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ra1l [238]
3 years ago
11

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 r

ate of 5 cm/s . How fast is the x -coordinate of the point changing at that instant? cm/s
Mathematics
1 answer:
REY [17]3 years ago
7 0

Answer:

<em>The x-coordinate is changing at 10 cm/s</em>

Step-by-step explanation:

<u>Rate of Change</u>

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

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