Find the radius of a right cone with the slant height 21m and the surface area 168pi m^2
Since the lateral area is A = Pi(r)s, 168 = 3.1416(r)21. Solve for r.
To find the radius of a right cone, we can use the formula for the surface area of a cone:
Surface Area = πr(r + l),
where r is the radius and l is the slant height.
Given that the surface area is 168π m^2 and the slant height is 21 m, we can substitute these values into the formula:
168π = πr(r + 21).
We can simplify this equation by canceling out the π on both sides:
168 = r(r + 21).
Expanding the equation gives:
168 = r^2 + 21r.
Now, let's rearrange the equation to solve for the radius:
r^2 + 21r - 168 = 0.
This is a quadratic equation. To solve it, we can factor it or use the quadratic formula. Let's use the quadratic formula:
r = (-b ± sqrt(b^2 - 4ac)) / 2a,
where a = 1, b = 21, and c = -168.
Plugging in these values, we get:
r = (-21 ± sqrt(21^2 - 4(1)(-168))) / 2(1).
Calculating this expression gives us two possible solutions:
r = (-21 ± sqrt(441 + 672)) / 2.
r = (-21 ± sqrt(1113)) / 2.
Therefore, the radius of the right cone is given by:
r = (-21 + sqrt(1113)) / 2 = approximately 6.76 m,
or
r = (-21 - sqrt(1113)) / 2 = approximately -27.76 m.
Since a negative radius does not make sense in this context, we can conclude that the radius of the right cone is approximately 6.76 m.
To find the radius of a right cone, we need to use the slant height and the surface area of the cone.
The formula for the surface area of a cone is given by:
SA = πr(r + l), where SA is the surface area, r is the radius, and l is the slant height.
Given that the surface area is 168π m^2 and the slant height is 21 m, we can substitute these values into the formula:
168π = πr(r + 21)
Simplifying the equation by canceling out the common factor π:
168 = r(r + 21)
Expanding the equation:
r^2 + 21r - 168 = 0
Now we can solve this quadratic equation to find the possible values for the radius.
Factoring the quadratic equation, we get:
(r - 7)(r + 24) = 0
Setting each factor equal to zero:
r - 7 = 0 --> r = 7
r + 24 = 0 --> r = -24
Since the radius of a cone cannot be negative, the only valid solution is r = 7.
Therefore, the radius of the cone is 7 meters.