The radius of the cone is:
The total area of cone is given by:
where r is the radius of the cone and l is the slant height of the cone.
and the lateral area of cone is given by:
Hence, from equation (1) and (2) we have:
i.e.
i.e.
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Explanation:
I'll use the quadratic formula to find the roots or x intercepts. This slight detour allows us to factor without having to use guess-and-check methods.
The equation is of the form ax^2+bx+c = 0
This leads to...
Now use those roots to form these steps
Refer to the zero product property for more info.
Therefore, the original expression factors fully to (4x-5)(3x+1)
Use the FOIL rule to expand it out and you should get 12x^2-11x-5 again.
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We did that factoring so we could find the side lengths of the rectangle.
I'm using the fact that area = length*width
The order of length and width doesn't matter.
From here, we can then compute the perimeter of the rectangle
P = 2(L+W)
P = 2(4x-5+3x+1)
P = 2(7x-4)
P = 14x - 8
Unless I am misreading, the question, or don't know all of it, there can be an infinite number of similar shapes. Three are give to you at the bottom of this answer, but there are many more possibilities.
