A 5ft tall person is walking toward a light 20ft off the ground at 8 ft/sec. What is the rate of change of the length of the persons shadow when they are 15ft away from the light? What is the speed of the tip of the shadow moving?
using similar triangles, if the person is x feet away, and the shadow has length s,
s/5 = (x+s)/20
4s = x+s
x = 3s
so, dx/dt = 3 ds/dt
that means ds/dt = -8/3 ft/s
So, the shadow is shrinking as the man is walking, so the tip is moving at the rate of -(8 + 8/3) = -32/3 ft/s
what does the minus sign mean?
To find the rate of change of the length of the person's shadow, we can use similar triangles.
Let's denote the person's height as "h", the distance between the person and the light as "x", and the length of the shadow as "s". From the problem, we have h = 5 ft and dx/dt = 8 ft/sec (the rate of change of x).
Using similar triangles, we can establish the following relationship:
h / s = (h + x) / x
Substituting the known values, we have:
5 / s = (5 + x) / x
Cross-multiplying, we get:
5x = s(5 + x)
To solve for s, we can rearrange the equation:
s = (5x) / (5 + x)
Now, we need to find ds/dt, the rate of change of s with respect to time. We can differentiate both sides of the equation with respect to t:
ds/dt = d((5x) / (5 + x)) / dt
Using the quotient rule, we have:
ds/dt = ((5 + x)(5 * dx/dt) - (5x)(1 * dx/dt)) / (5 + x)^2
Simplifying the equation, we get:
ds/dt = (40x) / (5 + x)^2
Now, we can substitute x = 15 ft into the equation to find the rate of change of the length of the person's shadow when they are 15 ft away from the light:
ds/dt = (40 * 15) / (5 + 15)^2
= 600 / 20^2
= 600 / 400
= 1.5 ft/sec
Therefore, the rate of change of the length of the person's shadow when they are 15 ft away from the light is 1.5 ft/sec.
To find the speed of the tip of the shadow moving, we need to determine how fast the shadow is growing in length.
Since the person's shadow is the sum of the person's height and the distance from the person to the light (s = h + x), we can differentiate both sides of the equation with respect to t:
ds/dt = dh/dt + dx/dt
From the information given in the problem, we know that dx/dt = 8 ft/sec. The rate of change of the person's height, dh/dt, is not given, so we cannot determine the exact speed of the tip of the shadow.
However, we can say that the speed of the tip of the shadow is the sum of the person's walking speed and the rate of change of the length of the shadow. So, the speed of the tip of the shadow is 8 ft/sec + 1.5 ft/sec = 9.5 ft/sec (approximately).
To find the rate of change of the length of the person's shadow, we can use similar triangles. Let's call the person's height h, the distance from the person to the light x, and the length of the shadow y.
The ratio of the person's height to their shadow length is constant, and is equal to the ratio of the distance from the person to the light to the length of the shadow. This can be written as:
h/y = x/y
We can solve this equation for y to get:
y = (h*x)/y
Next, we can differentiate both sides of the equation with respect to time (t) to find the rates of change:
d(y)/dt = d([(h*x)/y])/dt
To find d(y)/dt, we need to find the rates of change of h, x, and y with respect to time.
Since the person's height is constant, the rate of change of h with respect to t is zero: dh/dt = 0.
The rate of change of x with respect to t is given as 8 ft/sec: dx/dt = 8 ft/sec.
To find d(y)/dt, we can differentiate the equation for y with respect to t:
d(y)/dt = [(h*(d(x)/dt)*y - x*(d(y)/dt))/(y^2)]
Now we have all the values we need except for d(y)/dt. Let's solve the equation for d(y)/dt:
d(y)/dt = [(h*(dx/dt)*y - x*(d(y)/dt))/(y^2)]
To find d(y)/dt, we need to know the values of h, x, and y at the given point when the person is 15 ft away from the light. Let's substitute these values into the equation:
h = 5 ft
x = 15 ft
y = (h*x)/y = (5*15)/20 = 3.75 ft
Now we can calculate d(y)/dt:
d(y)/dt = [(5*(8)*3.75 - 15*(d(y)/dt))/(3.75^2)]
To solve for d(y)/dt, we need to isolate the variable d(y)/dt. Let's rearrange the equation:
d(y)/dt = [40*3.75 - 15*(d(y)/dt)]/(3.75^2)
Multiplying both sides by (3.75^2), we get:
d(y)/dt * 3.75^2 = 150 - 15*(d(y)/dt)
Expanding the equation, we have:
(3.75^2)*d(y)/dt + 15*(d(y)/dt) = 150
Now we can solve for d(y)/dt:
(14.0625)*d(y)/dt + 15*(d(y)/dt) = 150
(29.0625)*d(y)/dt = 150
d(y)/dt = 150 / 29.0625
Using a calculator, we can find that d(y)/dt is approximately 5.15 ft/sec.
To find the speed of the tip of the shadow moving, we need to determine the rate of change of the distance from the person to the light (x) with respect to time (t), which we already know is 8 ft/sec.
Therefore, the speed of the tip of the shadow moving is 8 ft/sec.