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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. A. 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? B. 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? C. 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?

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Answer:

Step-by-step explanation: The ladder, the wall and the ground form a right triangle, with hypotenuse 26 ft  (the lenght of the ladder), the height "y" one leg, and the other one "x" distance between from the wall and the base of the ladder.  Then

x²  +  y²   =  (26)²    (1)   Pytaghoras theorem

x²  +  y²   =   676

In that case  x  and  y  are fuction of t (tey are moving) so we derive from t

and get:

2x Dx/dt  + 2y Dy/dt   =  0

x Dx/dt  =  - y  Dy/dt    =  0

We know that   Dx/dt  =  3 f/min, and the question is to find Dy/dt  when x is equal to 10 feet apart from the wall so by subtitution

10* (3)  = - y  Dy/dt      

From equation 1 when   x= 10      y ²  +  100  =   676    y  = √576

y  =  24 ft

Then

30  = - 24 Dy/dt            ⇒   Dy/dt   =  -30/24   = - 5/4 ft/min

First question   Dy/dt  =  -  5/4  ft/min

Second question  no.  rate will increase as x is increasing

Third question:

Yes answer will be always negative. The ladder is falling down, and the movement is in opposite direction to axis y.  In actual terms ladder lenght is fixed 26 and in the right side x is increasing then to keep Pythagoras theorem the other side have to decrease.

A what rate the top of the ladder strike the ground. We have to evaluate the point  x  =  26 then

3*26  =  -  Dy/dt  

A  =  1/2 x*y  

DA/dt   =  1/2 Dx/dt *y

DA/dt   =( 1/2 )* 3 * 24  

DA/dt   = 36 ft  

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