(A) Let the position of the first cyclist be $A$ and the second cyclist be $B$ at time $t$. Let the distance between them be $x$.Given $OA = 4t$ and $

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(A) Let the position of the first cyclist be $A$ and the second cyclist be $B$ at time $t$. Let the distance between them be $x$.Given $OA = 4t$ and $OB = 3t$.Using the Law of Cosines in $\triangle OAB$:$x^2 = (OA)^2 + (OB)^2 - 2(OA)(OB) \cos(120^{\circ})$$x^2 = (4t)^2 + (3t)^2 - 2(4t)(3t) \left(-\frac{1}{2}\right)$$x^2 = 16t^2 + 9t^2 + 12t^2$$x^2 = 37t^2$$x = \sqrt{37}t$The rate at which they are separating is $\frac{dx}{dt}$.$\frac{dx}{dt} = \frac{d}{dt}(\sqrt{37}t) = \sqrt{37} \text{ km/h}$.

Class 12 Mathematics Applications of Derivatives Rate of Change of Quantities

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Two cyclists are moving from a junction of two roads inclined at an angle of $120^{\circ}$ to each other,with velocities of $4 \text{ km/h}$ and $3 \text{ km/h}$ respectively. The rate at which they are separating from each other after $1 \text{ hour}$ is ..... $\text{km/h}$.

A
$\sqrt{37}$
B
$37$
C
$13$
D
$\sqrt{13}$

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Applications of Derivatives

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Rate of Change of Quantities

The side of a square piece is increasing at a rate of $4 \text{ cm/min}$. Find the rate at which the area is increasing when the side is $8 \text{ cm}$ long.

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The rate of change of the volume of a sphere with respect to its radius when $r = 7 \ cm$ is . . . . . . .

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$A$ particle moving from a fixed point on a straight line travels a distance $S$ metres in $t$ seconds. If $S = t^3 - t^2 - t + 3$,then the distance (in metres) travelled by the particle when it comes to rest is:

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$A$ spherical ball of salt is dissolving in water in such a manner that the rate of decrease of the volume at any instant is proportional to its surface area. Prove that the radius is decreasing at a constant rate.

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Two cyclists are moving from a junction of two roads inclined at an angle of $120^{\circ}$ to each other,with velocities of $4 \text{ km/h}$ and $3 \text{ km/h}$ respectively. The rate at which they are separating from each other after $1 \text{ hour}$ is ..... $\text{km/h}$.

Similar Questions

The side of a square piece is increasing at a rate of $4 \text{ cm/min}$. Find the rate at which the area is increasing when the side is $8 \text{ cm}$ long.

$A$ stone is dropped into a quiet lake and waves move in circles at the speed of $5 \text{ cm/s}$. At the instant when the radius of the circular wave is $8 \text{ cm}$,how fast is the enclosed area increasing?

The rate of change of the volume of a sphere with respect to its radius when $r = 7 \ cm$ is . . . . . . .

$A$ particle moving from a fixed point on a straight line travels a distance $S$ metres in $t$ seconds. If $S = t^3 - t^2 - t + 3$,then the distance (in metres) travelled by the particle when it comes to rest is:

$A$ spherical ball of salt is dissolving in water in such a manner that the rate of decrease of the volume at any instant is proportional to its surface area. Prove that the radius is decreasing at a constant rate.

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