To find the slant height [tex]\( l \)[/tex] of a cone given the surface area [tex]\( S = 500 \pi \text{ ft}^2 \)[/tex] and the radius [tex]\( r = 15 \text{ ft} \)[/tex], we can use the correct formula for the surface area of a cone. The complete surface area [tex]\( S \)[/tex] of a cone includes the lateral (side) surface area and the base surface area:
[tex]\[ S = \pi r (l + r) \][/tex]
Since the surface area [tex]\( S = 500 \pi \text{ ft}^2 \)[/tex] and the radius [tex]\( r = 15 \text{ ft} \)[/tex], we can substitute these values into the formula:
[tex]\[ 500 \pi = \pi \cdot 15 (l + 15) \][/tex]
First, we can simplify the equation by canceling out [tex]\(\pi\)[/tex] from both sides:
[tex]\[ 500 = 15(l + 15) \][/tex]
Next, solve for [tex]\( l \)[/tex]:
[tex]\[ 500 = 15l + 225 \][/tex]
Subtract 225 from both sides:
[tex]\[ 275 = 15l \][/tex]
Now, divide both sides by 15 to isolate [tex]\( l \)[/tex]:
[tex]\[ l = \frac{275}{15} = 18.3333 \text{ ft} \][/tex]
Therefore, the slant height [tex]\( l \)[/tex] of the cone is:
[tex]\[ l = 18.3333 \text{ ft} \][/tex]
