a tower 7 metres high stands on top of building which is 9 metres high. An observer at the bottom of the building notices that, as she walks away from the building, the angle theta which the tower subtends at her eyes seems to increase in size for a certain distance and then to decrease. determine what position of x maximises the angle theta.
To determine the position of x that maximizes the angle theta, we need to understand the geometry of the situation. Let's begin by drawing a diagram.
First, we have the distance from the observer to the base of the building, which we can call "d". As the observer walks away from the building, the distance between the observer and the base of the tower (which is also the top of the building) increases, forming a right-angled triangle.
Let's label the height of the tower as "h" (7 meters) and the height of the building as "H" (9 meters).
Now, let's determine the relationship between the distances and angles in order to find the position of x that maximizes the angle theta.
1. Determine the overall distance between the observer and the tower (including the building):
This distance can be found using the Pythagorean theorem: d² = x² + H².
2. Find the distance between the observer and the top of the tower:
This distance is given by: (d + h)² = x² + H².
3. Determine the angle theta using trigonometry:
The tangent of theta is equal to the opposite side (h) divided by the adjacent side (x), or tan(theta) = h/x.
4. Express the tangent in terms of d and solve for theta:
Since tan(theta) = h/x and we know the relationship between x and d, we can substitute x² + H² for d² to obtain:
tan(theta) = h/sqrt(x² + H²).
Now, we must find the position of x that maximizes the angle theta. To do this, we need to differentiate the tangent function with respect to x and set it equal to zero. Then solve for x.
Differentiating the tangent function, we get:
d(tan(theta))/dx = (d/dx)(h/sqrt(x² + H²)) = -h(x² + H²)^(-3/2)(2x) = -2hx/(x² + H²)^(3/2).
Setting the derivative equal to zero, we have:
-2hx/(x² + H²)^(3/2) = 0.
Since h is non-zero, the only solution to this equation is x = 0.
Hence, the position of x that maximizes the angle theta is x = 0, which means the observer needs to be at the bottom of the tower (right at the base of the building). At this position, theta will be at its maximum angle.