(6+2y)/(3*sqrt(9-y^2)) = Ae^(kt)
How do you show the steps to figure what y equals?
To find the value of y, we need to isolate it on one side of the equation. Let's go through the steps:
Step 1: Start by multiplying both sides of the equation by the denominator, which is 3*sqrt(9 - y^2):
(6 + 2y) = Ae^(kt) * (3*sqrt(9 - y^2))
Step 2: Divide both sides of the equation by Ae^(kt):
(6 + 2y) / (Ae^(kt)) = 3*sqrt(9 - y^2)
Step 3: Square both sides of the equation to eliminate the square root:
[(6 + 2y) / (Ae^(kt))]^2 = [3*sqrt(9 - y^2)]^2
Step 4: Simplify the equation:
(6 + 2y)^2 / (Ae^(kt))^2 = 9*(9 - y^2)
Step 5: Expand the squared terms:
(36 + 24y + 4y^2) / (A^2 * e^(2kt)) = 81 - 9y^2
Step 6: Multiply both sides by (A^2 * e^(2kt)) to remove the denominator:
36 + 24y + 4y^2 = (81 - 9y^2) * (A^2 * e^(2kt))
Step 7: Distribute and rearrange the equation:
36 + 24y + 4y^2 = 81A^2 * e^(2kt) - 9y^2 * A^2 * e^(2kt)
Step 8: Combine like terms:
4y^2 + 24y + 36 = 81A^2 * e^(2kt) - 9A^2y^2 * e^(2kt)
Step 9: Rearrange the equation by bringing all terms to one side:
(4 + 9A^2 * e^(2kt))y^2 + (24 + 9A^2 * e^(2kt))y + (36 - 81A^2 * e^(2kt)) = 0
Step 10: This is a quadratic equation in terms of y. We can solve for y by using the quadratic formula:
y = (-b ± sqrt(b^2 - 4ac)) / 2a
Comparing this equation to our quadratic equation in Step 9:
a = (4 + 9A^2 * e^(2kt))
b = (24 + 9A^2 * e^(2kt))
c = (36 - 81A^2 * e^(2kt))
Plug in the values of a, b, and c into the quadratic formula, and you will have the solutions for y.