A piece of plexiglass is in the shape of a semi-circle with radius 2m. Determine the dimensions of the rectangle with the greatest area that can be cut from the piece of plexiglass?
To determine the dimensions of the rectangle with the greatest area that can be cut from the piece of plexiglass, we can break down the problem into smaller steps:
Step 1: Visualize the situation
Imagine a semi-circle with a radius of 2m. This can be represented as half of a circle, where the curved part is part of the circumference and the straight line segment forms the diameter of the circle.
Step 2: Identify the rectangle
To find the dimensions of the rectangle, we need to visualize how it can be inscribed within the semi-circle. We can see that the rectangle should have its longer side equal to the diameter of the semi-circle and its shorter side equal to the radius, as shown below:
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Step 3: Determine the dimensions
The diameter of the semi-circle is twice the radius, so in this case, it would be 2m * 2 = 4m. The length of the rectangle is equal to this diameter, so it would be 4m. The width of the rectangle is equal to the radius, so it would be 2m.
Therefore, the rectangle with the greatest area that can be cut from the piece of plexiglass has dimensions of 4m by 2m.
To determine the dimensions of the rectangle with the greatest area that can be cut from the piece of plexiglass, we need to find the length and width of the rectangle.
1. Start by visualizing the semicircle and the rectangle inside it.
- The semicircle has a radius of 2m.
- The rectangle will be inscribed within the semicircle, with two of its sides along the curved edge of the semicircle.
2. Recall that the area of a rectangle is given by the formula: Area = length × width.
3. Let's label the length and width of the rectangle as L and W, respectively.
4. Based on the geometry of the problem, we can determine the relationship between L and W.
- The length of the rectangle, L, will be equal to the diameter of the semicircle, which is twice the radius: L = 2 × 2m = 4m.
- The width of the rectangle, W, will be equal to the radius of the semicircle: W = 2m.
5. Calculate the area of the rectangle using the formula: Area = length × width.
- Substituting the values we found, Area = 4m × 2m = 8m^2.
Therefore, the maximum area of the rectangle that can be cut from the piece of plexiglass is 8 square meters, with dimensions of 4 meters (length) and 2 meters (width).
Draw a rectangle in a semicircle, let its base be 2x and its height be y
draw a line from the centre to the vertex of the rectangle.
That should be a right-angled triangle with sides x, y, and 2 and
x^2 + y^2= 4 ---> - √(4-x^2)
Area of rect. = 2xy
= 2x(x4-x^2)^(1/2)
d(Area) = 2x(1/2)(4 - x^2)^(-1/2)(-2x) + 2(4 - x^2)^(1/2)
= 0 for a max of Area
this solves to x = 2/√3, subbing that back gives us
y = √8/√3 or 2√2/3
so the dimensions of largest area is 4/√3 by 2√2/√3