Question

A particle moves along the curve below. y = √ 8 + x 3 As it reaches the point ( 1 , 3 ) , the y -coordinate is increasing at a rate of 5 cm/s . How fast is the x -coordinate of the point changing at that instant? cm/s

Answer 1

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:

$\displaystyle \frac{dy}{dt}=\frac{dy}{dx}\cdot \frac{dx}{dt}$

We have

$y=\sqrt{8+x^3}$

Or, equivalently

$y^2=8+x^3$

We need to know the rate of change of x respect to t. We'll use implicit differentiation:

$\displaystyle 2y\cdot \frac{dy}{dt}=3x^2\cdot \frac{dx}{dt}$

Solving for dx/dt

$\displaystyle \frac{dx}{dt}=\frac{2y\cdot \frac{dy}{dt}}{3x^2}$

Plugging in the values x=1, y=3, dy/dt=5

$\displaystyle \frac{dx}{dt}=\frac{2(3)\cdot 5}{3\cdot 1^2}=10$

The x-coordinate is changing at 10 cm/s