Answer:
θ =
[7/x] -
[2/x]
x0 = 3.74 feet
Explanation:
A rectangular billboard 5 feet in height stands in a field so that its bottom is 6 feet above the ground. A nearsighted cow with eye level at 4 feet above the ground stands x feet from the billboard. Thus:
The vertical angle subtended by the billboard at cow's eye (θ) is equal to the difference between the angles subtended by the billboard at the bottom (θ
) and top (θ
) of the cow's eye level.
θ = θ
- θ
=
[(5+6-4)/x] -
[(6-4)/x] =
[7/x] -
[2/x]
To determine the distance x0 that maximizes θ. We will conduct the derivative with respect to the variable 'x'. Therefore;
dθ/dx = [1/(1+(7/x)^2)](-7/x)^2 -[1/(1+(2/x)^2)](-2/x)^2]
Rearranging the equation, we have:
7x^2 (1+4/x^2) = 2x^2 (1+49/x^2)
7(x^2 + 4) = 2(x^2 + 49)
7x^2 - 2x^2 = 98 - 28
5x^2 = 70
x^2 = 14
x0 = sqrt (14) = 3.74 feet