Question

A 26 foot ladder is sliding down a vertical wall. The base of the ladder is sliding away from the wall at a rate of 3 feet per minute. At what rate is the ladder sliding down the wall when the base of the ladder is 10 feet from the wall? Would you expect the same rate when the ladder is any distance from the wall? Would you expect your answer to be negative when the base is any distance from the wall? Is there a physical reason why the rate is negative? At what rate the top of the ladder strike the ground? If you consider the triangle formed by the ladder, the wall, and the ground, at what rate is the area of the triangle changing when the base of the ladder is 10 feet from the wall? What does the sign of your answer (whether it is positive or negative) say about the area of the triangle at that moment? Would you expect the rate at which the area is changing to have the same sign (positive or negative) when the base of the ladder is any distance from the wall? At what rate is the angle between the ladder and the ground changing when the base of the ladder is 10 feet from the wall? Would expect the angle to be decreasing at a more rapid rate when the ladder is closer to a vertical position or a horizontal position? All explanations must be neatly hand written or typed. The supporting math work must be neat and suitable for scanning. The final document should be saved and submitted as a single .pdf file.

Answer 1

A.

The rate of sliding down of the top of the ladder depends on both the value of x and y, ie the current position of the ladders end from the bottom of the wall. Therefore the value of rate will be changing as the ladder slides down

The rate of sliding of the top of the ladder is negative at any distance from the bottom because of the fact that the distance between the top of the ladder and the bottom of the wall is decreasing as the ladders is sliding down. Therefore dy/dt will always be negative.

Because dy/dt depends on 1/y, the first derivative of the position of the top is undefined at the time when ladder hits the ground. As y approaches 0, the velocity of the top grows without bound, so in theory the top is going infinitely fast when it hits the ground.

B.

The rate of change of area is positive at the moment. Thus it can be concluded that the area is increasing with respect to time.

The rate of area change depends on x,y,dx/dt and dy/dt. Thus the rate of area change will vary at each position of the ladder.Since dx/dt is fixed so we can say that dA/dt will depend on only x, y and dy/dy. dy/dt initially starts with 0 and is ever increasing(in absolute value; its sign remains negative). Thus initially the value of dA/dt will be positive as y.dx/dt will be greater than x.dy/dt, but after a point of time x.dy/dt will be larger than y.dx/dt, at which point dA/dt will become negative and area will start decreasing.

C.

d?/dt = 1/(26 * cos?) dy/dt

rate is directly proportional to dy/dt

so as dy/dt increases, so does d?/dt.

And dy/dt increases as the ladders comes down. Therefore rate of change of angle will be greater when the ladder is at horizontal position.