HERE IS A VERY LONG AND DETAILED SOLUTION. HAVE FUN!
Solve for n:
100/n = 10 + 2.46 1/sqrt(n)
10 + 2.46 1/sqrt(n) = 10 + (123 1/sqrt(n))/50:
100/n = 10 + (123 1/sqrt(n))/50
Bring 10 + (123 1/sqrt(n))/50 together using the common denominator 50 sqrt(n):
100/n = (123 + 500 sqrt(n))/(50 sqrt(n))
Cross multiply:
5000 sqrt(n) = n (123 + 500 sqrt(n))
Subtract n (123 + 500 sqrt(n)) from both sides:
5000 sqrt(n) - n (123 + 500 sqrt(n)) = 0
5000 sqrt(n) - (123 + 500 sqrt(n)) n = 5000 sqrt(n) - 123 n - 500 n^(3/2):
5000 sqrt(n) - 123 n - 500 n^(3/2) = 0
Simplify and substitute x = sqrt(n):
5000 sqrt(n) - 123 n - 500 n^(3/2) = 5000 sqrt(n) - 123 (sqrt(n))^2 - 500 (sqrt(n))^3 = -500 x^3 - 123 x^2 + 5000 x = 0:
-500 x^3 - 123 x^2 + 5000 x = 0
Factor x and constant terms from the left hand side:
-x (500 x^2 + 123 x - 5000) = 0
Multiply both sides by -1:
x (500 x^2 + 123 x - 5000) = 0
Split into two equations:
x = 0 or 500 x^2 + 123 x - 5000 = 0
Substitute back for x = sqrt(n):
sqrt(n) = 0 or 500 x^2 + 123 x - 5000 = 0
Square both sides:
n = 0 or 500 x^2 + 123 x - 5000 = 0
Divide both sides by 500:
n = 0 or x^2 + (123 x)/500 - 10 = 0
Add 10 to both sides:
n = 0 or x^2 + (123 x)/500 = 10
Add 15129/1000000 to both sides:
n = 0 or x^2 + (123 x)/500 + 15129/1000000 = 10015129/1000000
Write the left hand side as a square:
n = 0 or (x + 123/1000)^2 = 10015129/1000000
Take the square root of both sides:
n = 0 or x + 123/1000 = sqrt(10015129)/1000 or x + 123/1000 = -sqrt(10015129)/1000
Subtract 123/1000 from both sides:
n = 0 or x = sqrt(10015129)/1000 - 123/1000 or x + 123/1000 = -sqrt(10015129)/1000
Substitute back for x = sqrt(n):
n = 0 or sqrt(n) = sqrt(10015129)/1000 - 123/1000 or x + 123/1000 = -sqrt(10015129)/1000
Raise both sides to the power of two:
n = 0 or n = (sqrt(10015129)/1000 - 123/1000)^2 or x + 123/1000 = -sqrt(10015129)/1000
Subtract 123/1000 from both sides:
n = 0 or n = (sqrt(10015129)/1000 - 123/1000)^2 or x = -123/1000 - sqrt(10015129)/1000
Substitute back for x = sqrt(n):
n = 0 or n = (sqrt(10015129)/1000 - 123/1000)^2 or sqrt(n) = -123/1000 - sqrt(10015129)/1000
Raise both sides to the power of two:
n = 0 or n = (sqrt(10015129)/1000 - 123/1000)^2 or n = (-123/1000 - sqrt(10015129)/1000)^2
100/n ⇒ 100/0 = ∞^~
10 + 2.46 1/sqrt(n) ⇒ 10 + 2.46 1/sqrt(0) = ∞^~:
So this solution is incorrect
100/n ⇒ 100/(-123/1000 - sqrt(10015129)/1000)^2 = 100000000/(123 + sqrt(10015129))^2 ≈ 9.25175
10 + 2.46 1/sqrt(n) ⇒ 10 + 2.46 1/sqrt((-123/1000 - sqrt(10015129)/1000)^2) ≈ 10.7483:
So this solution is incorrect
100/n ⇒ 100/(sqrt(10015129)/1000 - 123/1000)^2 = 100000000/(sqrt(10015129) - 123)^2 ≈ 10.8088
10 + 2.46 1/sqrt(n) ⇒ 10 + 2.46 1/sqrt((sqrt(10015129)/1000 - 123/1000)^2) ≈ 10.8088:
So this solution is correct
The solution is:
Answer: |n = (sqrt(10015129)/1000 - 123/1000)^2=9.251749460185566
