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# Rate of change and the chain rule

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A rocket ship is speeding away from the sun. In convenient units, the radiation R inside the ship is given by
R = 1/x^2, where x is the distance to the sun. This distance x, as a function of time t in hours, is given by x = 12t+ 2.

How far is the ship from the sun when the radiation level is decreasing at a rate of 3 units per hour, when dR/dt =-3?
What does x=___?

https://brainmass.com/math/optimization/rate-change-chain-rule-518482

#### Solution Preview

Use the chain rule here:
dR/dt=dR/dx dx/dt.

Now, with dR/dx=-2/x^3 and dx/dt=12, then we have the following:
dR/dt=-24/x^3

When dR/dt=-3, we see ...

#### Solution Summary

Given the distance to the sun of a rocket as a function of time, and the radiation inside the ship as a function of the distance to the sun, we find the distance to the sun when the radiation level is decreasing at a rate of 3 units per hour

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