A man 5.5 ft tall walks away from a lamp post 10 ft high at the rate of 8 ft/s.

(a) How fast does his shadow lengthen?
(b) How fast does the tip of the shadow move?
Can you please tell me what method/s you used to come up with a solution?

I wasn't sure about the sketch I made so I really don't know where to start....

http://www.jiskha.com/display.cgi?id=1439263714

a triangle has base of 1.6 inches and an altitude of 8 inches. find the dimensions of the largest rectangle that can be inscribed in the triangle if the base of the rectangle coincides with the base of the triangle.

To solve this problem, we can use similar triangles and related rates. Here's a step-by-step explanation:

Step 1: Draw a diagram
Start by drawing a diagram to represent the situation. Draw a vertical line to represent the lamp post, and label it with its height (10 ft). Then draw another line to represent the man's height (5.5 ft), and a line extending from the man's feet to the tip of the shadow.

Step 2: Identify the variables
Let's use the following variables:
- h: height of the lamp post (given as 10 ft)
- s: length of the shadow
- x: distance between the man and the lamp post
- y: distance between the man's feet and the tip of the shadow

We are given that the man is walking away from the lamp post at a rate of 8 ft/s.

Step 3: Set up the similar triangles
The triangle formed by the man, his shadow, and the lamp post is similar to the triangle formed by the lamp post, its shadow, and the ground. This means that we can set up a proportion to relate the lengths of the sides of the triangles:

h / s = (h + y) / x

Step 4: Find an equation relating the variables
To find an equation that relates the variables, we need to eliminate y. We can use the Pythagorean theorem to relate the lengths of the sides of the triangle formed by the man, his shadow, and the ground:

(s + y)^2 = x^2

Simplifying and substituting the value for y from the similar triangles equation, we get:

s^2 = x^2 - (h/x * s)^2

Step 5: Differentiate the equation with respect to time
To find how fast the shadow lengthens, we need to differentiate the equation with respect to time. Let's use the chain rule:

d/dt(s^2) = d/dt(x^2) - d/dt((h/x * s)^2)

2s * ds/dt = 2x * dx/dt - (h^2/x^3) * s^2 * ds/dt

Step 6: Solve for ds/dt (rate of shadow lengthening)
Rearrange the equation to isolate ds/dt:

2s * ds/dt = 2x * dx/dt - (h^2/x^3) * s^2 * ds/dt

(2s + (h^2/x^3) * s^2) * ds/dt = 2x * dx/dt

ds/dt = (2x * dx/dt) / (2s + (h^2/x^3) * s^2)

Substitute the given values: x = 8 ft/s, s = 10 ft, and h = 5.5 ft.

ds/dt = (2 * 8 ft/s * dx/dt) / (2 * 10 ft + (5.5^2 / 8^3) * 10 ft^2)

Simplifying the expression, we find the rate at which the shadow lengthens.

To find the speed at which the tip of the shadow moves, we need to find dy/dt (rate of change of y). This can be obtained by differentiating the similar triangles equation d/dt(h/s) = d/dt((h + y)/x) with respect to time and solving for dy/dt.