Answer to Question #196807 in Calculus for Hamza baig

Question

Q : 1 The distance x meters moved by a partide int seconds is given by x = t + 3t + 4 . Find the velocity and accelerating after 3 seconds .

Q : 2 The radius of a circle is increasing uniformly at the rate of 3 cm per second . Find the rate at which its area is increasing when radius is 10cm .

Q : 3 If displacement is s = sin2t , find , its acceleration .

Answer 1

Q : 1 ) x = t + 3t + 4 ........................Equation(1)

We know that velocity and acceleration are given by:

v = $\dfrac{dx}{dt}$

a = $\dfrac{d^2x}{dt^2}$

Differentiating equation (1) with respect to t, we have

v = $\dfrac{dx}{dt}$ = 4 ............................................Equation (2)

Differentiating Equation (2) with respect to t, we have

a = $\dfrac{d^2x}{dt^2}$ = 0

Hence, we see that the velocity at any time is constant and is equal to 4 m/s and acceleration is equal to 0.

Q : 2) $\dfrac{dr}{dt}$ = 3 cm/s

Now, area of a circle is given by

A = $\pi$ $*$ r² ..................Equation(3)

Differentiating the above equation with respect to t, we have

$\dfrac{dA}{dt}$ = $\pi$ $*$ 2 $*$ r $*$ $\dfrac{dr}{dt}$

Substituting the value of r and $\dfrac{dr}{dt}$ in the above equation, we have

$\dfrac{dA}{dt}$ = $\pi$ $*$ 2 $*$ 10 $*$ 3 cm² / s

$\dfrac{dA}{dt}$ = 60$\pi$ cm²/s = 188.4 cm² / s

Q : 3) s = $\sin2t$ ................Equation(4)

Now, acceleration is given by

a = $\dfrac{d^2s}{dt^2}$

Differentiating Equation (1) twice with respect to t, we have

a = $\dfrac{d^2s}{dt^2}$ = $\dfrac{d}{dt}(2cos 2t)$

a = $\dfrac{d^2s}{dt^2}$ = - 4 $\sin2t$

a = - 4 $\sin2t$