Height the designers of a water park are creating a new...

HEIGHT The designers of a water park are creating a new slide and have sketched some preliminary drawings. The length of the ladder is 30 feet, and its angle of elevation is $60^\circ$ (see figure). (a) Find the height $h$ of the slide. (b) Find the angle of depression $\theta$ from the top of the slide to the end of the slide at the ground in terms of the horizontal distance $d$ the rider travels. (c) The angle of depression of the ride is bounded by safety restrictions to be no less than $25^\circ$ and not more than $30^\circ$. Find an interval for how far the rider travels horizontally.

Transcript

00:01 Here we have the design of a water slide in a water theme park.

00:05 So the design is that actually in the phase of designing this, so they need to calculate some values in order to fix the design of this particular prime.

00:20 So what they have fixed is that they have a ladder to reach the top of the water slide with the length of which is 30 feet and the angular inclination of this ladder is 60.

00:31 So they need to find the height of this particular point or this particular platform on which that is at the top of the water slide.

00:43 And from the there is an angle of depression theta through which the slide goes down and the horizontal distance travelled by the user or the rider is d.

00:57 So what they need is that first of all we need to find the height that is it's and we need to find the height that is it's and we need to find find the angle theta in terms of d that is we need to find the relation between theta and d that is the angle of depression of the water slide and b is the horizontal distance give it mind it's not the actual distance traveled it's the horizontal distance traveled okay and in the next part what they have given is that there is a restriction that is the angle of theta or the angle of depression has a minimum at a maximum maximum value that is it can't go more than 20, sorry, more than 30 degree and it shouldn't be less than 25 degree due to safety regulations.

01:46 So by keeping that in mind, you need to find the range of the distance, the range of values that we can have or the values that we can have so that the horizontal distances that can travel that a rider can travel depending on the angle that is set, that is between 25 and 30.

02:06 So first let's find it.

02:07 Now this is the representation of the actual figure.

02:11 I don't think you need to draw this rectangle because it doesn't play any role in this calculation.

02:17 But i just drew this because it just represents the action figure that is you go to up, then there's a platform, then you go down.

02:26 In the calculation you don't need to consider this.

02:29 So let's name this.

02:31 That is a, b, c, b.

02:36 E and f so let's find it's so here this is a right time right so here also we have a 9 degree and here also we have an so this is a right triangle and we need find h so we are given with the ab so h is equal to x is equal to bf we have angle a is equal to 60 degree and we have ab as equal to 30.

03:16 So here we can apply the properties of trigonometry, right? we need to connect these two values using one of the trigonometric functions, right? so here this is 16, this is the opposite side and this is the hypothesis.

03:32 The function that connects the opposite side and hybridness is sign.

03:36 So we just need to find, this need to play sign role here.

03:39 Sorry sign here so sign a is equal to sign a is equal to bf that is the opposite side divided by a b that is the high corpus right now that means sign 60 is equal to bf is which is what we need to find and ab so that means h is equal to 30 sine 60 right now what is sign 60 right now what is 16 is equal to 30 multiplied by sine 60 is root 3 by 2 right so once you do that we'll get h is equal to when you divide 30 and 2 we'll get 15 root 3 so that means it's the maximum high sorry the height of this platform is 15 road 3 feet or if you want you can do the calculations and write at 25 point around 25 .98 feet.

04:52 So it's or the height of the platform is 50 root 3 or 25 .98 feet.

04:58 So we've done that...

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