Height of the ladder on the wall in decreasing at the rate of ≈ 6m/s
Finding a function's derivative, or rate of change, is the process of differentiation in mathematics. The practical approach of differentiation can be performed utilizing only algebraic operations, three fundamental derivatives, four principles of operation, and an understanding of how to manipulate functions, in contrast to the theory's abstract character.
let ym be the height of wall at which ladder touches
Also let the foot of ladder be xm away from the wall
then by the pythagoras theorem we have ,
[tex]x^{2} +y^{2} =[/tex] 64 [ length of ladder = 8 m ]
⇒ y = [tex]\sqrt{64 - x^{2} }[/tex]
then , the rate of change of height (y) with respect to time (t) is given by
[tex]\frac{dy}{dt}=\frac{-x}{\sqrt{64 - x^{2} }} . \frac{dx}{dt}[/tex]
it is given that [tex]\frac{dx}{dt}[/tex] = 5 m/s
∴ [tex]\frac{dy}{dt}=[/tex] [tex]\frac{-5x}{\sqrt{64 - x^{2} }}[/tex]
now when x = 6m , we have :
[tex]\frac{dy}{dt} = \frac{-5 * 6}{\sqrt{64 - 6^{2} }}[/tex]
= -30 / 5 .3
= -5.660377358490566
therefore height of the ladder on the wall in decreasing at the rate of ≈ 6m/s
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