3. You work for a defense contractor that is making long...

3. You work for a defense contractor that is making long range missiles. Their trajectory self corrects for atmospheric conditions (as well as the rotation of the Earth). After several tests in many different atmospheric conditions, you can model the maximum range of the projectile as $R = 27800 - 5t^2 - 6ht - 3h^2 + 400t + 300h$ where $R$ is the range in km, $t$ is the temperature in $^\circ C$ and $h$ is the percentage humidity. Find the optimal temperature and humidity value that the missile has the maximum range.

Transcript

00:01 Application of calculus is missiles.

00:02 That's a very common thing to talk about because trajectories are easy to conceive of in terms of these functions that we're working with.

00:12 So here we're given this function r, in terms of a range of a missile.

00:18 And we're supposed to optimize the atmospheric conditions for controlling the missile.

00:24 Where h is the humidity and t is the temperature.

00:29 And so we're going to do is we're going to take the derivative of each of this function with respect to each variable.

00:35 And just like in a single variable calculus, where that tangent equals to zero, that was a bad tangent line, is what we're looking for to optimize each of the variables here.

00:53 So r of t, let's do that.

00:56 So it's this with respect to t, so that 27 ,000 goes away, so you get minus 10.

01:03 T, by the power rule, minus 6h, the minus 3h square goes away.

01:10 It's a constant with respect to t.

01:12 Then i get plus 400, and 300h goes away with respect to t.

01:19 And the derivative with respect to h.

01:23 Let's see, that 27 ,000 goes away, the t goes away, we have minus six, minus six t.

01:38 And then let's see with respect to h we get minus 6h by the power rule the 400 t goes away with respect to t because we're different or with respect to h then we have plus 300 so we set them equal to zero so we get minus 10 t minus 6h plus 400 and then that equals zero and i'm going to connect conveniently write rh underneath it.

02:10 So we get minus 6t minus 6h plus 300.

02:17 So it becomes a system in two variables, h and t.

02:20 They want to solve for one of them...

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