## watchmath 5 years ago Find the height of the lamp post (see the attachment).

1. watchmath

2. myininaya

did you make this problem up or did you find it somewhere

3. watchmath

It is from a book :). We give this as an assignment for our student :).

4. myininaya

i think i'm fixing to figure out lol

5. myininaya

if i can figure out P i'm done

6. myininaya

so far i have y1=2x0/y0 where P=(-x0,y0) this is the way i'm suppose to go about it right?

7. myininaya

and where L representing the point of the bulb of the lamp post at (3,y1)

8. myininaya

so found the slope of the line containing P and the x intercept (-5,0) and got m=(y1-0)/(3+5)=y1/8 and i found the derivative of the ellipse which is y'=-x/4y i evaluated y' at (-x0,y0) and got y'=x0/4y0 so this has to be equal to y1/8 so we have y1=2x0/y0

9. myininaya

omg omg P=(-1,1) so y1=2

10. myininaya

11. myininaya

the height is 2 :)

12. myininaya

ty so much for the problem :) i don't know if i went about the long way or not but i feel good about it

13. watchmath

Good job Miyaniya :) So you like it?

14. myininaya

yes im going to use it wow it took while for me to figure how to get P lol that was my problem for the longest time

15. myininaya

is this way that you were thinking about finding the height? or is there a shorter way?

16. watchmath

Here what I will do to find $$P$$. Let $$P=(a,b)$$. Using $$(a,b)$$ and $$(-5,0)$$ the slope is $$m=\frac{b}{a+5}$$. Using implicit differentiation the slope is $$m=-\frac{a}{4b}$$. Setting the two to be equal, we have $$4b^2=-a^2-5a$$. But since $$(a,b)$$ is on the ellipse, $$4b^2=5-a^2$$. So $$-a^2-5a=-a^2+5$$ which implies $$a=-1$$. It follows that $$4b^2=5-a^2=4$$. So $$b=\pm 1$$. But $$b$$ is above the $$x$$-axis. So $$b=1$$. Hence $$P=(-1,1)$$. Let $$h$$ be the height that we are looking for. Using similar triangle we have $$\frac{h}{5+3}=m=\frac{1}{4}$$. Then $$h=2$$.

17. myininaya

ok nice. you have less text than i have writing lol

18. myininaya

thank you so much watchmath. i'm totally going to give this to my calculus students and let them work on it as a group project

19. watchmath

My pleasure :)