The surface area of a right cone is 55π cm squared. Find the radius if the slant height is 6cm.
S.A. = πr(r + 6)
55π = πr² + 6πr
55 = r² + 6r
r² + 6r - 55 = 0
(r + 11)(r - 5) = 0
r = 5 cm
To find the radius of the right cone, we need to use the formula for the surface area of a cone and solve for the radius.
The surface area of a cone formula is given by:
S = πr(r + l),
where S is the surface area, r is the radius, and l is the slant height.
In this case, we have:
S = 55π cm^2,
l = 6 cm.
Substituting the given values into the formula, we have:
55π = πr(r + 6).
Expanding and rearranging the equation, we get:
55π = πr^2 + 6πr.
Dividing both sides of the equation by π, we have:
55 = r^2 + 6r.
Now, we have a quadratic equation. Setting the equation equal to zero, we have:
r^2 + 6r - 55 = 0.
To solve for the radius, we can factor the quadratic equation or use the quadratic formula. Factoring the equation, we have:
(r + 11)(r - 5) = 0.
Setting each factor equal to zero, we have two possible solutions:
r + 11 = 0, or r - 5 = 0.
Solving each equation, we find:
r = -11 or r = 5.
Since a radius cannot be negative, the valid solution is:
r = 5 cm.
Therefore, the radius of the right cone is 5 cm.
To find the radius of the right cone, we will need to use the formula for the surface area of a cone and some algebraic manipulation.
The formula for the surface area of a cone is given by:
Surface Area = πr(r + L)
Where r is the radius of the base of the cone and L is the slant height.
In this case, we are given that the surface area is 55π cm² and the slant height is 6 cm. Plugging in these values into the formula, we get:
55π = πr(r + 6)
To simplify the equation, we can cancel out the π on both sides:
55 = r(r + 6)
Now, expand the equation:
55 = r² + 6r
Rearrange the equation to make it a quadratic equation:
r² + 6r - 55 = 0
Now, we can solve this quadratic equation for r. We can either factorize it or use the quadratic formula.
Let's use the quadratic formula:
r = (-b ± √(b² - 4ac)) / (2a)
In this case, a = 1, b = 6, and c = -55.
Plugging these values into the formula, we get:
r = (-6 ± √(6² - 4(1)(-55))) / (2(1))
Simplifying further:
r = (-6 ± √(36 + 220)) / 2
r = (-6 ± √256) / 2
r = (-6 ± 16) / 2
We have two possible solutions:
r₁ = (-6 + 16) / 2 = 5 cm
r₂ = (-6 - 16) / 2 = -11 cm
Since the radius cannot be negative, the radius of the right cone is 5 cm.