Find an equation of variation where y varies directly as x and inversely as the square of w, and y=24 when x=8 and w=3.
To find an equation of variation, we need to express the relationship between the variables involved. In this case, we are told that y varies directly as x and inversely as the square of w.
A direct variation means that two variables are proportional to each other. Mathematically, we can write this as y = kx, where k is the constant of variation.
An inverse variation means that two variables are inversely proportional to each other. Mathematically, we can write this as y = k/w^2, where k is the constant of variation.
Now let's use the given information to solve for the constant of variation, k. We are given that y = 24 when x = 8 and w = 3.
Using the direct variation equation, we have 24 = k * 8. Solving for k, we find that k = 3.
Using the inverse variation equation, we have 24 = 3 / (3^2). Simplifying, we get 24 = 3/9. Multiplying both sides by 9, we get 24 * 9 = 3. Therefore, k = 72.
Now, we can combine the direct and inverse variation equations to give the equation of variation in this problem.
y = kx * k/w^2
y = 72x/w^2
So, the equation of variation in this case is y = 72x/w^2.