#Calc1: Related Rates Question

10 messages · Page 1 of 1 (latest)

mint prismBOT
uneven meadow
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Your approach is sound, and the answer is close to correct 🙂

You have taken the jet has travelled 10 mi, so your P = 10 mi... but the jet s 10 mi from the station, so it shouldbe your C = 10 mi.

reposting the solution (with minor corrections) below...

cobalt roseBOT
uneven meadow
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Here is a solution, though it doesn't match the options given...

Note that I think you have over-complicated the calculus

And, a suggestion, maybe choose more meaningful variables,

like $d$ and $D$ for distances (choosing $D$ for the larger distance)?

Anyway...

If I take the distance travelled by the jet in $t$ hours as $d$ miles,

measured from $t = 0$ corresponding to directly overhead, then $d = 490t$

and making $D$ the distance from the radar station at time $t$ gives

$D^2 = 2^2 + d^2$ by Pyth, so $D = \sqrt{4 + (490t)^2}$ as $D > 0$.

$\frac{dD}{dt} = \frac{1}{2}\left[4 + (490t)^2\right]^{-\frac{1}{2}} \times \frac{d}{dt}\left[4 + (490t)^2\right] \qquad \text{using the Chain Rule}$

$\implies \frac{dD}{dt} = \frac{490^2t}{\sqrt{4 + (490t)^2}}$

Now, when $D = 10\ \text{mi}: \qquad (490t)^2 = D^2 - 4 = 96 \implies t = \frac{\sqrt{96}}{490}\ \text{h}$

Thus, $\frac{dD}{dt} = \frac{490^2 \times \frac{\sqrt{96}}{490}}{\sqrt{4 + 96}} = 196\sqrt{6}\ \text{mi h}^{-1} \approx 480.0999...\ \text{mi h}^{-1}$

If you want a clear but more sophisticated calculus approach,

I would implicitly differentiate from the Pythagoras' Theorem form:

$D^2 = 4 + (490t)^2 \implies 2D\frac{dD}{dt} = 0 + 490^2 \times 2t$

giving $\frac{dD}{dt} = \frac{490^2t}{D} \qquad \text{. . . . . (1)}$

$D = 10 \implies \frac{100 - 4}{490^2} = t^2$ and so, in (1):

$\frac{dD}{dt} = \frac{490^2}{10} \times \frac{\sqrt{96}}{490} = 49\sqrt{96} = 196\sqrt{6}\ \text{mi h}^{-1} \approx 480.0999...\ \text{mi h}^{-1}$

cobalt roseBOT
uneven meadow
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Fair enough, but making $d = 10$ might have been an easier-to-spot error...

cobalt roseBOT
uneven meadow
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It is a discipline to pause to consider variable choice, but it is worth developing as it makes your work easier for you to check and easier for someone else to read / follow / mark... and it increases the chances of minor errors being missed by a marker as you have helped them to go faster! 🙂

mint prismBOT
#
Solved

Post marked as solved by @stark carbon.

Use .unsolved if this was a mistake.

uneven meadow
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Ok, but (even if only for tactical / strategic reasons, making life easier for your markers is beneficial for you. 🙂