Find the radius of a right cone with a slant height of 22 and surface area of 168pi
To find the radius of a right cone with a given slant height and surface area, we can use the formulas for the slant height and surface area of a cone.
The slant height of a cone is the distance from the tip of the cone to any point on the perimeter of the base. In this case, the slant height is given as 22.
The formula for the slant height of a cone is given by the Pythagorean theorem:
`slant height^2 = radius^2 + height^2`
Given the slant height (22) and the surface area (168π), we can use the formula for the surface area of a cone to relate the radius, height, and slant height:
`surface area = π * radius * (radius + slant height)`
Substituting the given values, we have:
`168π = π * radius * (radius + 22)`
Dividing both sides by π, we get:
`168 = radius * (radius + 22)`
Expanding the equation, we have:
`168 = radius^2 + 22*radius`
Rearranging the equation and setting it equal to zero, we have a quadratic equation:
`radius^2 + 22*radius - 168 = 0`
Now we can solve this quadratic equation to find the values of the radius. We can either factor the quadratic equation or use the quadratic formula.
Factoring the quadratic equation, we have:
`(radius - 6)(radius + 28) = 0`
So we have two possible solutions for the radius: radius = 6 or radius = -28.
Since radius cannot be negative, we discard the negative value and conclude that the radius of the cone is 6 units.