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 is, so that the
angle subtended at the woman’s eye by the sign is a maximum?
Make a diagram, sideview, woman on left, wall on right.
let the line from the woman's eyes to the wall be x ft
draw line from there eyes to the top of the picture
let the angle of elevation of her eyes to the bottom of the picture be A
let the angle between top and bottom of picture be B
so we want angle B to be a maximum
angle B = angle (A+B - A)
tan B = tan(A+B - A)
= (tan(A+B) - tanA)/(1 + tan(A+B)tanA)
= (5/x - 2/x)/(1 + (5/x)(2/x))
= (3/x)/(x^2 + 10)/x^2)
= 3x/(x^2+10)
d(tanB)/dx = (3(x^2+10) - 3x(2x))/(x^2+10)^2
= 0 for a max of B
3x^2 + 30 - 6x^2 = 0
x^2 = 10
x = √10
To solve this problem, we can use the concept of optimization, which involves finding the maximum or minimum of a given function. In this case, we need to find the distance from the wall where the angle subtended at the woman's eye by the sign is a maximum.
Let's denote the distance from the wall to the woman as "x". To find the angle, we can use the concept of similar triangles.
First, let's set up the problem:
The height of the sign is 3 feet.
The base of the sign is 2 feet above the eye level of the woman.
Now, let's consider the triangle formed by the woman's eye, the base of the sign, and the top of the sign. Since the triangles are similar, we have:
(1) (height of sign) / (distance from the wall) = (height of woman's eye) / (distance from the wall + x)
Using the given values, we have:
3 / x = (height of eye) / (x + 2)
We can now solve for x to find the distance from the wall:
Cross-multiplying, we get:
3(x + 2) = x(height of eye)
Expanding and rearranging the equation:
3x + 6 = x(height of eye)
Now, we can differentiate both sides of the equation with respect to x to find the maximum angle:
(differentiating)
3 = (height of eye) - x(d(height of eye) / dx)
Simplifying, we get:
x(d(height of eye) / dx) = (height of eye) - 3
Now, we can solve for x by rearranging the equation:
x = (height of eye - 3) / (d(height of eye) / dx)
Once we have the value for x, we can substitute it back into the equation (1) to find the maximum angle subtended at the woman's eye.
Note: In this problem, the height of the woman's eye is not given. To obtain a specific numerical answer, the height of the woman's eye would need to be provided.