Question

A billboard d = 85 feet wide is perpendicular to a straight road and is 40 feet from the road (see figure). Find the point on the road at which the angle θ subtended by the billboard is a maximum. (Round your answer to two decimal places.)
x = ft

Answer 1

tan(a) = 40/x
tan(a + o) = (40+85)/x = 125/x

Answer 2

Now I'm stuck :P.

Answer 3

take the arctan of bot sides...

so that you can write an equation for \(\theta\)
\[\theta=???\]

Answer 4

\(\theta\) as a function of x

Answer 5

well rifght now I have tan(a+o).

Answer 6

I gotta get rid of the tan(a) huh..

Answer 7

\[\large a= \tan^{-1} \left( 40 \over x \right)\]

\[\large a+\theta = \tan^{-1} \left( 125\over x \right)\]

\[\large\theta =(a+\theta)-a\]

Answer 8

so hmmm...

Answer 9

arctan(125/x) -arctan(40/x)?

Answer 10

yeah

Answer 11

hmmm

Answer 12

I'm still confused :P.

Answer 13

what part

Answer 14

what exactly we are doing lol :P. Maybe as we go along I'll get it..

Answer 15

Just no sure where this is going to end up...

Answer 16

\[\Large \theta =\tan^{-1}\left( 125 \over x \right)-\tan^{-1}\left( 40 \over x \right)\]

so...
\[\Large {d \theta \over dx} = \dots?\]

Answer 17

seems like it's going ot be some UGLYness...

Answer 18

iknow ... groan

Answer 19

125/(x^2 * 15625?)

Answer 20

err +

Answer 21

- (40/(x^2+1600)?

Answer 22

lol I guess there was more to this....

\[\frac{1210(x^2-50000)}{(x^2+1600) (x^2) (1562500)}\]

Answer 23

the fawk..

Answer 24

i'm not good at that i used wolfram alpha

you flipped the signs ...
it should be -125/.... +40/...

Answer 25

so did I lol... Wolfman FTW!

Answer 26

Okies so I'm not retypijng that, lol link plz?

Answer 27

http://www.wolframalpha.com/input/?i=differentiate+arctan%28125%2Fx%29-arctan%2840%2Fx%29

http://www.wolframalpha.com/input/pod.jsp?id=MSP48681a22hb622d3hg3b500001446aa99b6f3bb56&s=45&button=1

Answer 28

and now we solve for x which I'm not going to attempt :P.

Answer 29

actually... I think it'as sqrt(5000?)

Answer 30

yeah that works :P 70.71

Answer 31

yes

Answer 32

now onto the next question, we must hurry!

Answer 33

:P

Answer 34

http://openstudy.com/study#/updates/4fe6893ce4b02c91101b32a8