how do i solve for r if i have 100= (pi)(r) (square root of r^2+25)?

Question

thise comes from :the surface area formula: S=(pi)(r) (square root of r^2 + h^2) where the h=5 and s=100 how do i solve for r ??? please include steps to follow, thanks!!!

Answer 1

100=πr√(r²+25)
Square both sides:
10,000=π²r²(r²+25)
Distribute:
10,000=π²r⁴+25π²r²
Divide by π²:
10,000/π² = r⁴+25r²
Add 625/4:
10,000/π² + 625/4 = r⁴+ 25r² + 625/4
Factor right side:
10,000/π² + 625/4 = (r²+25/2)²
Take the square root of both sides:
±√(10,000/π² + 625/4) = r²+25/2
Subtract 25/2:
r²= -25/2 ± √(10,000/π² + 625/4)
Take the square root again:
r=±√(-25/2 ± √(10,000/π² + 625/4))
This gives your four possible solutions for the radius. However, note that if the ± sign on the inner square root is negative, you will get an imaginary solution, which in the context of radii is nonsensical. Similarly, the ± sign on the outer square root is also not negative, because that would mean you have a negative radius, which is also nonsensical. Thus your solution is:
r=√(-25/2 + √(10,000/π² + 625/4))

Answer 2

Square everything to get rid of the radical:
100^2 = ((pi)^2)*(r^2)(r^2 + 25)
r^4 + 25 r^2 - (100/pi)^2 = 0

r^2 = (-25 + sqrt(25^2+4*(100/pi)^2)/2, (r^2 may not be -)
r = sqrt((sqrt(625 + ((200/pi)^2) - 25)/2)

from here it's just a matter of correct keystrokes on your calculator. It can be done by hand , but it gets pretty tedious.

Answer 3

take square on both side:
10000/pi^2=(r^2)(r^2+25)

r^4 + 25r^2 - 10000/pi^2 = 0

solve as a quadratic equation for r^2

then solve for r

Answer 4

(pi)(r)(r^2+25)^1/2=100
(pi)^2r^2(r^2+25)=10000
r^4+25r^2=10000/2*3.14*3.14
r^4+25r^2=507.12
r^4+25r^2-507.12=0
now put r^2=t
t^2+25t-507.12=0
find t using the quadratic formula
substitute it for x^2 and find x by taking the square root

Answer 5

100= (pi)(r) (square root of r^2+25)
square both side
10,000 =(pi)^2 (r)^2{r^2+25}
10,000 =9.87 (r)^2{r^2+25}
10,000/9.87 = {r^4+25 r^2}
{r^4+25 r^2 -1013}=0
r^2= { -25 +/- sq rt[(25)^2 -(4)(1)(-1013)] }/2
r^2= { -25 +/- sq rt[(625) +4052] }/2
r^2= { -25 +/- sq rt[4677] }/2
r^2= { -25 +/- [68.39] }/2
r^2= { -25 +/- [68.39] }/2
r^2= 21.695 or -46.695
neglecting (-)sign because square of negative number cannot be real
Therefore
r^2 = 21.695 or r=+/-(4.6578)approximately

Answer 6

r = 4.66