Suppose 4x2 + 16y2 = 100, where x and y are functions of t.(a)If dydt = 14, find dxdt when x = 3 and y = 2.dxdt = (b)If dxdt = 4, find dydt when x = −3 and y = 2.
Question
Suppose 4x2 + 16y2 = 100, where x and y are functions of t.(a)If dydt = 14, find dxdt when x = 3 and y = 2.dxdt = (b)If dxdt = 4, find dydt when x = −3 and y = 2.
Solution
(a) To find dx/dt when x = 3 and y = 2, we first differentiate the given equation with respect to t.
The derivative of 4x^2 is 8x(dx/dt) and the derivative of 16y^2 is 32y(dy/dt). The derivative of 100 is 0. So, we get:
8x(dx/dt) + 32y(dy/dt) = 0
Substitute x = 3, y = 2, and dy/dt = 14 into the equation:
83(dx/dt) + 322*14 = 0
24(dx/dt) + 896 = 0
Solving for dx/dt gives:
dx/dt = -896 / 24 = -37.33
(b) To find dy/dt when x = -3 and y = 2, we substitute x = -3, dx/dt = 4 into the differentiated equation:
8*-34 + 322*(dy/dt) = 0
-96 + 64(dy/dt) = 0
Solving for dy/dt gives:
dy/dt = 96 / 64 = 1.5
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