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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=___?

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

## SOLUTION This solution is FREE courtesy of BrainMass!

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 that -3=-24/x^3, so in x^3=8 form, where x=2.

Alternatively, one can write R in terms of t, by replacing the expression for x:
R=1/(12t+2)^2
Then dR/dt=-24/(12t+2)^3, so dR/dt=-3 implies that -24/(12t+2)^3=-3, or equivalently (12t+2)^3=8.
Solving for t gives only one value, t=0. Replacing the value of t in the expression for x gives x=2.

Note that the method that uses the chain rule is faster. You may consider that when solving problems of the same kind.

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

Â© BrainMass Inc. brainmass.com December 24, 2021, 10:54 pm ad1c9bdddf>
https://brainmass.com/math/optimization/rate-change-chain-rule-518482