Alexei wants to hang a mirror in his boat and put a frame around it. The mirror and frame must have an area of 19.25 square feet. The mirror is 3 feet wide and 5 feet long. Which quadratic equation can be used to determine the thickness of the frame, x?
2x2 + 8x − 19.25 = 0
3x2 + 5x − 19.25 = 0
4x2 + 16x − 4.25 = 0
4x2 + 16x + 15 = 0
We can start by finding the area of just the mirror, which is 3 feet x 5 feet = 15 square feet. We know that the total area of the mirror and frame is 19.25 square feet, so the area of just the frame must be 19.25 - 15 = 4.25 square feet.
Let's call the width of the frame "x". Since the mirror is 3 feet wide, the total width of the mirror and frame will be 3 + 2x (one extra x on each side). Since the mirror is 5 feet long, the total length of the mirror and frame will be 5 + 2x. Therefore, the area of the mirror and frame can be expressed as:
(3 + 2x)(5 + 2x) = 19.25
Expanding this equation gives:
15 + 13x + 4x^2 = 19.25
Subtracting 19.25 from both sides gives:
4x^2 + 13x - 4.25 = 0
Therefore, the correct quadratic equation to determine the thickness of the frame, x, is:
4x^2 + 13x - 4.25 = 0
To determine the quadratic equation that can be used to find the thickness of the frame, x, we'll need to find the area of the frame.
The total area of the mirror and frame is given as 19.25 square feet. The mirror has dimensions of 3 feet width and 5 feet length.
Let's denote the thickness of the frame as x.
The area of the mirror is length multiplied by width, which gives us: 3 * 5 = 15 square feet.
Now, to find the area of the frame, we subtract the area of the mirror from the total area:
Area of frame = Total area - Area of mirror
Area of frame = 19.25 - 15
Area of frame = 4.25 square feet
The equation to determine the thickness of the frame, x, in terms of area is:
x * (3 + 2x) = 4.25
This equation can be simplified to:
3x + 2x^2 = 4.25
Rearranging the terms, we get:
2x^2 + 3x - 4.25 = 0
Therefore, the correct quadratic equation that can be used to determine the thickness of the frame, x, is:
2x^2 + 3x - 4.25 = 0
So, the answer is 2x^2 + 3x - 4.25 = 0.