Let $z ^ { 2 } = x ^ { 2 } + y ^ { 2 }$ and assume that $z 0$ If $\frac { d x } { d t } = - 25$ and $\frac { d y } { d t } = - \frac { 50 } { 3 }$ when $x = 200$ and $y = 150 \text {, }$ then $\frac { d z } { d t }$ is

A)-40
B)-30
C)30
D)40
E)50

Question

Let

z ^ { 2 } = x ^ { 2 } + y ^ { 2 }

and assume that

z > 0

If

\frac { d x } { d t } = - 25

and

\frac { d y } { d t } = - \frac { 50 } { 3 }

when

x = 200

and

y = 150 \text {, }

then

\frac { d z } { d t }

is

A)-40
B)-30
C)30
D)40
E)50

Quizplus · Accepted Answer

Given $z^2 = x^2 + y^2$, differentiate both sides with respect to $t$ to find $\frac{dz}{dt}$. This gives $2z\frac{dz}{dt} = 2x\frac{dx}{dt} + 2y\frac{dy}{dt}$. Substituting the given values, $x = 200$, $y = 150$, $\frac{dx}{dt} = -25$, and $\frac{dy}{dt} = -\frac{50}{3}$, and solving for $\frac{dz}{dt}$ yields $\frac{dz}{dt} = -30$.

Let z2=x2+y2z ^ { 2 } = x ^ { 2 } + y ^ { 2 }z2=x2+y2

Let $z ^ { 2 } = x ^ { 2 } + y ^ { 2 }$