The sides of the triangle show increase in such a way that

dz/dt = 1 and dx/dt = 3 * dy/dt.
At the instant when x = 12 and y = 5, what is the value of dx/dt?

Hard to say, since you don't describe the relationship between a and the other variables. No diagrams are shown here.

But, given that you have 5 and 12, I'd suspect that z is the hypotenuse of a 5-12-13 right triangle, so that
z^2 = x^2 + y^2
z dz/dt = x dx/dt + y dy/dt
13 * 1 = 12 dx/dt + 5* 1/3 dx/dt
13 = (12 + 5/3) dx/dt
13 * 5/41 = dx/dt

To find the value of dx/dt, we need to use the given relationship between dx/dt and dy/dt. We know that dx/dt = 3 * dy/dt.

Given that dz/dt = 1, we also know that dz/dt = dx/dt + dy/dt.

Substituting dx/dt = 3 * dy/dt into the previous equation, we get:

1 = 3 * dy/dt + dy/dt

Combining like terms, we have:

1 = 4 * dy/dt

Now, let's solve for dy/dt:

dy/dt = 1/4

Since we are given that x = 12 and y = 5, we can use these values to find dx/dt:

dx/dt = 3 * dy/dt
= 3 * (1/4)
= 3/4

Therefore, at the instant when x = 12 and y = 5, the value of dx/dt is 3/4.

To find the value of dx/dt, we are given that dz/dt = 1 and dx/dt = 3 * dy/dt. However, we need more information to determine the exact value of dx/dt at the instant when x = 12 and y = 5.

We can use the information given to determine the relationships between dz/dt, dx/dt, and dy/dt. Let's start by finding the relationship between dz/dt and dx/dt.

Since dz/dt represents the rate of change of the side z, it is independent of both dx/dt and dy/dt. Therefore,

dz/dt = 1

Next, we're given that dx/dt = 3 * dy/dt. This tells us that the rate at which the side x is changing is three times the rate at which the side y is changing. In other words,

dx/dt = 3 * dy/dt

Now let's find dx/dt at the instant when x = 12 and y = 5.

Using the relationship we found above, we can substitute the given values into the equation:

dx/dt = 3 * dy/dt
dx/dt = 3 * (dy/dt) (1)
dx/dt = 3 * (1/3 * dx/dt) (2)

Since we know dz/dt = 1, we can substitute it into equation (1):

dx/dt = 3 * (1/3 * dx/dt)
dx/dt = dx/dt

Now, we have dx/dt = dx/dt, which means there is no unique value of dx/dt at the given instant (x = 12, y = 5). The value of dx/dt cannot be determined solely with the given information.

Therefore, the value of dx/dt is unknown at the instant x = 12 and y = 5.