Question:

A ladder $5\,m$ long is leaning against a wall. The bottom of the ladder is pulled along the ground away from the wall, at the rate of $2m/sec$. The speed at which its height on the wall decreases when the foot of the ladder is $4\, m$ away from the wall is

Updated On: May 12, 2024
  • $\ce{ \frac{3}{8} m /sec }$
  • $\ce{ \frac{8}{3} m /sec }$
  • $\ce{ \frac{5}{3} m /sec }$
  • $\ce{ \frac{2}{3} m /sec }$
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The Correct Option is B

Solution and Explanation

Let $AB$ be the ladder of length 5 m.
We are given, $\frac{dx}{dt} = 2 \, {m / sec}$
In $\Delta ABC,$
$AB^2 = AC^2 + BC^2$
$\Rightarrow\:\: (5)^2 = x^2 + h^2$ ....(i)
Differentiating (i) w.r.t. t,
we get
$0 = 2x \frac{dx}{dt} + 2h \frac{dh}{dt}$
$\Rightarrow \:\:\frac{dh}{dt} = \frac{-x}{h} \frac{dx}{dt}$ .....(ii)
Now, from (i), when $x = 4$
$h^2 = 25 - 16 \:\:\: \Rightarrow \, h^2 = 9 \:\:\: \Rightarrow \:\: h = 3 \, m$
From (ii)
$\left[ \frac{dh}{dt} \right] = \frac{-4}{3} \times 2 = \frac{-8}{3}$
$\because$ The negative sign shows the height decreases and decreasing rate is $\frac{8}{3} { m/sec}$

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Concepts Used:

Application of Derivatives

Various Applications of Derivatives-

Rate of Change of Quantities:

If some other quantity ‘y’ causes some change in a quantity of surely ‘x’, in view of the fact that an equation of the form y = f(x) gets consistently pleased, i.e, ‘y’ is a function of ‘x’ then the rate of change of ‘y’ related to ‘x’ is to be given by 

\(\frac{\triangle y}{\triangle x}=\frac{y_2-y_1}{x_2-x_1}\)

This is also known to be as the Average Rate of Change.

Increasing and Decreasing Function:

Consider y = f(x) be a differentiable function (whose derivative exists at all points in the domain) in an interval x = (a,b).

  • If for any two points x1 and x2 in the interval x such a manner that x1 < x2, there holds an inequality f(x1) ≤ f(x2); then the function f(x) is known as increasing in this interval.
  • Likewise, if for any two points x1 and x2 in the interval x such a manner that x1 < x2, there holds an inequality f(x1) ≥ f(x2); then the function f(x) is known as decreasing in this interval.
  • The functions are commonly known as strictly increasing or decreasing functions, given the inequalities are strict: f(x1) < f(x2) for strictly increasing and f(x1) > f(x2) for strictly decreasing.

Read More: Application of Derivatives