Find the missing measure for a right circular cone given...Find r if T.A.= 12 pi and L.A = 8 pi.
To find the missing measure for a right circular cone, we need to use the formulas for the total surface area (T.A.) and lateral area (L.A.) of a cone.
The formula for the total surface area of a cone is given by T.A. = πr^2 + πrl, where r is the radius of the base and l is the slant height of the cone.
The formula for the lateral area of a cone is given by L.A. = πrl.
Given that T.A. = 12π and L.A. = 8π, we can set up the following equations:
12π = πr^2 + πrl (Equation 1)
8π = πrl (Equation 2)
Since both equations involve πrl, we can solve Equation 2 for rl and substitute it into Equation 1. Let's solve Equation 2 first:
8π = πrl
Divide both sides by π:
8 = rl
Now substitute rl = 8 into Equation 1:
12π = πr^2 + (8)(r)
Simplify the equation:
12π = πr^2 + 8r
Now, rearrange the equation to have it in standard quadratic form:
0 = πr^2 + 8r - 12π
To further simplify, divide all terms by π:
0 = r^2 + 8r - 12
Now we have a simple quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula.
If we factor the quadratic equation, we get:
0 = (r - 2)(r + 6)
Setting each factor equal to zero, we find two possible values for r:
r - 2 = 0 -> r = 2
r + 6 = 0 -> r = -6
Since we are dealing with a physical measurement (the radius), we discard the negative value. Therefore, the missing measure for the radius of the cone, r, is 2.