Question

A particle moves along the curve $y=x^{3/2}$ in the first quadrant in such a way that it distance from the origin increases at the rate of $11$ units per second. The value of $dx/dt$ when $x=3$

• A. $4$
• B. $\frac{9}{2}$
• C. $\frac{3\sqrt{3}}{2}$
• D. None of these

Answer 1

The correct option is A $4$

The distance D of the point from the origin is given by

$D^2=x^2+y^2$

from given, we have,

$D^2=x^2+(x^{\frac{3}{2}}{)}^2$

$D^2=x^2+x^3$

$2D\frac{dD}{dt}=(2x+3x^2)\frac{dx}{dt}$

$2\sqrt{x^2+x^3}\frac{dD}{dt}=(2x+3x^2)\frac{dx}{dt}$

$2\sqrt{3^2+3^3}\left(11\right)\frac{dD}{dt}=\left(2\right(3)+3(3{)}^2)\frac{dx}{dt}$

$\frac{dx}{dt}=4$