A cone of radius r height h and slant height l is given. find the total surface area in terms of r and h. if the surface area 50pi find l as a function of r.?

In a right cone, the distance between the vertex and any point on the boundary of the base is called the slant height l of a cone. If you draw a right cone, the slant height l, height h, and the radius r of the base form a right triangle. Therefore, the Pythagorean Theorem applies h^2 + r^2 = l^2. This relationship helps derive formulas for the lateral and surface area of a right cone.

The lateral area: L.A. = (pi)(r)(l)

The surface area: S.A. = (pi)(r)(l) +(pi)(r^2) = (pi)(r)(l + r)

Now, you need to find the surface area in terms of r and h. So you need to express l in term of h and r. The Pythagorean Theorem above gave you what you need. So substitute √(h^2 + r^2)for l into the surface area formula: S.A. = (pi)(r)[√(h^2 + r^2) + r]

If S.A. = 50 pi, then

50pi = (pi)(r)(l + r) divide by pi to both sides

50 = rl + r^2 subtract r^2 to both sides

50 - r^2 = rl divide by r to both sides

50/r - r = l