Problem with angles and differentiation

[TW Cantor]

Homework Statement

A "big screen" AB of height L is placed at a height h above a point C on the ground as shown (in the attachment). A person wishes to sit on the ground at a point D in order to watch a video on the big screen. She wishes to sit at a horizontal distance x away from C which gives her the best view. This means that she wishes to sit at a value of x which maximises the viewing angle . If L = 16.2m and h = 7.7m, find this value of x.

Homework Equations

The Attempt at a Solution

i know i have to get an equation which i can then differentiate but I am not sure how to get this equation and what i would then differentiate with respect to.

 

[tiny-tim]

Hi TW Cantor!

Find the angle ADB as a function of x, and then differentiate …

what do you get? ​

 

[TW Cantor]

umm i got cos(θ)=((x^2)+184.03)/((sqrt((x^2)+571.21)*(sqrt((x^2)+59.29))

is that the sort of thing i should get getting for cos in terms of x?

 

[tiny-tim]

How did you get that?

 

[TW Cantor]

haha obviously not the right way then. i tried to find the lengths of AD and BD in terms of x, and then used the cosine rule to get theta in terms of x?

 

[tiny-tim]

Oh I see it now! …

yes that looks ok …

now minimise cosθ (the whole thing). ​

 

[TW Cantor]

sorry does that mean inverse cos of the whole thing?

 

[tiny-tim]

no! stop making it complicated!

you want the largest θ …

that'll be the smallest cosθ …

put y = cosθ (if that makes you happier) …

now y is a function of x, so use dy/dx to find a minimum for y (= cosθ)

 

[TW Cantor]

ok, so i say that y=((x^2)+184.03)/((sqrt((x^2)+571.21)*(sqrt((x^2)+59.29))
i then say dy/dx=0 and use that to calculate a value for x?

 

[TW Cantor]

this method worked :-) i got x=13.57, thanks a lot for your help :-) :-)

 

[tiny-tim]

… i then say dy/dx=0 and use that to calculate a value for x?

(have a square-root: √ and try using the X² tag just above the Reply box )

Yes.

Except, now that I look at it again, it would be easier to use y² instead of y (to avoid those nasty square-roots ).

 

[TW Cantor]

yeah it did come out as quite a nasty differential :-( but i got the right answer so its fine :-)