A particle moves along the circle $x ^ { 2 } + y ^ { 2 } = 25$ . If x increases at a rate 2 units per minute, at what rate is y changing when $x = 3$ ?
A) $2 \text { unit } / \min$

B) $\frac { 3 } { 2 } \text { unit } / \mathrm { min }$
C) $3 \text { unit } / \mathrm { min }$
D) $\frac { 1 } { 4 } \text { unit } / \mathrm { min }$
E) $- 2 \text { unit } / \min$
F) $- \frac { 3 } { 2 } \text { unit } / \mathrm { min }$

G) $- 3 \text { unit } / \min$

H) $- \frac { 1 } { 4 } \text { unit } / \mathrm { min }$

Question

A particle moves along the circle $x ^ { 2 } + y ^ { 2 } = 25$ . If x increases at a rate 2 units per minute, at what rate is y changing when $x = 3$ ?
A) $2 \text { unit } / \min$

B) $\frac { 3 } { 2 } \text { unit } / \mathrm { min }$ 
C) $3 \text { unit } / \mathrm { min }$
D) $\frac { 1 } { 4 } \text { unit } / \mathrm { min }$
E) $- 2 \text { unit } / \min$ 
F) $- \frac { 3 } { 2 } \text { unit } / \mathrm { min }$

G) $- 3 \text { unit } / \min$

H) $- \frac { 1 } { 4 } \text { unit } / \mathrm { min }$

Quizplus · Accepted Answer

Given the equation of the circle $x^2 + y^2 = 25$, we differentiate both sides with respect to time $t$ to find the rate at which $y$ is changing when $x = 3$. Differentiating, we get $2x \frac{dx}{dt} + 2y \frac{dy}{dt} = 0$. Given $\frac{dx}{dt} = 2$ units/min and substituting $x = 3$, we solve for $\frac{dy}{dt}$ when $x = 3$. Substituting $x = 3$ into the circle's equation, we find $y = \pm 4$. Choosing the positive or negative value of $y$ depends on the direction of motion, but since we're interested in the rate of change of $y$, we'll proceed with the calculation which yields the same magnitude for $\frac{dy}{dt}$ in either case. Substituting $x = 3$ and $\frac{dx}{dt} = 2$ into the differentiated equation, we solve for $\frac{dy}{dt}$, which gives $\frac{dy}{dt} = -\frac{3}{2}$ units/min, indicating that $y$ is decreasing at a rate of $\frac{3}{2}$ units per minute when $x = 3$.

A Particle Moves Along the Circle x2+y2=25x ^ { 2 } + y ^ { 2 } = 25x2+y2=25

A Particle Moves Along the Circle $x ^ { 2 } + y ^ { 2 } = 25$