El largo de una pintura rectangular es 8 pulgadas menor que el doble de su ancho. Si el marco tiene 4 pulgadas de ancho y un área de 816 pulgadas cuadradas, hallar las dimensiones de la pintura sin el marco.

To find the dimensions of the painting without the frame, let's assign variables to the width and length of the painting.

Let's say the width of the painting is "x" inches. According to the problem, the length of the painting is 8 inches less than twice the width, so we can express the length as 2x - 8.

Now, we need to consider the frame. The frame has a width of 4 inches on all sides, which means it reduces the width and length of the painting by 8 inches in total (4 inches on each side). Therefore, the width of the painting including the frame would be x + 8, and the length would be 2x - 8 + 8, which simplifies to 2x.

To find the area of the painting including the frame, we multiply the width and length: (x + 8)(2x) = 816.

Expanding the equation, we have 2x^2 + 8x = 816.

Rearranging the equation to make it easier to solve, we get 2x^2 + 8x - 816 = 0.

To solve this quadratic equation, we can either factor it or use the quadratic formula. In this case, factoring is not straightforward, so we'll use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 2, b = 8, and c = -816. Substituting these values into the formula, we get:

x = (-8 ± √(8^2 - 4(2)(-816))) / (2(2))

Simplifying the equation further:

x = (-8 ± √(64 + 6528)) / 4

x = (-8 ± √(6592)) / 4

Now, we need to find the values of x that satisfy the equation. We can take the positive square root to find one solution:

x = (-8 + √(6592)) / 4

x ≈ 17.39

Since the width of the painting cannot be negative, we disregard the negative solution.

Therefore, the width of the painting, including the frame, is approximately 17.39 inches.

To find the length of the painting, we substitute this value of x into the expression for the length: 2x - 8.

Length = 2(17.39) - 8

Length ≈ 26.78

So, the dimensions of the painting without the frame are approximately 17.39 inches by 26.78 inches.

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