Calculus/Calculus

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Question
Find dy/dx at (-2/3, 0) if    x3y2 - 2ey = 3x + y.
Enter the exact value (as a decimal).



Six-foot tall Barry is performing his last show at the world-famous copa-cabana lounge. The stage is 20 feet deep and has a spot light attached to the front edge (aimed back toward Barry) that casts a shadow on the back wall of the stage. Barry is singing and dancing his way toward the front edge of the stage at a rate of 5 feet per min. Find the rate at which the height of Barry's shadow is changing when he is 8 feet from the back wall.

Enter your answer to the nearest 0.0001 ft/min.

Answer
It can be seen that 3xy + 2xy - 2e^y = 3 + y'.
Subtract 3 from both sides, then put in both x=-2/3 and y=0.

Speed is 5'/m, distance is 8' from back wall, light is 20' from back wall, height is 6'.
The distance from the front of the stage is given by D(t) = 12 - 5t and his height is 6'.

It is known that the shadow height H(t) is given by H(t)/20 = 6/D(t)
since those are the height and base of two similar triangles.

Taking the derivative gives H'(t)/20 = -6D'(t)/D(t).

We know at his point that D(t)=12 and D'(t)=-5 (since it is decreasing the distance).
Put thses in, mulitiply both sides by 20, and will give the speed that the shadow is growing at.  Note that we have a negative times a negatives, so the result is positive.
This means the shadow is increasing in height.  

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