A statue 10 feet high is standing on a base 13 feet high. If an observer's eye is 5 feet above the ground, how far should he stand from the base in order that the angle between his lines of sight to the top and bottom of the statue be a maximum?
If the angles to the bottom and top are a and b, then from a distance x we have
tan(a) = 8/x
tan(b) = 18/x
We want the angle c=b-a, so
c = arctan(18/x)-arctan(8/x)
dc/dx = 10(144-x^2)/((x^2+8^2)(x^2+18^2))
dc/dx=0 at x=12
Sorry for asking, but why did tan(a) become 8/x and tan(b) 18/x?
because that's the definition of tan(a): opposite/adjacent.
Did you draw a diagram of the setup? That's the first step to all of these problems.
Thank you so much
Well, I must say it's quite the sight to see someone trying to maximize their angles while observing a statue. But never fear, I'm here to help! Let's get cracking on this mathematical masterpiece.
First, let's assume that our observer is standing at a distance "x" from the base of the statue. Now, if we draw a little triangle, we can see that the observer's line of sight will create an angle with the top of the statue and another angle with the bottom of the statue.
We want to find the value of "x" that maximizes the angle between these lines of sight. To do that, we'll need to take a few steps. Let's start by establishing the given values:
The statue is 10 feet high, and the base is 13 feet high. The observer's eye is 5 feet above the ground.
Now, let's imagine that the observer is looking straight ahead. The line of sight from the observer's eye to the top of the statue will form one angle with the horizontal line, while the line of sight from the observer's eye to the bottom of the statue will form another angle.
Considering these angles, we can use some trigonometry here. Let's call the angle formed by the line of sight to the top of the statue "θ." We can find this angle using the tangent function since we know the height of the statue (10 feet) and the distance from the observer to the statue (x).
tan(θ) = Height of the statue (10ft) / Distance from the observer to the statue (x)
Now let's call the angle formed by the line of sight to the bottom of the statue "φ." We can find this angle using the tangent function as well, but this time using the height difference between the statue and the base (13 - 10 = 3 feet) and the same distance from the observer to the statue (x).
tan(φ) = Height difference (3ft) / Distance from the observer to the statue (x)
Okay, now we're ready to maximize these angles! We need to differentiate both equations with respect to "x" and set the derivatives equal to zero to find the maximum values.
Differentiating tan(θ) = 10 / x with respect to "x" gives us:
sec^2(θ) * dθ/dx = -10 / x^2
Differentiating tan(φ) = 3 / x with respect to "x" gives us:
sec^2(φ) * dφ/dx = -3 / x^2
Setting these derivatives equal to zero, we get:
-10 / x^2 = 0 (for dθ/dx)
-3 / x^2 = 0 (for dφ/dx)
Hmm... something doesn't quite add up here. It seems like there's no solution for maximizing these angles. Perhaps we should rethink our approach.
Or maybe we should just tell the observer to stand at a distance where they can comfortably view the statue without worrying about maximizing angles. After all, sometimes the best view is just a matter of personal perspective!
To find the distance at which the angle between the observer's lines of sight to the top and bottom of the statue is a maximum, we need to consider the geometry of the situation.
Let's start by visualizing the scenario. We have a statue standing vertically on a base, and the observer's eye is located 5 feet above the ground. We want to find the distance at which the angle between the lines of sight to the top and bottom of the statue is a maximum.
Now, let's break down the problem. We have a right triangle formed by the observer, the statue, and the base. The height of the statue is 10 feet, and the height of the base is 13 feet.
Let's assume that the observer stands at a distance x from the base. We want to find the value of x that maximizes the angle between the observer's lines of sight to the top and bottom of the statue.
Let's call the angle between the line of sight to the top of the statue and the horizontal line θ1, and the angle between the line of sight to the bottom of the statue and the horizontal line θ2.
To find θ1 and θ2, we can use trigonometry. In a right triangle, the tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height of the statue (10 feet), and the adjacent side is the distance from the observer to the base (x).
So, we have:
tan(θ1) = 10 / x
tan(θ2) = 23 / x
Now, we want to find the distance x that maximizes the difference between θ1 and θ2. In other words, we want to find the maximum value of (θ1 - θ2).
To find the maximum value of (θ1 - θ2), we can take the derivative of (θ1 - θ2) with respect to x and set it equal to zero. Then we solve the resulting equation for x.
Differentiating (θ1 - θ2) with respect to x, we get:
d(θ1 - θ2) / dx = (dθ1 / dx) - (dθ2 / dx)
Using the chain rule, we can find the derivatives of θ1 and θ2 with respect to x:
(dθ1 / dx) = (-10 / x^2) (Since tan'(x) = 1 + tan^2(x))
(dθ2 / dx) = (-23 / x^2) (Since tan'(x) = 1 + tan^2(x))
Now, we set the derivative equal to zero and solve for x:
(-10 / x^2) - (-23 / x^2) = 0
-10 + 23 = 0
13 = 0
The equation -10 + 23 = 0 has no solution. This means that the derivative is never equal to zero, and therefore, there is no maximum value for (θ1 - θ2).
In other words, there is no specific distance from the base at which the angle between the observer's lines of sight to the top and bottom of the statue is maximized.