A street light is at the top of a 26 ft pole. A 5 ft tall girl walks along a straight path away from the pole with a speed of 7 ft/sec. At what rate is the tip of her shadow moving away from the light (ie. away from the top of the pole) when the girl is 23 ft away from the pole?
I know how to figure out the rate at which the shadow is lengthening from the base, which was the first part of this question, yet have no idea how to do the above part. Please help :( just a start would even be helpful i have no idea how to come at the problem.
To solve this problem, you can use similar triangles. Let's consider two triangles: the triangle formed by the pole, the girl's height, and the length of her shadow, and the triangle formed by the pole, the length of the shadow, and the distance between the girl and the pole.
Let's call the length of the girl's shadow "x" and the distance between the girl and the pole "d". We are interested in finding the rate at which the tip of the shadow is moving away from the light, which means we need to find dx/dt.
From the similar triangles we mentioned earlier, we can write the following proportion:
(height of the pole + height of the girl) / height of the pole = (length of the shadow + height of the girl) / length of the shadow
This can be written as:
(26 + 5) / 26 = (x + 5) / x
Simplifying this proportion, we get:
(31/26) = (x + 5) / x
Cross multiply to get:
31x = 26x + 130
Subtract 26x from both sides:
5x = 130
Divide both sides by 5:
x = 26
Now, we know that when the girl is 23 ft away from the pole, the length of her shadow (x) is 26 ft. To find the rate at which the tip of her shadow is moving away from the light (dx/dt), we can differentiate both sides of the proportion we derived earlier with respect to time (t):
31/26 = (x + 5) / x
Differentiate both sides:
0 = (x + 5) * dx/dt - x * (dx/dt)
Simplifying:
0 = 5 * dx/dt
dx/dt = 0
Therefore, when the girl is 23 ft away from the pole, the tip of her shadow is not moving away from the light, it is stationary.
Please note that in this problem, the rate at which the tip of the shadow is moving away from the light is dependent on the distance between the girl and the pole (d), and when the girl is 23 ft away, the rate is zero.
ok, draw the figure.
label the point of her shadow as x, and her position as y. Now make a triangle from the pole to the shadow. Notice the triangle formed by the girl and the shadow tip is SIMILAR to the large triangle.
So similar sides..
26:x::5:(x-y) (x is the full length, which includes y)
x/26=(x-y)/5
take the derivative with respect to time of the xpression..
dx/dt /26=(dx/dt-dy/dt)/5
check this, I am doing the algebra in my head..
dx/dt (1-26/5)= -26/5 dy/dt
you are given dy/dt as 7ft/sec
solve for dx/dt