Class 12 Mathematics Application of Derivatives: NCERT Solutions for Question 10
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 2 cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall?
Let y m be the height of the wall at which the ladder touches. Also, let the foot of the ladder be x maway from the wall.
Then, by Pythagoras theorem, we have:
x2 + y2 = 25 [Length of the ladder = 5 m]
\begin{align}\Rightarrow y = \sqrt{25 - x^2}\end{align}
Then, the rate of change of height (y) with respect to time (t) is given by,
\begin{align}\frac{dy}{dx} = \frac{-x}{\sqrt{25 - x^2}}.\frac{dx}{dt}\end{align}
It is given that
\begin{align}\frac{dx}{dt}= 2 \; cm/s.\end{align}
\begin{align}\therefore\frac{dy}{dt} = \frac{-2x}{\sqrt{25 - x^2}}\end{align}
Now, when x = 4 m, we have:
\begin{align}\therefore\frac{dy}{dt} = \frac{-2 \times 4}{\sqrt{25 - 4^2}}=-\frac{8}{3}\end{align}
Hence, the height of the ladder on the wall is decreasing at the rate of
\begin{align}\frac{8}{3}\end{align}
Popular Questions of Class 12 Mathematics
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(i) Symmetric but neither reflexive nor transitive.
(ii) Transitive but neither reflexive nor symmetric.
(iii) Reflexive and symmetric but not transitive.
(iv) Reflexive and transitive but not symmetric.
(v) Symmetric and transitive but not reflexive. - Q:- Determine whether each of the following relations are reflexive, symmetric and transitive:
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(ii) Relation R in the set N of natural numbers defined as
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(iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as
R = {(x, y): y is divisible by x}
(iv) Relation R in the set Z of all integers defined as
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(v) Relation R in the set A of human beings in a town at a particular time given by
(a) R = {(x, y): x and y work at the same place}
(b) R = {(x, y): x and y live in the same locality}
(c) R = {(x, y): x is exactly 7 cm taller than y}
(d) R = {(x, y): x is wife of y}
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- Q:- Show that the relation R in R defined as R = {(a, b): a ≤ b}, is reflexive and transitive but not symmetric.
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3 Comment(s) on this Question
Thank u so much
need to change the units?
how do u got dy/dx
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