# calculus

posted by .

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!!(:

• calculus -

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'

## Similar Questions

1. ### Trig

A billboard rests on top of a base. From a point level with the bottom of the base, the angle of elevation of the bottom of the billboard is 26 degrees and the top of the billboard is 31 degrees. From a point 21 meters further away, …
2. ### Calculus

A sign 3 feet high is placed on a wall with its base 2 feet above the eye level of a woman attempting to read it. How far from the wall should she stand to get the best view of the sign, that is, so that the angle subtended at the …
3. ### Calculus

Application of Derivatives and Integrals: A sign 3 feet high is placed on a wall with its base 2 feet above the eye level of a woman attempting to read it. How far from the wall should she stand to get the best view of the sign, that …
4. ### 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?
5. ### calculus

An auditorium with a flat floor has a large screen on one wall. The lower edge of the screen is 3 feet above eye level and the upper edge is 10 feet above eye level. Assume you walk away fron the screen at a rate of 3 ft/s while looking …
6. ### Calculus (math)

A painting in an art gallery has height h and is hung so its lower edge is a distance d above the eye of an observer (as in the figure). How far from the wall should the observer stand to get the best view?
7. ### Calculus (math)

A painting in an art gallery has height h and is hung so its lower edge is a distance d above the eye of an observer. How far from the wall should the observer stand to get the best view?
8. ### math

A picture is 5 feet high and the eye level of an observer is 2 feet below the bottom edge of the picture. How far from the picture should the observer stand if he wants to maximize the angle subtended by the picture?
9. ### science

A scenery in your drawing room has height h and is hung in the wall so that its lower edge is at a distance d above the eye of an observer. How far from the wall should the observer stand to get the best view?
10. ### science

A scenery in your drawing room has height h and is hung in the wall so that its lower edge is at a distance d above the eye of an observer. How far from the wall should the observer stand to get the best view?

More Similar Questions