(5 points) A street light is at the top of a 18 ft pole. A 5 ft tall girl walks along a straight path away from the pole with a speed of 6 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 22 ft away from the pole?

2.31 ft/sec

Step-by-step explanation:

In the picture attached, a representation of the problem is shown, where w(t) is the walking distance (as a function of time) and s(t) is the position of the tip of her shadow (as a function of time).

We want to find the derivative of s(t).

From triangles similarity:

s(t)/5 = [w(t) + s(t)]/18

18s(t) = 5w(t) + 5s(t)

13s(t) = 5w(t)

w(t) = 13/5*s(t)

We know that her speed is 6 ft/sec, that is:

d(w(t))/dt = 6 ft/sec

From the previous relationship:

d(w(t))/dt = 13/5*d(s(t))/t

Replacing:

6 = 13/5*d(s(t))/t

d(s(t))/t = 5*6/13

d(s(t))/t ≈ 2.31 ft/sec

x = central angle apc = measure of minor arc ac

the two angles x and 30 add to 180, so x+30 = 180 and x = 180-30 = 150. the measure of minor arc ac is 150 degrees.

what we need to do from here is find the circumference and then multiply it by x/360 since there are 360 degrees in a circle. in this case, x = 150, so x/360 = 150/360 = 15/36 = 5/12. the shortest distance along the curve from a to c is 5/12 of the distance around the entire circle (aka circumference)

c = circumference

c = pi*d

c = pi*24

c = 24pi

now multiply this by 5/12 to get

distance along minor arc ac = (5/12)*(circumference)

distance along minor arc ac = (5/12)*(24pi)

distance along minor arc ac = 10pi

the formula you can use is

l = (x/360)*pi*d

with

l = length of minor arc

x = central angle

d = diameter

which essentially is the steps we took though a bit differently.

the least value out of the data shown is 5, maybe if you post a picture with it i could a little more.

step-by-step explanation: