A 5.5 foot tall woman walks at 6ft/s torward a street light that is 16.5 ft above the ground. what is the rate of change of the length of her shadow when she is 14ft from the street light? At what rate is the tip of her shadow moving?
How do I get the equation for this and how do I work it out?
two similar triangles
First the stationary situation
x/5.5 = (x+14)/16.5
5.5(x+14) = 16.5 x
11 x = 77
x = 7
now start moving
x = length of shadow
y = woman to pole
(x+y)/16.5 = x/5.5
16.5 x = 5.5 x + 5.5 y
11 x = 5.5 y
2 x = y
so
2 dx/dt = dy /dt
if dy/dt = -6 then dx/dt = -3 ft/s
then for tip
d tip/dt = dy/dt + dx/dt
= -3-6 = -9 ft/s
Well, isn't this an illuminating question! Let's shed some light on it, shall we?
First, let's label the important quantities. Let's call the height of the woman h, the distance between her and the street light x, and the length of her shadow s. We're given that h is 5.5 feet, the rate at which she's walking is 6 ft/s, and the height of the street light is 16.5 feet.
Now, we want to find the rate of change of the length of her shadow when she is 14 feet from the street light. Let's call this rate ds/dt (the derivative of s with respect to t, which represents time).
To find the equation relating x, h, and s, we can use similar triangles. We have two similar triangles here: one formed by the woman, her shadow, and the street light, and another formed by the woman, her shadow, and the ground.
The ratio of the height of the woman to the length of her shadow is the same as the ratio of the height of the street light to the distance between the woman and the street light. So we can write:
h/s = (16.5)/(x + s)
Now we can solve this equation for s:
s = (h * (x + s)) / 16.5
To find ds/dt, we differentiate both sides of this equation with respect to t (time):
ds/dt = (dh/dt * (x + s) + h * ds/dt) / 16.5
We're given that dh/dt (the rate at which the woman's height is changing) is 0 because she's not growing or shrinking, just walking. We can substitute dh/dt = 0 into the equation, then plug in the given values to solve for ds/dt when x = 14.
Once we have ds/dt, we can use similar triangles again to find the rate at which the tip of her shadow is moving. The ratio of the length of her shadow to the distance between her and the street light is the same as the ratio of the length of her shadow to the distance between the street light and its tip. So we can write:
s/x = (s + ds/dt)/(x + dt/dt)
Now we can solve this equation for dt/dt:
dt/dt = (x * ds/dt) / (s - ds/dt)
Substitute the given values and calculated ds/dt from the previous part to find dt/dt when x = 14.
I hope this explanation didn't cast too much shade on your understanding!
To solve this problem, we can use similar triangles to relate the height of the woman to the length of her shadow.
Let's denote:
- h: height of the woman (5.5 ft)
- x: distance between the woman and the street light
- y: length of the shadow
- y' = dy/dt: rate of change of the length of the shadow
- x' = dx/dt: rate at which the woman is moving towards the street light
We need to find y' when x = 14 ft and also find x' when x = 14 ft.
Using the similar triangles, we have the following proportion:
h/y = (h + 16.5)/(y + 16.5)
To find the equation relating x and y, we can use the Pythagorean theorem with the right-angled triangle formed by the woman, the tip of her shadow, and the street light:
x^2 + y^2 = (y + 16.5)^2
Now, we can differentiate both equations with respect to time (t) and solve for y' and x'.
Differentiating the first equation, we get:
0 = (h + 16.5)y' - hy' (since the derivative of a constant is zero)
=> hy' = (h + 16.5)y'
=> y' = h/(h + 16.5)
Differentiating the second equation implicitly, we have:
2x(x') + 2y(y') = 2(y + 16.5)(y') (since the derivative of t^2 is 2t and the derivative of a constant is zero)
To find x' when x = 14 ft, substitute the given values into the equation and solve for x':
2(14)(x') + 2(y)(y') = 2(y + 16.5)(y')
28(x') + 2(5.5)(y') = 2(y + 16.5)(y')
To find y' when x = 14 ft, substitute the given values into the equation and solve for y':
2(14)(x') + 2(y)(y') = 2(y + 16.5)(y')
Now you can substitute the given values (h = 5.5 ft, x = 14 ft) into the equations to find y' and x'.
To solve this problem, we can use similar triangles and rates of change. Let's break it down step by step:
Step 1: Set up a diagram
Draw a diagram to visualize the situation. Draw a vertical line to represent the street light, a horizontal line to represent the ground, and a slanted line to represent the woman's shadow. Label the woman's height as 5.5 feet, the distance between the woman and the street light as 14 feet, and the height of the street light as 16.5 feet.
Step 2: Identify similar triangles
Notice that the two triangles formed in the diagram are similar. The larger triangle is formed by the woman's height, the length of her shadow, and the height of the street light. The smaller triangle is formed by the woman's height, the length of her shadow that is changing, and the distance between her and the street light.
Step 3: Write the proportion
Since the triangles are similar, we can write a proportion to relate their corresponding sides. The proportion is:
(Length of shadow)/(Distance to street light) = (Height of street light)/(Woman's height)
Let's plug in the given values:
Length of shadow = x (since it is the unknown we want to find)
Distance to street light = 14 feet
Height of street light = 16.5 feet
Woman's height = 5.5 feet
The proportion becomes:
x/14 = 16.5/5.5
Step 4: Solve for x
To solve for x, cross-multiply the proportion and then divide:
x = (14 * 16.5) / 5.5
x = 42 feet
So, when the woman is 14 feet away from the street light, the length of her shadow is 42 feet.
Step 5: Calculate the rate of change
To find the rate of change of the length of her shadow, we need to differentiate the equation we obtained in Step 4 with respect to time. However, since no time values are given in the problem, we can assume that the rate of change is with respect to time t:
d(x)/dt = 0
This means that the rate of change of the length of her shadow when she is 14 feet from the street light is 0. The length of her shadow remains constant.
Step 6: Find the rate of change of the tip of her shadow
To find the rate of change of the tip of her shadow, we need to consider the rate at which the woman is moving toward the street light.
Given that the woman is walking at 6 ft/s, the rate of change of the tip of her shadow can be found by differentiating the equation x = 42 with respect to time t:
d(x)/dt = d(42)/dt = 0
Therefore, the rate of change of the tip of her shadow is also 0. The tip of her shadow does not move as she walks toward the street light.
In summary:
- The length of her shadow is 42 feet when she is 14 feet from the street light.
- The rate of change of the length of her shadow when she is 14 feet from the street light is 0.
- The rate of change of the tip of her shadow as she walks toward the street light is also 0.