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An object is moving in clockwise direction around the unit circle x2 + y2 = 1. As it passes through the point (1/2, √3/2), its y-coordinate is decreasing at the rate of 3 units per second. The rate at which the x-coordinate changes at this point is (in units per second)
- a)2
- b)3√3
- c)√3
- d)2√3
Correct answer is option 'B'. Can you explain this answer?
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An object is moving in clockwise direction around the unit circle x2+ ...
Understanding the Problem
An object is moving clockwise around the unit circle defined by the equation x² + y² = 1. At the point (1/2, √3/2), the y-coordinate is decreasing at a rate of 3 units per second. We need to find the rate at which the x-coordinate changes at this point.
Given Information
- Point: (1/2, √3/2)
- Rate of change of y-coordinate: dy/dt = -3 units/second (negative indicates decrease)
Using the Unit Circle Equation
Since the object is constrained to the unit circle, we can use implicit differentiation of the equation x² + y² = 1 with respect to time t:
1. Differentiate: d(x²)/dt + d(y²)/dt = 0
2. Apply the chain rule: 2x(dx/dt) + 2y(dy/dt) = 0
3. Simplifying gives: x(dx/dt) + y(dy/dt) = 0
Substituting Known Values
At the point (1/2, √3/2):
- x = 1/2
- y = √3/2
- dy/dt = -3
Substituting these values into the equation:
(1/2)(dx/dt) + (√3/2)(-3) = 0
Solving for dx/dt
Rearranging the equation:
(1/2)(dx/dt) = (√3/2)(3)
Multiply both sides by 2:
dx/dt = 3√3
Conclusion
The rate at which the x-coordinate changes at the point (1/2, √3/2) is 3√3 units per second, which corresponds to option 'B'.
An object is moving clockwise around the unit circle defined by the equation x² + y² = 1. At the point (1/2, √3/2), the y-coordinate is decreasing at a rate of 3 units per second. We need to find the rate at which the x-coordinate changes at this point.
Given Information
- Point: (1/2, √3/2)
- Rate of change of y-coordinate: dy/dt = -3 units/second (negative indicates decrease)
Using the Unit Circle Equation
Since the object is constrained to the unit circle, we can use implicit differentiation of the equation x² + y² = 1 with respect to time t:
1. Differentiate: d(x²)/dt + d(y²)/dt = 0
2. Apply the chain rule: 2x(dx/dt) + 2y(dy/dt) = 0
3. Simplifying gives: x(dx/dt) + y(dy/dt) = 0
Substituting Known Values
At the point (1/2, √3/2):
- x = 1/2
- y = √3/2
- dy/dt = -3
Substituting these values into the equation:
(1/2)(dx/dt) + (√3/2)(-3) = 0
Solving for dx/dt
Rearranging the equation:
(1/2)(dx/dt) = (√3/2)(3)
Multiply both sides by 2:
dx/dt = 3√3
Conclusion
The rate at which the x-coordinate changes at the point (1/2, √3/2) is 3√3 units per second, which corresponds to option 'B'.
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An object is moving in clockwise direction around the unit circle x2+ y2= 1. As it passes through the point (1/2,√3/2), its y-coordinate is decreasing at the rate of 3 units per second. The rate at which the x-coordinate changes at this point is (in units per second)a)2b)3√3c)√3d)2√3Correct answer is option 'B'. Can you explain this answer?
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