<span>X is increasing at the rate of 12 units per minute.
Without the actual diagram of the triangle, this question is impossible to answer. However, doing a google search using your exact wording has given me several search results, some of which have a drawing of the triangle. I am going to assume that those drawings match the triangle you were given. If this is not the case, then this answer will be wrong, but should provide you enough information to be able to solve the problem on your own.
First, the triangle is a right triangle with the hypotenuse having a length of 5 and X being the side opposite to angle theta.
Now we need to create an equation that defines X in terms of theta. Since X is opposite to theta, and we know the length of the hypotenuse, the sine function comes to mind. So
sin A = x/5
5sin A = x
x = 5 sin A
Now let's calculate the first derivative of that equation with respect to time. So
d/dt[ x ] = d/dt[ 5 sin A ]
dx/dt = 5 d/dt[ sin A ]
dx/dt = 5 cos(A) * d/dt[ A ]
dx/dt = 5 cos(A) * dA/dt
Now we've been given dA/dt which is 3 radians per minute. But we need to calculate cos(A) when X equal 3. Since we know the hypotenuse, we can use the Pythagorean theorem to get the length of the adjacent side and from there calculate the cosine. So
sqrt(5^2 - 3^2) = sqrt(25 - 9) = sqrt(16) = 4.
And since cosine = adjacent/hypotenuse, we have 4/5 = 0.8. Now substitute the known values into the equation and solve.
dx/dt = 5 cos(A) * dA/dt
dx/dt = 5 * 0.8 * 3
dx/dt = 12
Therefore X is increasing at the rate of 12 units per minute at the moment X equals 3.
Note: If the triangle you've been given does not have a hypotenuse of 5. Then THIS ANSWER IS WRONG.
If you've been given a hypotenuse, the general equation will be:
dx/dt = H * cos(A) * dA/dt
Where H = given hypotenuse, dA/dt = rate of change in theta. So
dx/dt = H * cos(A) * 3
dx/dt = H * sqrt(H^2 - 3^2)/H * 3
dx/dt = sqrt(H^2 - 9) * 3
dx/dt = 3*sqrt(H^2 - 9)
So just plug in whatever value of H you have and you'll get the answer. And if you plug in 5, you'll see that the answer is still 12.</span>
