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:

part a

x     y

0     8

1     7.5

2     7

3     6.5

4     6

5     5.5  

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part b

it is a function because for every x value there is only one y value

part   c

for every x value there is only one y value

x       y

0       8

1       7.6

2       7.2

3       6.8

step-by-step explanation:

simplifying

5(x + -6) = 2(x + 3)

reorder the terms:

5(-6 + x) = 2(x + 3)

(-6 * 5 + x * 5) = 2(x + 3)

(-30 + 5x) = 2(x + 3)

reorder the terms:

-30 + 5x = 2(3 + x)

-30 + 5x = (3 * 2 + x * 2)

-30 + 5x = (6 + 2x)

solving

-30 + 5x = 6 + 2x

solving for variable 'x'.

move all terms containing x to the left, all other terms to the right.

add '-2x' to each side of the equation.

-30 + 5x + -2x = 6 + 2x + -2x

combine like terms: 5x + -2x = 3x

-30 + 3x = 6 + 2x + -2x

combine like terms: 2x + -2x = 0

-30 + 3x = 6 + 0

-30 + 3x = 6

add '30' to each side of the equation.

-30 + 30 + 3x = 6 + 30

combine like terms: -30 + 30 = 0

0 + 3x = 6 + 30

3x = 6 + 30

combine like terms: 6 + 30 = 36

3x = 36

divide each side by '3'.

x = 12

simplifying

x = 12

this should .