Answer:
In order to maximize [tex]\theta[/tex], the cow must stand 3.74 ft away from the billboard, approximately.
Explanation:
The vertical angle subtended by the billboard at the cow's eye can be calculated from
[tex]\theta = \arctan{\frac{7}{x}} - \arctan{\frac{2}{x}}.[/tex]
Now, maximaxing [tex]\theta[/tex], we have
[tex]\theta_{max} = \frac{d\theta}{dx}|_{x=x_0}=0\\-\frac{7}{x_0^2+49} - \left(\frac{2}{x_0^2+4}\right) = 0\\\frac{7}{x_0^2+49} - \frac{2}{x_0^2+4} = 0.[/tex]
Solving the equation for [tex]x_0[/tex]
[tex]x_0 = \pm\sqrt{14} \approx 3.74.\\[/tex]
Which can be interpretated as the cow standing either one or the other side from the billboard. The next plot confirms that [tex]x_0[/tex]'s positive root, gives a maximum for the angle [tex]\theta[/tex].
The definition of supported angle and trigonometry allows to find the results for the supported angle and the distance for the maximum angle are:
a) Supported angle is: [tex]\theta =tan^{-1} \frac{7}{x} - tan^{-1} \frac{2}{x}[/tex]
b) Distance for maximum angle is: x₀ = 3.74₀ ft
Given parameters
To find
In optics the angle that the body occupies in the eye is called the supported angle, this is found when trigonometry, in general the greater the angle the greater the perception of the size of the object.
In the attached we can see a scheme of the system, for simplicity we place our zero at the height of the eyes of the cow, the height are:
fence bottom y₀ = 6 -4 = 2 ft
Fence height y = 11 -4 = 7 ft
Let's find the two angles using trigonometry
Bottom
tan θ₁ = [tex]\frac{yo}{x}[/tex]
Upper part
tan θ₂ = [tex]\frac{y}{x}[/tex]
The supported angle is
Δθ =θ₂ - θ₁ = [tex]tan^{-1} \frac{y}{x} - tan^{-1} \ \frac{y_o}{x}[/tex]
[tex]\Delta \theta = tan^{-1} \frac{7}{x} - tan^{-1} \frac{2}{x}[/tex]
This is the angle the cow looks at.
b) To maximize this angle we use the mathematical properties that the maximum or minimum of a function can find by setting its first derivative equal to zero.
[tex]\theta = \Delta \theta \\\frac{d \theta}{dx} = 0[/tex]
We use that the tabule derivative is
f (x) = tan⁻¹ [tex]\frac{a}{x}[/tex]
[tex]\frac{df}{dx} = - \frac{a}{x^2 + a^2}[/tex]
We carry out the derivatives
[tex]- \frac{7}{x^2+ 7^2} + \frac{2}{x^2 + 2^2} \\x^2 +4 = \frac{2}{7} (x^2 + 49)[/tex]
[tex]x^2 ( 1 - \frac{2}{7}) = \frac{2}{7} \ 49 - 4 \\0.71429 x^2 = 10\\x= \sqrt{ \frac{10}{0.71429} }[/tex]
x₀ = 3.74 ft
For this distance the supported angle is maximum.
In conclusion using the definition of supported angle and trigonometry we can find the results for the supported angle and the distance for the maximum angle are:
a) Supported angle [tex]\theta = tan^{-1} \frac{7}{x} - tan^{-1} \frac{2}{x}[/tex]
b) Distance for maximum angle is: x₀ = 3.74 ft
Learn more here: brainly.com/question/24977842
