Question

Suppose that z= x3y2, where both x and y are changing with time. At a certain instant when x= 1 and y=2, x is decreasing at the rate of 2 units/s, and y is increasing at the rate of 3 units/s. How fast is z changing at this instant?
Is z increasing or decreasing?​

Answer 1

$\huge\bold{\gray{\sf{Answer:}}}$

$\bold{Explanation:}$

$z = {x}^{3} {y}^{2}$

we differentiate 'z' with respect to the time.Here,we differentiate it by multiplication method:

$= > \frac{dz}{dt} = {3x}^{2} \frac{dx}{dt} {y}^{2} + {x}^{3} 2y \frac{dy}{dt}$

Given that when x =1 ,x is decreasing at the rate of 2units /s

It implies that:

=>$\left.\dfrac{dx}{dt}\right |_{x = 1 }= - 2\:units/s$

when y =2,y is increasing at the rate of 3units/s

=>$\left. \dfrac{dy}{dt}\right |_{y = 2 }= 3\:units/s$

Z change at this instant:-

=>$\left. \dfrac{dz}{dt} \right |_{x = 1,y = 2 }=( 3 {x}^{2} \frac{dx}{dt} |x = 1) {y}^{2} + {x}^{3} (2y \frac{dy}{dt} |y = 2)$

=>$\left. \dfrac{dz}{dt} \right |_{(x = 1)(y=2)}= 3 {(1)}^{2} ( - 2) {(2)}^{2} + {(1)}^{3}(4)(3)$

$= - 12units/s$

$\therefore$z is decreasing

Answer 2

Answer:

answer is

-12units/s

z is decreasing