A point is moving along the curve xy=12. When the point is at (4,3), the x-coordinate decreases at the rate of 2cm/sec. How fast is the y-coordinate changing at that point?
dy/dx=-y/x is my change rate so far, should i substitute the coordinates (x,y) to the equation or just use the decrease rate for x to get the answer?
To find how fast the y-coordinate is changing at the point (4, 3), you need to substitute the coordinates (x, y) into the equation dy/dx = -y/x and then use the decrease rate for x to calculate the rate of change for y.
Given that the equation is xy = 12, substitute x = 4 and y = 3 into the equation:
4 * 3 = 12
Now, we can differentiate both sides of the equation with respect to time t to find dy/dt:
d/dt (4 * y) = d/dt (12)
4 * dy/dt = 0
Since the x-coordinate is decreasing at a rate of 2 cm/sec, we can say that dx/dt = -2 cm/sec.
Substituting this into the equation, we get:
4 * dy/dt = -2
Now, solve for dy/dt:
dy/dt = -2/4
So, the y-coordinate is changing at a rate of -0.5 cm/sec at the point (4, 3).
To find how fast the y-coordinate is changing at the point (4, 3), you can use the rate of change formula $\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$. In this case, $\frac{dx}{dt}$ represents the rate at which the x-coordinate is changing.
Given that the x-coordinate decreases at a rate of 2 cm/sec, we can substitute $\frac{dx}{dt} = -2$ into the formula.
Now, to find $\frac{dy}{dx}$, we need to differentiate the equation $xy = 12$ implicitly with respect to x.
Differentiating both sides of the equation with respect to x, we get:
$\frac{d(xy)}{dx} = \frac{d(12)}{dx}$
Applying the product rule and chain rule, we get:
$y \cdot \frac{dx}{dx} + x \cdot \frac{dy}{dx} = 0$
Simplifying this equation, we have:
$\frac{dy}{dx} = -\frac{y}{x}$
Substituting the given x and y values from the point (4, 3), we have:
$\frac{dy}{dx} = -\frac{3}{4}$
Finally, substituting $\frac{dx}{dt} = -2$ and $\frac{dy}{dx} = -\frac{3}{4}$ into the rate of change formula, we get:
$\frac{dy}{dt} = \left(-\frac{3}{4}\right) \cdot \left(-2\right) = \frac{3}{2}$
Therefore, the y-coordinate is changing at a rate of $\frac{3}{2}$ cm/sec at the point (4, 3).
the rate at which x changes is given as 2 cm/sec
which is dx/t.
so you have to find the derivative with respect to time (t), you found the rate of change of y with respect to x
xy = 12
x dy/dt + y dx/dt = 0
now plug in our given stuff.
4 dy/dt + 3(2) = 0
4dy/dt = -6
dy/dt = -6/4 = -3/2 cm/sec