Answer:
The x-coordinate is changing at 10 cm/s
Step-by-step explanation:
Rate of Change
Suppose two variables x and y are related by a given function y=f(x). If they both change with respect to a third variable (time, for instance), the rate of change of them is computed as the derivative using the chain rule:
[tex]\displaystyle \frac{dy}{dt}=\frac{dy}{dx}\cdot \frac{dx}{dt}[/tex]
We have
[tex]y=\sqrt{8+x^3}[/tex]
Or, equivalently
[tex]y^2=8+x^3[/tex]
We need to know the rate of change of x respect to t. We'll use implicit differentiation:
[tex]\displaystyle 2y\cdot \frac{dy}{dt}=3x^2\cdot \frac{dx}{dt}[/tex]
Solving for dx/dt
[tex]\displaystyle \frac{dx}{dt}=\frac{2y\cdot \frac{dy}{dt}}{3x^2}[/tex]
Plugging in the values x=1, y=3, dy/dt=5
[tex]\displaystyle \frac{dx}{dt}=\frac{2(3)\cdot 5}{3\cdot 1^2}=10[/tex]
The x-coordinate is changing at 10 cm/s
