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.