1a sam is standing on the roof of a building thats next to...

1a.) Sam is standing on the roof of a building that's next to another taller building. The distance between the two buildings is 70m. From where Sam is, the angle of depression to the base of the other building is 45° and the angle of elevation to the top of the building is 30°. What is the height of the taller building? Include a diagram in your answer. 1b.) The top of a flagpole sways back and forth in high winds. The top sways 10 cm to the right (+) and 10 cm to the left (-) of its resting position. At t = 0s, the pole was momentarily at its resting position, then started to move left. Find the equation of a function that models the distance the pole is from its resting position in terms of time if one complete cycle takes 6 minutes. 1c.) A healthy adult's lungs contain from 2200mL to 2800mL of air during normal breathing. An inhale-exhale cycle takes approximately 4 seconds. At time t = 0s, there is 2500mL of air in the lungs. Write a sinusoidal equation V(t) that models the volume of air in the lungs over time in seconds.

Transcript

00:01 Let's say we are standing on top of a building.

00:04 So let me use my art skills to draw a stick figure representation of me.

00:09 And there's a larger building right next to it.

00:13 And we're trying to find the height of this larger building.

00:16 So we're trying to find out what this height is.

00:19 And the information that they're giving us is that this is, these two buildings are 70 meters across from each other.

00:30 So this is 70 meters.

00:32 What's also given is that the angle of elevation to the building is 30 degrees.

00:39 So angle of elevation means that your line is going up into the right.

00:45 So we have an angle of elevation from here, up to the top of the building.

00:50 That's 30 degrees.

00:52 And then we have an angle of depression.

00:54 And depression means that your line is going down into the right.

00:57 You have an angle of depression to the base of the building.

01:00 So we have another line going down here, and that's 45 degrees.

01:07 And this is all they give us, and they want us to find what the height of our building is.

01:13 Well, this basically forms two right triangles, right? we have a right triangle here.

01:18 We have a right triangle here.

01:20 And if we split those off, we have our top right triangle here in green.

01:25 We have a 30 -degree angle, and we have our 70 -meter side on the bottom.

01:31 And we're trying to find this side, which i will call h1.

01:34 And this is a trigonometry problem.

01:38 We can either do this by using the properties of a 30, 60, 90 triangle, or if we don't remember that, we can just use the effect that like this side is opposite and this side is adjacent.

01:53 So we can write tangent of 30 equals our opposite side over in adjacent.

02:01 And the reason that i'm getting that is because of sokatoa, right? this is what we use to get out all of our sides in like order.

02:12 And because we have an opposite side and in adjacent side, we use toa, which is tangent.

02:19 So if i'm solving this for h1, i would just, i would multiply both sides by 70 to get 70 tangent 30.

02:31 And then if i put that in the calculator, that's approximately 40 .14 meters.

02:37 I'm just rounding that off.

02:40 So that is going to be the length of the top part.

02:43 So this length here, which is h1.

02:48 Now our bottom part, h2, which is this part of the height, that's going to be our triangle on the bottom here.

02:56 All right.

02:57 So we have 45 degrees.

02:59 We have our 70s side and we're trying.

03:02 So i find this.

03:03 This is another tangent problem because we don't have the hypotenuse.

03:08 So we have tangent of 45 equals h2, which is opposite, over 70, which is again adjacent.

03:19 So we multiply that.

03:20 We got 70 times tangent of 45, right, equals h2.

03:27 And that basically gives us back a 70.

03:32 So we have 70.

03:34 Equals h2, right? and the reason why like doing this in this matter is important is because if they were like a different angles, this same process works, right? i could also use it by just knowing that this is a 45 -90 triangle, which means that the legs have to be equal, but like i'm trying to show you a process that will work even if the angles are different.

04:05 So what that means is our final height, right? our total height, which i'll call h total, right? this is h total.

04:12 It's going to be equal to the sum of h1 and h2.

04:17 So that's just 40 .14 plus 70.

04:22 That's going to be 1 .10 .14 meters.

04:27 So that will be the height of our building.

04:30 All right.

04:31 So that's problem number one.

04:35 All right.

04:35 So let's try out problem number two now.

04:38 So our second problem it basically says that we have a flagpole.

04:46 Right? so let's draw a flagpole here...