# calculus

a billboard 20ft tall is located on top of a building with its lower edge 60 ft above the level of a viewers eye.
how far from a point directly below the sign should a viewer stand to maximize the angle (theta) between the lines of sight of the top and bottom of the billboard.

hint: let theta=a-b then tan (theta)=tan(a-b)=(tan(a)-tan(b))/(1+tan(a)tan(b)) differentiate implicitly with respect to x and set d(theta)/dx=0.

im going to try to draw the drawing but no promises on how it'll turn out.
theta=t
........a_
......./|20|
....../ |..|
...../t/|b.|
..../\/.|..|
..././..|.60
,,/\/...|..|
./\/\a..|..|
0./.\b\.|..|
|-----x-|c i hope that makes since!!(:

1. 0
2. 0
3. 122
1. Using the hint,

tanθ = tan(a-b) = (tan a - tan b)/(1 + tana tanb)

If the viewer is at a distance x,

tan a = 80/x
tan b = 60/x

tanθ = (80/x - 60/x)/(1 + 80/x * 60/x)
tanθ = 20x/(x^2 + 4800)

sec^2θ θ' = 20(4800-x^2)/(4800 + x^2)^2
θ' = 20(4800-x^2)/((4800 + x^2)^2 * sec^2θ)
Now,
sec^2θ = 1 + tan^2θ = 1 + 40x^2/(4800+x^2)^2
so,
θ' = 20(4800-x^2)/(4800 + x^2)^2 * (4800+x^2)^2/(40x^2 + (4800+x^2)^2)
= 20(4800-x^2)/(40x^2 + (4800+x^2)^2)

θ' = 0 when 4800 = x^2 or,
x = 40√3 = 69.38'

1. 0
2. 0
posted by Steve

## Similar Questions

1. ### physics

A rock is shot vertically upward from the edge of the top of a tall building.The rock reaches its maximum height above the top of the building 1.90 s after being shot. Then, after barely missing the edge of the building as it

asked by kristy on September 12, 2013
2. ### Pre- Calculus

An antenna is located at the top of a building 1000 ft tall. From a point on the same horizontal plane as the base of the building, the angles of elevation from the top and the bottom of the antenna are 65.8 degrees and 62.6

asked by Connie on April 27, 2012
3. ### physics

In a scene in an action movie, a stunt man jumps from the top of one building to the top of another building 3.8 m away. After a running start, he leaps at an angle of 18◦ with respect to the flat roof while traveling at a speed

asked by abby on October 9, 2018
4. ### Physics

In a scene in an action movie, a stunt man jumps from the top of one building to the top of another building 3.5 m away. After a running start, he leaps at an angle of 14 with respect to the flat roof while traveling at a speed

asked by Travis on October 4, 2011
5. ### Physics

An objectis thrown upward from the edge of a tall building with a velocity of 10m/s. Where will the object be 3s after it is thrown? Take g = 10m/s^2 15m above the top of the building 30m below the top of the building 15m below

asked by Lawrence on October 13, 2011
6. ### Maths

Standing on top of a slope of 5 degrees, i see the top of a building and an angle of elevation of 35degrees15'. i am 160cm tall and it is 12m from the building base. How tall is the building?

asked by R on June 9, 2015
7. ### Physics!!!!!!

In a scene in an action movie, a stunt man jumps from the top of one building to the top of another building 5.0 m away. After a running start, he leaps at an angle of 10◦ with respect to the flat roof while traveling at a speed

asked by Anonymous on December 5, 2016
8. ### physics

In a scene in an action movie, a stunt man jumps from the top of one building to the top of another building 4.4 m away. After a unning start, he leaps at an angle of 17◦ with respect to the flat roof while traveling at a speed

asked by Kacie on December 4, 2016
9. ### physics

In a scene in an action movie, a stunt man jumps from the top of one building to the top of another building 4.9 m away. After a running start, he leaps at an angle of 17 ◦ with respect to the flat roof while traveling at a

asked by Anonymous on November 8, 2013
10. ### Physics

In a scene in an action movie, a stunt man jumps from the top of one building to the top of another building 5.0 m away. After a running start, he leaps at an angle of 10◦ with respect to the flat roof while traveling at a speed

asked by Anonymous on December 4, 2016

More Similar Questions