(6+2y)/(3*sqrt(9-y^2)) = Ae^(kt)
How do you show the steps to figure what y equals?
To solve for y in the equation (6+2y)/(3*sqrt(9-y^2)) = Ae^(kt), we can follow these steps:
Step 1: Isolate the square root term
Multiply both sides of the equation by 3*sqrt(9-y^2):
(6+2y) = Ae^(kt) * 3 * sqrt(9-y^2)
Step 2: Simplify the equation
Combine like terms on the left side:
6 + 2y = 3Ae^(kt) * sqrt(9-y^2)
Step 3: Isolate the square root term again
Subtract 6 from both sides of the equation:
2y = 3Ae^(kt) * sqrt(9-y^2) - 6
Step 4: Square both sides of the equation
To eliminate the square root, we square both sides:
(2y)^2 = (3Ae^(kt) * sqrt(9-y^2) - 6)^2
Simplifying the right side gives:
4y^2 = (3Ae^(kt))^2 * (9-y^2) - 2 * 6 * 3Ae^(kt) * sqrt(9-y^2) + 6^2
Step 5: Simplify and expand the equation
Expand the right side using the distributive property, and square the terms:
4y^2 = 9A^2e^(2kt)(9-y^2) - 36Ae^(kt) * sqrt(9-y^2) + 36
Expand the terms further:
4y^2 = 81A^2e^(2kt) - 9A^2e^(2kt)y^2 - 36Ae^(kt) * sqrt(9-y^2) + 36
Step 6: Collect and Combine like terms
Rearrange the equation by collecting similar terms:
4y^2 + 9A^2e^(2kt)y^2 = 81A^2e^(2kt) - 36
Simplify the right side:
(4 + 9A^2e^(2kt))y^2 = 81A^2e^(2kt) - 36
Step 7: Divide both sides by (4 + 9A^2e^(2kt))
Divide both sides of the equation by (4 + 9A^2e^(2kt)):
y^2 = (81A^2e^(2kt) - 36) / (4 + 9A^2e^(2kt))
Step 8: Take the square root of both sides
Take the square root of both sides to solve for y:
y = ± sqrt((81A^2e^(2kt) - 36) / (4 + 9A^2e^(2kt)))
So, by following these steps, y can be determined.