A particle moves along a curve y = frac{2x^3 | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Given the curve equation is $y = \frac{2x^3 - 1}{3}$.We are given that the rate of change of the $y$-coordinate is $18$ times the rate of change o

Vedclass · Accepted Answer

$(-3, -\frac{55}{3}), (3, \frac{53}{3})$

Vedclass · Answer

(D) Given the curve equation is $y = \frac{2x^3 - 1}{3}$.We are given that the rate of change of the $y$-coordinate is $18$ times the rate of change of the $x$-coordinate,which implies $\frac{dy}{dt} = 18 \frac{dx}{dt}$.Dividing both sides by $\frac{dx}{dt}$,we get $\frac{dy}{dx} = 18$.Now,differentiate the curve equation with respect to $x$:$\frac{dy}{dx} = \frac{d}{dx} (\frac{2x^3 - 1}{3}) = \frac{1}{3} \cdot 6x^2 = 2x^2$.Equating the derivative to $18$:$2x^2 = 18 \implies x^2 = 9 \implies x = \pm 3$.For $x = 3$,$y = \frac{2(3)^3 - 1}{3} = \frac{54 - 1}{3} = \frac{53}{3}$.For $x = -3$,$y = \frac{2(-3)^3 - 1}{3} = \frac{-54 - 1}{3} = -\frac{55}{3}$.Thus,the points are $(3, \frac{53}{3})$ and $(-3, -\frac{55}{3})$.Comparing this with the given options,the correct option is $D$.

$A$ particle moves along a curve $y = \frac{2x^3 - 1}{3}$. The points on the curve at which the $y$-coordinate is changing $18$ times as fast as the $x$-coordinate are

Similar Questions

If a balloon lying at an altitude of $30 \ m$ from an observer at a particular instant is moving horizontally at the rate of $1 \ m/s$ away from him, then the rate at which the balloon is moving away directly from the observer at the $40^{\text{th}}$ second is (in $m/s$)

$A$ spherical iron ball of $10 \text{ cm}$ radius is coated with a layer of ice of uniform thickness that melts at the rate of $50 \text{ cm}^3/\text{min}$. If the thickness of the ice is $5 \text{ cm}$,then the rate at which the thickness of the ice decreases is:

If the rate of change of the slope of the tangent drawn to the curve $y=x^3-2x^2+3x-2$ at the point $(2,4)$ is $k$ times the rate of change of its abscissa, then $k=$

In a Maha Kumbh, a drone camera is moving along the curve $3y = x^3 - 3$. The camera captures high-quality pictures when the $y$-coordinate changes $9$ times as fast as the $x$-coordinate. Which of the following is a precise position of the drone at that instant?

The radius of a cylinder is increasing at the rate of $3 \, m/sec$ and its altitude is decreasing at the rate of $4 \, m/sec$. The rate of change of volume when the radius is $4 \, m$ and the altitude is $6 \, m$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module