A point is moving along the circle
$x ^ { 2 } + ( y - 3 ) ^ { 2 } = 5$
In such a way that its x-coordinate is decreasing at a rate of 2 units per second. At what rate is the y-coordinate decreasing at the instant when the point has reached $( - 2,4 )$

A)8 units per second
B) 6 units per second
C) 3.5 units per second
D) 5 units per second
E) 4 units per second

Question

A point is moving along the circle  
  $x ^ { 2 } + ( y - 3 ) ^ { 2 } = 5$  
In such a way that its x-coordinate is decreasing at a rate of 2 units per second. At what rate is the y-coordinate decreasing at the instant when the point has reached $( - 2,4 )$  
 
A)8 units per second
B) 6 units per second
C) 3.5 units per second
D) 5 units per second
E) 4 units per second

Quizplus · Accepted Answer

Given the circle's equation $x^2 + (y - 3)^2 = 5$, we differentiate both sides with respect to time $t$ to find the relationship between the rates of change of $x$ and $y$. This gives us $2x\frac{dx}{dt} + 2(y - 3)\frac{dy}{dt} = 0$. We are told $\frac{dx}{dt} = -2$ units per second (since $x$ is decreasing), and we need to find $\frac{dy}{dt}$ when the point is at $(-2, 4)$.Substituting $x = -2$, $y = 4$, and $\frac{dx}{dt} = -2$ into the differentiated equation, we get:$$2(-2)(-2) + 2(4 - 3)\frac{dy}{dt} = 0$$$$8 + 2\frac{dy}{dt} = 0$$$$2\frac{dy}{dt} = -8$$$$\frac{dy}{dt} = -4$$Therefore, the y-coordinate is decreasing at a rate of 4 units per second when the point is at $(-2, 4)$.

A Point Is Moving Along the Circle x2+(y−3)2=5x ^ { 2 } + ( y - 3 ) ^ { 2 } = 5x2+(y−3)2=5

A Point Is Moving Along the Circle $x ^ { 2 } + ( y - 3 ) ^ { 2 } = 5$