A carpenter is to use 12m of wood to construct a picture frame.what is the maximum area of the frame
I will assume your frame is in the shape of a rectangle.
(A circle would give you more area)
The largest area is obtained when your rectangle is a square.
So your length is 12 m and you will need 4 equal sides.
How long is each side ? (ignore length lost due to saw cut)
How do you find the area of a square?
Let me know what you get so I can check it
To find the maximum area of the picture frame using 12 meters of wood, we need to determine the dimensions that maximize the area of a rectangle. Let's assume the length of the frame is x meters.
1. Determine the width of the frame:
Since the frame is a rectangle, we can write the equation: 2(width) + 2(length) = 12m.
Given that the length is x meters, 2(width) + 2x = 12m.
Simplifying the equation, we get 2(width) = 12m - 2x.
Dividing both sides by 2, we have width = 6m - x.
2. Calculate the area of the frame:
The area of a rectangle is given by the formula: Area = length × width.
So, the area of the frame is A = x × (6m - x).
3. To find the maximum area, we need to find the vertex (maximum point) of the parabolic curve that represents the area function.
Since the area is given by a quadratic equation, we can rewrite it in the form of A = -x^2 + 6x.
4. To find the x-coordinate of the vertex, we use the formula: x = -b / (2a), where a is the coefficient of x^2 and b is the coefficient of x in the equation.
In our case, a = -1 and b = 6.
Plugging in these values, we get x = -6 / (2*(-1)) = -6 / -2 = 3.
5. Substituting x = 3 back into the area equation A = -x^2 + 6x, we get A = -(3)^2 + 6(3) = -9 + 18 = 9.
Therefore, the maximum area of the picture frame is 9 square meters.
To find the maximum area of the frame, we need to determine the shape that will give us the maximum area with the given amount of wood (12m in this case).
Let's assume the picture frame has a rectangular shape. To find its maximum area, we need to consider the dimensions that will use up the entire 12m of wood.
Let's say the width of the frame is "w" meters. In that case, the length of the frame would be (12 - 2w) meters since we need to subtract the lengths of the two sides that are adjacent to the width.
The area of a rectangle is given by the formula: Area = length * width.
So, the area of the picture frame, A, can be expressed as:
A = w * (12 - 2w)
A = 12w - 2w^2
To find the maximum area, we need to find the value of w that maximizes the equation A. We can do this by finding the vertex of the quadratic equation.
The vertex of a quadratic equation in the form of Ax^2 + Bx + C can be found using the formula:
x = -B / (2A)
In our case, A = -2, B = 12, and C = 0.
w = -12 / (2 * -2)
w = -12 / -4
w = 3
We have found that the width of the frame that maximizes the area is 3 meters.
Substituting this value of w back into the area equation:
A = 12w - 2w^2
A = 12 * 3 - 2 * 3^2
A = 36 - 2 * 9
A = 36 - 18
A = 18
Therefore, the maximum area of the picture frame is 18 square meters.