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Anit [1.1K]
3 years ago
9

The coordinates of a particle in the metric xy-plane are differentiable functions of time t with:

Mathematics
1 answer:
frez [133]3 years ago
7 0

Answer:

The rate of change of the distance \frac{ds}{dt} when x = 9 and y = 12 is -7.6\:\frac{m}{s}.

Step-by-step explanation:

This is an example of a related rate problem. A related rate problem is a problem in which we know one of the rates of change at a given instant \frac{dx}{dt} and we want to find the other rate \frac{dy}{dt} at that instant.

We know the rate of change of x-coordinate and y-coordinate:

\frac{dx}{dt}=-2\:\frac{m}{s} \\\\\frac{dy}{dt}=-8\:\frac{m}{s}

We want to find the rate of change of the distance \frac{ds}{dt} when x = 9 and y = 12.

The distance of a point (x, y) and the origin is calculated by:

s=\sqrt{x^2+y^2}

We need to use the concept of implicit differentiation, we differentiate each side of an equation with two variables by treating one of the variables as a function of the other.

If we apply implicit differentiation in the formula of the distance we get

s=\sqrt{x^2+y^2}\\\\s^2=x^2+y^2\\\\2s\frac{ds}{dt}= 2x\frac{dx}{dt}+2y\frac{dy}{dt}\\\\\frac{ds}{dt}=\frac{1}{s}(x\frac{dx}{dt}+y\frac{dy}{dt})

Substituting the values we know into the above formula

s=\sqrt{9^2+12^2}=15

\frac{ds}{dt}=\frac{1}{15}(9(-2)+12(-8))\\\\\frac{ds}{dt}=\frac{1}{15}\left(-18-96\right)\\\\\frac{ds}{dt}=\frac{1}{15}(-114)=-\frac{38}{5}=-7.6\:\frac{m}{s}

The rate of change of the distance \frac{ds}{dt} when x = 9 and y = 12 is -7.6\:\frac{m}{s}

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