Inquiry about rate of change of distance: angle sun makes w/ ground is increasing at 0.26* / min.

[coooool222]

Basically all i want to know is if the distance from the sun to shadow (the hypotenuse) is changing or constant.

i tried to use the equation cos (theta) = length of shadow (which is x) / the hypotenuse (distance)

i took derivates of both sides but i do believe the hypotenuse(distance) is changing with time so i cannot use it rather i would need to use
tan theta (length of building)/length of shadow

 

[blamocur]

Basically all i want to know is if the distance from the sun to shadow (the hypotenuse) is changing or constant.

Why do you want to know that?

i tried to use the equation cos (theta) = length of shadow (which is x) / the hypotenuse (distance)

Can you express the dependency of the length of shadow ('x') on angle [imath]\theta[/imath] ?
Can you write an expression for the rate of change of the length of shadow, i.e., [imath]\frac{dx}{d\theta}[/imath] ?
Can you write an expression for [imath]\frac{dx}{dt}[/imath] ?

 

[blamocur]

i tried to use the equation cos (theta) = length of shadow (which is x) / the hypotenuse (distance)

At this point you don't know the length of the hypotenuse, but you do know the length of the vertical side.

 

[The Highlander]

View attachment 37291
Basically all i want to know is if the distance from the sun to shadow (the hypotenuse) is changing or constant.

i tried to use the equation cos (theta) = length of shadow (which is x) / the hypotenuse (distance)

i took derivates of both sides but i do believe the hypotenuse(distance) is changing with time so i cannot use it rather i would need to use
tan theta (length of building)/length of shadow

@coooool222,

Have you now solved this problem or have you just given up on it?

I wouldn't be surprised if you had just opted for the latter because it seems to me that an answer to the question (as presented) may not be possible.

The question tells us that at some initial time, let's call it \(\displaystyle t_0\),
the length of the shadow cast by the building is 35'.

This would be the blue line in the diagram (as I have modified it) opposite.

I have also added the green line for the (84') height of the building and the red line, striking the ground at the angle θ, represents the sun's rays that will be creeping gradually towards the building thus shortening over time the length of the building's shadow which length we may call x.



I was puzzled by your mention of the use of the Cosine ratio, as I don't see its relevance here!

The measure of the angle θ would clearly be obtained by use of the Tangent ratio, ie:-

[math]tan~θ = \frac{84}{35}\implies tan~θ = 2.4\implies θ = tan^{-1} (2.4) \approx 1.176~radians.[/math]
but the question also states that θ is increasing at 0.26° per minute, which is a constant rate of increase.

Thus, at \(\displaystyle t_1\) (ie: 1 minute after \(\displaystyle t_0\)) θ will have increased by \(\displaystyle \frac{0.26\times \pi}{180}\) to a new, larger, angle (say, \(\displaystyle \theta_1\)) and, of course, this angle (as \(\displaystyle \theta_n\) ?) will continue to increase by the same amount every minute thereafter until the shadow disappears altogether.

Now, the length of the shadow at \(\displaystyle t_1\) may be calculated as: \(\displaystyle x_1=\frac{84}{tan~θ_1}~\left(θ_1=θ+\frac{0.26\times \pi}{180}\right)\)

However, as I'm sure you know, the Tangent function is not linear, therefore, as θ_n increases each minute (in fixed increments of c.1.176 min^-1) the denominator of that equation (tan θ_n) will get, progressively, slightly larger, which means that the amount the shadow shortens each minute will get marginally less as each minute passes.

I cannot, therefore, see how you can express the (required) answer in "inches per minute" (another constant rate).

If this analysis is flawed I would certainly be grateful if anyone could point out where I have gone wrong.

(PS: I get the shadow disappearing in just short of 87 minutes. )
(So I suppose you could say the length of the shadow is changing at an average rate of 4.83 in/min. )

 

[Dr.Peterson]

I cannot, therefore, see how you can express the (required) answer in "inches per minute" (another constant rate).

(So I suppose you could say the length of the shadow is changing at an average rate of 4.83 in/min. )

It asked,

At the moment in question ... At what rate is the [length of the] shadow changing?​

I'd take that to be asking for the instantaneous rate. This is, after all, a calculus question.