A=πr√(h^2-r^2)
make "r" the subject of the formula
show workings
#thanks
Very interesting
To make "r" the subject of the formula A = πr√(h^2 - r^2), we need to isolate "r" on one side of the equation.
Step 1: Square both sides of the equation to eliminate the square root:
(A^2) = (π^2)(r^2)(h^2 - r^2)
Step 2: Expand the equation:
A^2 = π^2(r^2)(h^2 - r^2)
Step 3: Distribute π^2 into the brackets:
A^2 = π^2(h^2r^2 - r^4)
Step 4: Rearrange the equation to isolate r:
r^4 - h^2r^2 + (A^2/π^2) = 0
Step 5: This is now a quadratic equation in terms of "r^2". Therefore, replace "r^2" with "x":
x^2 - h^2x + (A^2/π^2) = 0
Step 6: Solve the quadratic equation to find "x". You can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = -h^2, and c = (A^2/π^2), so the equation becomes:
x = (h^2 ± √(h^4 - 4(A^2/π^2))) / 2
Step 7: Since "r^2" cannot be negative (radius is always positive), we discard the negative root, and only consider the positive root:
x = (h^2 + √(h^4 - 4(A^2/π^2))) / 2
Step 8: Substitute "x" back with "r^2":
r^2 = (h^2 + √(h^4 - 4(A^2/π^2))) / 2
Step 9: Finally, take the square root of both sides to solve for "r":
r = √((h^2 + √(h^4 - 4(A^2/π^2))) / 2)
And there you have it! "r" is the subject of the formula A = πr√(h^2 - r^2).
Make r the subject of the formula in A=pie r square root of h rest to the power 2-r rest to the power 2
6c(3e+2d)
square both sides
A^2=(PI*r)^2 (h^2-r^2)
divide both sides by PI^2
(A/PI)^2 -r^2h^2 +r^4=0 define a new variable, u=r^2
u^2-uh^2+(A/PI)^2=0 now use the quadratic formula
u=(h^2+-sqrt(h^4-4A^2/PI^2)/2
but u=r^2, so for r, take the square root of u above.
Note there are two solutions, life is not so simple in these equations.