perimeter of the frame.
First, let's find the area of the painting:
Area of the painting = length * width = 16 cm * 9 cm = 144 square cm
Next, let's find the total area of the frame:
Total area of the frame = area of the painting + area of the frame = 144 square cm + 84 square cm = 228 square cm
The total area of the frame consists of the four sides of the frame, each with width x.
The width of the frame is the same for all four sides, so we can say that the length of one side of the frame is (16 cm + 2x) and the width is (9 cm + 2x).
Now, let's calculate the perimeter of the frame:
Perimeter of the frame = 2 * (length of one side + width of one side)
= 2 * [(16 cm + 2x) + (9 cm + 2x)]
= 2 * (25 cm + 4x)
We know that the area of the frame = Total area of the frame - Area of the painting:
228 square cm = (25 cm + 4x) * (16 cm + 2x) - 144 square cm
228 square cm = (400 cm^2 + 82x cm + 8x^2) - 144 square cm
228 square cm = 400 cm^2 + 82x cm + 8x^2 - 144 square cm
Simplifying the equation:
8x^2 + 82x - 144 = 0
Now we have a quadratic equation, which we can solve using factoring, completing the square, or the quadratic formula. After solving the equation, we will get the value of x, which represents the uniform width of the frame.