In triangle shown above, if $\theta$ increases at a constant rate of 3 radians per minute, at what rate is $x$ increasing in units per minute when $x$ equals $3$ units?
The answer is 12
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1$\begingroup$ Do you know how to set up related rates problems or implicit differentiation? $\endgroup$– turkeyhundtDec 8, 2014 at 3:42
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$\begingroup$ yes we are supposed to use related rates $\endgroup$– lemonadedogeDec 8, 2014 at 3:43
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$\begingroup$ i did $5cos(\theta)\theta'=x'$ $\endgroup$– lemonadedogeDec 8, 2014 at 3:44
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1$\begingroup$ Plug in what you know. You know $\theta '$ is 3, and you should be able to find $\theta$ since it's a right triangle and you know two sides (5 and 3) $\endgroup$– turkeyhundtDec 8, 2014 at 3:46
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2 Answers
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you can use the constraint $$x = 5 \sin \theta, \to \frac{dx}{dt} = 5\cos \theta \frac{d\theta}{dt} = 15 \cos \theta = 15 \times \frac 4 5=12 \, unit/min $$
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$$\theta =\arcsin(3/5)$$
$$x'=5\cos(\arcsin(3/5))\cdot3$$
$$x'=12 \text{ units/min}$$
answered Dec 8, 2014 at 4:00