How do I find the radius of a right cone with a slant of 22 and surface area of 168pi m^2?

You didn't say if the surface area included the base.

If it does:
πr^2 + πrs = 168π
divide by π
r^2 + 22r - 168 = 0
completing the square
r^2 + 22r + 121 = 168 + 121 = 289
(r+11)^2 = √289
r+11 = 17
r = 6

If you did not include the base,

πrs = 168π
22r = 168
r = 168/22 = 84/11

To find the radius of a right cone with a given slant height and surface area, you can follow these steps:

Step 1: Understand the formula for the surface area of a right cone.
The formula for the surface area of a right cone is given by:
Surface Area = π * r * (r + l)
where π is a mathematical constant approximately equal to 3.14159, r is the radius of the base, and l is the slant height.

Step 2: Plug in the given values.
In this case, the surface area is given as 168π m², and the slant height is given as 22 m. So we can write the formula as:
168π = π * r * (r + 22)

Step 3: Simplify the equation.
Remove the π from both sides, as it cancels out:
168 = r * (r + 22)

Step 4: Solve the quadratic equation.
Expand the equation:
r² + 22r - 168 = 0

Now, you can either factorize the quadratic equation or use the quadratic formula to find the values of r.
If you factorize the equation, you would get:
(r - 6)(r + 28) = 0

From this, you can see that the possible values for r are r = 6 and r = -28.
Since the radius cannot be negative, we discard the value r = -28.

Hence, the radius of the right cone is r = 6 meters.