Assume that x and y are differentiable functions of t. Find dy/dt using the given values.

xy=x+4 for x=4, dx/dt=-5

a. 0.20

b. 3.75

c. 1.25

d. 5.00

e. -0.25

from xy = x + 4

x dy/dt + y dx/dt = dx/dt + 0

when x = 4 , in original
4y = 4+4
y = 2

so when x=4, y=2 and dx/dt = -5

4dy/dt + 2(-5) = -5 + 0
4dy/dt = 5
dy/dt = 5/4 which is c)

is it C. or D.

To find dy/dt, we need to differentiate the given equation with respect to t. Let's start by differentiating both sides of the equation xy = x + 4 with respect to t.

d(xy)/dt = d(x)/dt + d(4)/dt

To find dy/dt, we need to find the derivative of xy with respect to t. Since x and y are both functions of t, we can apply the product rule of differentiation.

Using the product rule: d(xy)/dt = x * (d(y)/dt) + y * (d(x)/dt)

Now let's substitute the given values: x = 4, and dx/dt = -5.

d(xy)/dt = 4 * (d(y)/dt) + y * (-5)

The equation becomes:

4 * (d(y)/dt) - 5y = -20

To solve for dy/dt, we need to isolate d(y)/dt, so let's rearrange the equation:

4 * (d(y)/dt) = 5y - 20

Now divide both sides by 4:

(d(y)/dt) = (5y - 20)/4

Substitute x = 4 back into the equation:

(d(y)/dt) = (5(4) - 20)/4

Simplifying the equation:

(d(y)/dt) = (20 - 20)/4
(d(y)/dt) = 0/4
(d(y)/dt) = 0

Therefore, dy/dt = 0.

The correct option is a. 0.20.

Keep in mind that when given values are substituted into the equation, it is essential to compute the result correctly.