To find the radius (r) and height (h) of a cone using the slant height (l), you can use the Pythagorean Theorem and the formula for the lateral surface area of a cone.
The Pythagorean Theorem states that for a right triangle, the sum of the squares of the two shorter sides (legs) is equal to the square of the longest side (hypotenuse). In the case of a cone, the slant height (l) is the hypotenuse of a right triangle formed by the height (h) and the radius (r).
Therefore, we can write:
l^2 = r^2 + h^2
In addition, the lateral surface area of a cone can be calculated using the formula:
L = πrl
where L is the lateral surface area, π is the constant pi, r is the radius, and l is the slant height.
From this equation, we can solve for either r or h in terms of the other variable and the slant height l. For example, solving for r, we have:
r = L / (πl)
Substituting this expression for r into the Pythagorean Theorem equation, we get:
l^2 = (L^2 / π^2l^2) + h^2
Simplifying this equation, we get:
h^2 = l^2 - (L^2 / π^2l^2)
Taking the square root of both sides, we can solve for h:
h = √(l^2 - (L^2 / π^2l^2))
Similarly, we could solve for r using the equation for h instead.
In summary, to find the radius and height of a cone given the slant height, you can use the Pythagorean Theorem and the lateral surface area formula to derive equations for r and h in terms of the slant height l and the lateral surface area L.
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