Graphical understanding of derivatives, Mathematics

Graphical Understanding of Derivatives:

A ladder 26 feet long is leaning against a wall. The ladder begins to move such that the bottom end moves away from the wall at a constant velocity of 2 feet by per second.   What is the downward velocity of the top end of the ladder when the bottom end is 10 feet from the wall?


Begin with the Pythagorean Theorem for a right triangle:     a2 = c2 - b2

Obtain the derivative of both sides of this equation with respect to time t.  The c, representing the length of the ladder is a constant.

2a(da/dt) = -2b(db/dt)

a(da/dt) = -b (db/dt)

But, db/dt is the velocity at that the bottom end of the ladder is moving  away  from  the  wall,  equal  to  2  ft/s,  and  da/dt  is  the downward  velocity  of the top end of the ladder  along  the wall, that is the quantity  to be determined.  Set b equal to 10 feet, substitute the known values into the equation, and solve for a.

a2 = c2 - b2


2442_Graphical Understanding of Derivatives.png

a= 24 ft

a(da/dt) = -b (db/dt)

(da/dt) = -b/a (db/dt)

(da/dt) = -10 ft/24 ft(2 ft/s)

(da/dt) = -0.833 ft/s

Therefore, when the bottom of the ladder is 10 feet from the wall and moving at 2ft/sec., the top of the ladder is moving downward at 0.833 ft/s. (The negative sign denotes the downward direction.)

Posted Date: 2/11/2013 1:04:24 AM | Location : United States

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