A surveyor is walking at the rate of 4 ft/sec along the diameter of a circular courtyard. A light at one end of a diameter perpendicular to her path casts a shadow on the circular wall. How fast is the shadow moving along the wall when her distance to the center of the courtyard is r/2?
Please help.
Let's label a few things in our diagram.
O = center of circle
P,Q are the ends of the perpendicular diameter
T is where the surveyor is standing
S is the point on the circle where the shadow falls
θ = angle QPS
a is the radius of the circle
x is the distance TO
r is the distance PS
z is the distance PT
Now, angle dθ subtends a small arc of length ds
we want to find ds/dt: how fast s is moving along the circle
r = 2a cosθ
tanθ = (a-4t)/a, so
sec^2θ dθ/dt = -4/a
sec^2θ = 1+tan^2θ = 1 + ((a-4t)/a)^2 = 2/a^2 (8t^2-4at+a^2)
and cos^2θ = a^2/(2(8t^2-4at+a^2))
ds = r dθ
ds/dt = -8 cos^3θ = -8(a^2/(2(8t^2-4at+a^2)))^(3/2)
= -2√2 a^3 / (8t^2-4at+a^2)^(3/2)
Now, what is t when x = a/2?
t = a/8
so now we know that at that time,
ds/dt = -2√2 a^3 / (5a^2/8)^(3/2) = -64 / 5√5 ft/s
the negative sign indicates that the shadow's distance from PQ is decreasing
Better double-check my math!
Why did the surveyor bring a light to a circular courtyard? Because they wanted to shed some light on the situation! 🌞
But don't worry, I'm here to help you shed light on the problem too. Let's get started!
First, let's draw a diagram to visualize the situation. We have a circular courtyard, with the surveyor walking along its diameter. The light is at one end of a diameter perpendicular to her path, and it casts a shadow on the circular wall.
Now, let's use some math to solve this problem. We'll need to use the concept of related rates, which involves finding the rate of change of one variable with respect to another variable.
Let's call the radius of the circular courtyard "r". We are given that the surveyor is walking at a rate of 4 ft/sec along the diameter. So, the rate at which the surveyor is moving along the diameter is 4 ft/sec.
To find how fast the shadow is moving along the circular wall, we need to find the rate of change of the shadow's position with respect to time. Let's call this rate "ds/dt".
We are given that the surveyor's distance to the center of the courtyard is r/2. Let's call this distance "x".
Now, we can set up a relationship between x and s (the position of the shadow on the circular wall). Since the surveyor is walking along a diameter, we know that x and s are related by the Pythagorean theorem: (x^2) + (s^2) = (r^2).
Differentiating both sides of this equation with respect to time t, we get:
2x(dx/dt) + 2s(ds/dt) = 0
We want to find ds/dt, so let's rearrange the equation:
ds/dt = -x(dx/dt) / s
Now, we need to find dx/dt (the rate at which x is changing with respect to time). We know that x = r/2, and dx/dt is the rate at which the surveyor is moving along the diameter, which is 4 ft/sec.
Substituting these values into the equation, we find:
ds/dt = - (r/2)(4 ft/sec) / s
Simplifying:
ds/dt = -2r / s ft/sec
So, the shadow is moving along the wall at a rate of -2r/s ft/sec, where r is the radius of the circular courtyard, and s is the position of the shadow on the circular wall.
I hope that sheds some light on the problem for you!
To solve this problem, we can use related rates. Let's start by assigning some variables:
- Let x represent the distance between the surveyor and the center of the courtyard.
- Let s represent the position of the shadow on the circular wall.
- Let t represent time.
We are given that the surveyor is walking at a rate of 4 ft/sec. Hence, we can say that dx/dt = 4 ft/sec.
So, we want to find ds/dt, the rate at which the shadow is moving along the circular wall when the surveyor is at a distance of r/2 from the center.
Now, let's set up our equation. We know that the surveyor is walking along the diameter of the courtyard, and the position of the shadow is related to the position of the surveyor:
s^2 + (x - r/2)^2 = r^2
To differentiate both sides of the equation with respect to time (t), we get:
2s * ds/dt + 2(x - r/2) * dx/dt = 0
Substituting the values we know, we get:
2s * ds/dt + 2(x - r/2) * 4 = 0
Now, we need to solve for ds/dt. Rearranging the equation, we get:
ds/dt = -4(x - r/2) / (2s)
Since we want to find the rate at which the shadow is moving along the wall when the surveyor is at a distance of r/2 from the center, we can substitute x = r/2:
ds/dt = -4(r/2 - r/2) / (2s)
Simplifying further, we get:
ds/dt = 0 / (2s) = 0 ft/sec
Therefore, when the surveyor is at a distance of r/2 from the center of the courtyard, the shadow will not move along the wall.
To find the rate at which the shadow is moving along the wall, we need to use a concept called related rates.
Let's denote the distance from the surveyor to the center of the courtyard as distance "d" and the distance from the light to the center of the courtyard as "r".
From the problem, we know that the surveyor is moving at a rate of 4 ft/sec along the diameter of the courtyard. This means that the rate of change of distance "d" with respect to time (let's call it dd/dt) is 4 ft/sec.
We want to find the rate at which the shadow is moving along the wall when the surveyor's distance to the center of the courtyard is r/2. Let's call this rate ds/dt.
To relate these rates, we can use the fact that the surveyor, the light, and the shadow form similar triangles. In other words, the ratio of the surveyor's distance to the light's distance is equal to the ratio of the shadow's distance to the light's distance.
Let's denote the distance from the light to the shadow as "x". The similar triangles give us the equation:
x/r = d/(r/2)
To solve for x in terms of d, we can rearrange the equation:
x = (2d)/r
Now, let's differentiate both sides of the equation with respect to time (t):
dx/dt = (2(dd/dt))/r
Since we know dd/dt is 4 ft/sec and want to find ds/dt, we can substitute the known values into the equation:
dx/dt = (2(4 ft/sec))/r
Simplifying the equation further, we get:
dx/dt = 8/r ft/sec
So, the rate at which the shadow is moving along the wall when the surveyor's distance to the center of the courtyard is r/2 is given by ds/dt = 8/r ft/sec.
Therefore, the shadow is moving at a rate of 8/r ft/sec along the wall.