a woman is walking at the rate of 5 feet per second 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 the distance from the woman to the center of the courtyard is 1/2r , where r feet is the radius of the courtyard.
where is the wall? at the circumference of the courtyard?
To find how fast the shadow is moving along the wall, we need to use the concept of related rates. We'll break down the problem into smaller parts and use derivative rules to calculate the rate of change.
Let's denote the radius of the circular courtyard as "r" (in feet). The distance from the woman to the center of the courtyard is given as 1/2r. This means the line segment from the center of the courtyard to the woman's position forms a right-angled triangle with the radius as the hypotenuse.
We'll consider the following variables:
- x: The distance between the woman and the center of the courtyard.
- y: The length of the shadow cast on the circular wall.
- θ: The angle formed between the woman, the center of the courtyard, and the point where the shadow touches the wall.
To solve this problem, we need to find an equation relating x, y, and θ. We can use the properties of right-angled triangles to form this equation.
From the given information, we know that the woman is walking at a rate of 5 feet per second along the diameter of the courtyard. This means the rate of change of x (dx/dt) is 5 ft/s.
We want to find the rate at which the shadow is moving along the wall, which is the rate of change of y (dy/dt). To do this, we need to find an equation relating x, y, and θ in terms of their derivatives.
Considering the right-angled triangle formed:
- The length of the side opposite angle θ is y.
- The length of the side adjacent to angle θ is x.
- The hypotenuse of the triangle is the radius r of the courtyard.
Using the trigonometric function tangent, we can write:
tan(θ) = (opposite side) / (adjacent side)
= y / x
Taking the derivative of both sides with respect to time (t), we get:
sec²(θ) * dθ/dt = (dy/dt * x - y * dx/dt) / x²
Since we want to find dy/dt, we rearrange the equation as follows:
dy/dt = (sec²(θ) * x * dθ/dt + y * dx/dt) / x²
Now, let's substitute the known values. When the distance from the woman to the center of the courtyard is 1/2r, we have:
x = 1/2r
y = r (since the shadow touches the wall at the same height as the center of the courtyard)
Furthermore, we can find θ using trigonometric relations in a right-angled triangle:
tan(θ) = (opposite side) / (adjacent side)
= y / x
= r / (1/2r)
= 2
Since the tangent of θ is 2, we can solve for θ using the arctan function:
θ = arctan(2)
Now we have all the necessary information to calculate the rate at which the shadow is moving along the wall, dy/dt. We substitute the values into the derived equation:
dy/dt = (sec²(arctan(2)) * (1/2r) * dθ/dt + r * (5 ft/s)) / (1/2r)²
The secondary task is to find dθ/dt. Since the woman is walking along the diameter, the change in angle θ is related to dx/dt, which is known. The circumference of a circle is given by 2πr, so for every complete revolution of the woman, the angle θ changes by 2π radians.
Therefore, dθ/dt = (change in θ) / (change in time) = (2π) / (time for one revolution)
Finally, we can substitute the derived values of dθ/dt and solve for dy/dt.