A ladder 16 feet long is leaning against the wall of a tall building. The base of the ladder is moving away from the wall at a rate of 1 foot per second.

A) How fast is the top of the ladder moving down the wall when the base of the ladder is 8 feet from the wall?
B)Find the rate at which the angle theta between the ladder and the wall of the building is changing when the base of the ladder is 8 feet from the wall.
C) A right triangle is formed by the ladder, the wall, and the ground. How fast is the area of this triangle changing when the base of the ladder is 8 feet from the wall?

a. 0.588

b. 0.0722

c. 4.576 sqft/sec

Step-by-step explanation:

Let b and h denote the base and height as indicated in the diagram. By pythagoras theorem, because it is a right angle triangle.

It is given that

Now differentiate (1) with respect to t (time) :

The minus sign indicates that the value of h is actually decreasing. The required answer is 0.588.

b. From the diagram, infer that . When b = 8, then .

Differentiate the above equation w.r.t t

c. The area of the triangle is given by . Differentiating w.r.t t,

Plugging in b = 8, h = 13.856, ,

30 units^2

step-by-step explanation:

area is l × w

the length is 3

and the width is 10

3 × 10= 30 units squared