A particle moves on the circle x2+y2=25 in the xy-plane for time t≥0. At the time when the particle is at the point (3,4), dx/dt=6. What is the value of dy/dt at this time?

The movement of the particle on the circle is its displacement

The value of dy/dt at this time is -9/2

The equation of the circle is given as:

Differentiate with respect to time

Divide through by 2

At point (3,4);

x =3 and y = 4.

So, we have:

Also: dx/dt = 6

So, we have:

Multiply

Subtract 18 from both sides

Divide through by 4

Simplify

Hence, the value of dy/dt at this time is -9/2

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Step-by-step explanation:

Differentiation

We have a relationship between x and y as follows:

Both variables depend on time t for t≥0.

Differentiating with respect to time:

Applying the derivative of a power function:

Recall the derivative of a constant is 0.

Dividing by 2:

Solving for y':

At some specific time, we have:

x=3, y=4, dx/dt = x' = 6. Substituting:

Operating:

Simplifying:

infinitely many solutions

step-by-step explanation:

we have been given the equation;

2(2x + 3) = -4 + 2(2x + 5)

we need to determine the value of x.

2(2x + 3), we expand this expression to obtain 4x + 6

similarly 2(2x + 5) will become 4x + 10

now we have;

4x + 6 = -4 + 4x + 10

4x + 6 = 4x + 10 - 4

4x + 6 = 4x + 6

the expression on the left hand side is identical to the one on the right hand side. therefore, we are looking at just a single straight line whose equation is;

y = 4x + 6

consequently, we shall have infinitely many solutions