2y^-3 = 4e^(-x/2)/(1+e(-x/2))
my teacher gave this problem in my chem class and i don't know what he is really asking , does any one have an idea if i have to solve for y or what's going on.. how would this be done..
To better understand what your teacher is asking for, let's break down the equation:
2y^-3 = 4e^(-x/2) / (1+e^(-x/2))
Here's what each part of the equation means:
- The variable "y" is raised to the power of -3, which means we have to take the reciprocal of y cubed.
- The term on the right side consists of two parts: 4e^(-x/2) and 1+e^(-x/2), both of which involve the exponential function e^(-x/2).
Based on this information, it seems that your teacher wants you to solve for the value of y in terms of x. So, we will attempt to isolate y on one side of the equation.
Step 1: Let's start by getting rid of the fraction. We can do this by multiplying both sides of the equation by the denominator, (1 + e^(-x/2)):
(1 + e^(-x/2)) * (2y^-3) = 4e^(-x/2)
This simplifies to:
2y^-3 * (1 + e^(-x/2)) = 4e^(-x/2)
Step 2: Distribute the terms on the left side of the equation:
2y^-3 + 2y^-3 * e^(-x/2) = 4e^(-x/2)
Step 3: Now, let's isolate the term with y by subtracting 2y^-3 from both sides:
2y^-3 * e^(-x/2) = 4e^(-x/2) - 2y^-3
Step 4: To further simplify the equation, let's combine like terms and put everything on one side:
2y^-3 * e^(-x/2) + 2y^-3 = 4e^(-x/2)
2y^-3 * e^(-x/2) + 2y^-3 - 4e^(-x/2) = 0
Step 5: At this point, we have a quadratic equation in terms of y^-3. Let's substitute a variable, such as "z," for y^-3 to make our calculations easier:
2z * e^(-x/2) + 2z - 4e^(-x/2) = 0
Step 6: Rearrange the terms:
2z * e^(-x/2) - 4e^(-x/2) + 2z = 0
Step 7: Factor out e^(-x/2):
2z * e^(-x/2) - 2 * e^(-x/2) + 2z = 0
Step 8: Factor out 2z from the first and last term:
2z * (e^(-x/2) + 1) - 2 * e^(-x/2) = 0
Step 9: Divide both sides of the equation by (e^(-x/2) + 1):
2z = (2 * e^(-x/2)) / (e^(-x/2) + 1)
Step 10: Simplify the right side of the equation:
2z = 2e^(-x/2) / (e^(-x/2) + 1)
Step 11: Divide both sides of the equation by 2:
z = e^(-x/2) / (e^(-x/2) + 1)
Step 12: Substitute back z for y^-3:
y^-3 = e^(-x/2) / (e^(-x/2) + 1)
Step 13: Take the reciprocal of both sides:
y^3 = (e^(-x/2) + 1) / e^(-x/2)
Step 14: Simplify the right side by dividing both terms by e^(-x/2):
y^3 = (e^(-x/2) / e^(-x/2) + 1 / e^(-x/2))
y^3 = (1 + e^(x/2))
Step 15: Finally, take the cube root of both sides to solve for y:
y = (1 + e^(x/2))^(1/3)
So, the solution for y in terms of x is y = (1 + e^(x/2))^(1/3).