What is the value of dy/dt when dx/dt is 3 and x equals 4, and what is the value of dx/dt when dy/dt is 5 and x equals 12?

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To solve the given problem, we will use the concept of related rates, which is a key idea in calculus. We have two functions, x(t) and y(t), that are dependent on a parameter t. The task includes finding the rates of change of these functions at specific points.

a) Finding dy/dt when dx/dt = 3 and x = 4

Assume that the relationship between x and y can be expressed with a function, say, y = f(x). The chain rule tells us that:

dy/dt = (dy/dx) * (dx/dt)

We need to find dy/dx at the point where x = 4. Assuming we know the function y in terms of x, we could differentiate it accordingly.

For this particular question, we will assume that the derivative dy/dx at x = 4 is given or can be calculated. Let’s say we evaluated it and found that dy/dx = m.

Now, substituting the known values:

dy/dt = m * (dx/dt)

If dx/dt = 3, then we have:

dy/dt = m * 3

Thus, to find the exact value of dy/dt, we would plug in the value of m, which is dy/dx at x = 4.

b) Finding dx/dt when dy/dt = 5 and x = 12

Again, we use the chain rule:

dx/dt = (dx/dy) * (dy/dt)

We need to find dx/dy at the point where x = 12. Similar to the previous step, if we know the function y in terms of x, we differentiate it to find dx/dy.

Let’s say we evaluated it and found that dx/dy = n.

Now substituting the known values, we get:

dx/dt = n * (dy/dt)

Since dy/dt = 5, then:

dx/dt = n * 5

Thus, to find the exact value of dx/dt, we would plug in the value of n, which is dx/dy at x = 12.

In summary, the process involves:

  • Understanding the relationship between the variables x and y.
  • Applying the chain rule to connect rates of change.
  • Finding specific derivatives at the required points.

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