b In this part we are ask to find a formula for the radius r of a cone in terms of its surface area and slant height. As in Part A, let's use the formula for the surface area of a cone.
We will use the Quadratic Formula to solve the given quadratic equation.
ar^2+ br+ c=0 ⇕ r=- b± sqrt(b^2-4 a c)/2 a
Now we can identify the values of a, b, and c.
π r^2+(π l )r -S.A.=0 ⇕
π r^2+( π l )r +( -S.A.)=0
We see that a= π, b= πl, and c= -S.A.. Let's substitute these values into the Quadratic Formula.
The solutions for this equation are r= -πl±sqrt(π^2l^2+4πS.A.)2π. Let's separate them into the positive and negative cases.
r=-πl±sqrt(π^2l^2+4πS.A.)/2π
r_1=-Ď€l+sqrt(Ď€^2l^2+4Ď€S.A.)/2Ď€
r_2=-Ď€l-sqrt(Ď€^2l^2+4Ď€S.A.)/2Ď€
Note that in the second solution r_2, the numerator is negative and the denominator is positive. Therefore, r_2 will get us a negative solution. However, r_2 represents a radius, which must be positive. This tells us that the only solution is r_1= -Ď€l+sqrt(Ď€^2l^2+4Ď€S.A.)2Ď€.
