A spotlight on the ground is shining on a wall 24m away. If a woman 2m tall walks from the spotlight toward the building at a speed of 1.2m/s, how fast is the length of her shadow on the building decreasing when she is 4m from the building?
never mind i got it thanks
To find the rate at which the length of her shadow on the building is decreasing, we will use related rates.
Let's denote:
x = distance of the woman from the spotlight (in meters)
y = length of the shadow on the building (in meters)
According to the question, the woman is walking towards the building at a speed of 1.2 m/s. This means that the rate of change of x with respect to time (dx/dt) is -1.2 m/s since the distance is decreasing.
To find the rate at which the length of her shadow on the building is changing (dy/dt), we need to find an equation relating x, y, and their rates of change.
We can start by using similar triangles. The height of the woman and her shadow creates two similar right triangles. The ratios of corresponding sides are equal:
2/24 = (2-y)/y
Simplifying the equation, we get:
2y = 48 - 24y
26y = 48
y = 48/26
y ≈ 1.85 m
Now, we have the equation relating x and y:
2/24 = (2-y)/y
To find dy/dt, we need to differentiate both sides of the equation with respect to time (t):
d(2/24)/dt = d((2-y)/y)/dt
Simplifying the equation, we get:
0 = (-1/y^2) * dy/dt * 2 + (2-y)/y^2 * dy/dt
0 = (-2/y^2) * dy/dt + (2-y)/y^2 * dy/dt
Rearranging the terms, we get:
(2/y^2) * dy/dt = (-2+y)/y^2 * dy/dt
Now, we can substitute the given values:
(2/(1.85^2)) * dy/dt = (-2+1.85)/1.85^2 * dy/dt
Simplifying the equation, we get:
(2/3.4225) * dy/dt ≈ (-0.15)/3.4225 * dy/dt
dy/dt = (-0.15/3.4225) * (3.4225/2)
dy/dt ≈ -0.083 m/s
Therefore, the length of her shadow on the building is decreasing at a rate of approximately 0.083 m/s when she is 4 m from the building.
To find the rate at which the length of the shadow is decreasing, we can use related rates.
Let's start by visualizing the problem:
The spotlight represents a point source of light on the ground. The wall represents the building where a shadow is being cast. The woman's height can be represented by a vertical line extending from the ground to her head. As she moves towards the building, her shadow on the wall changes in length.
We are given the following information:
- Distance from the spotlight to the wall: 24m
- Woman's height: 2m
- Woman's speed towards the building: 1.2m/s
We need to find the rate at which the length of her shadow is decreasing when she is 4m away from the building.
Let's set up a diagram to represent the situation:
Wall
------------------------------------
| A | distance | B |
| (spotlight)|----------24m-------------| (woman) |
------------------------------------
Ground
Let's assign some variables:
- Let x represent the distance between the woman and point B (where the shadow touches the wall).
- Let y represent the length of the shadow.
To find the rate of change in the length of the shadow with respect to time, we need to differentiate the equation y = f(x, t), where t is time.
From the similar triangles formed by the woman, her shadow, and the distance between her and the wall, we can derive the equation:
y/x = (2m)/(24m)
Simplifying this equation, we get:
y = (x/12)m
Now, we have an equation relating y and x. To find the rate at which y is changing with respect to time (dy/dt), we need to differentiate this equation with respect to t:
dy/dt = d/dt ((x/12)m)
To differentiate, we will use the chain rule:
dy/dt = (1/12)m * dx/dt
Now, we can substitute the given values into the equation to find dy/dt when x = 4m:
dy/dt = (1/12)m * dx/dt
= (1/12)m * (1.2m/s)
Substituting the values, we get:
dy/dt = (1/12) * 1.2
= 0.1 m/s
Therefore, the length of her shadow on the building is decreasing at a rate of 0.1 m/s when she is 4m away from the building.