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Math Help - Find Angle Theta...Optimization

  1. #1
    Senior Member nycmath's Avatar
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    Find Angle Theta...Optimization

    The following question is from my single variable calculus textbook, chapter on optimization. I don't know how to get started. I know angle theta must be found. I also know that trigonometry is involved here.

    The range R of a projectile fired with an initial velocity V at an angle theta with a horizontal is R = [(V)^2 * sin2(theta)]/g.
    Find the angle theta such that the range is a maximum.
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  2. #2
    MHF Contributor ebaines's Avatar
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    Re: Find Angle Theta...Optimization

    Take the derivative of the function and set it to zero, then solve for theta. That will give you a point where R is either a local maximum or a local minimum. Then take the second derivative at that point: if the 2nd derivative is a negative value that means the function is accelerating downwards and the point is a local maximum; if the 2nd derivative is positive then the functionat that point is accelerating upwards and it's a local minimum.
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    Re: Find Angle Theta...Optimization

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    Senior Member nycmath's Avatar
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    Re: Find Angle Theta...Optimization

    I will do as you say and play with the function some more. One thing, can you find the first derivative and I' ll take it from there?
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    Re: Find Angle Theta...Optimization

    Quote Originally Posted by nycmath View Post
    I will do as you say and play with the function some more. One thing, can you find the first derivative and I' ll take it from there?
    R=\frac{v^2}{g} \sin(2\theta)\Rightarrow

    \frac{dR}{d\theta}=\frac{2 v^2}{g}\cos(2\theta)

    you really should have been able to do that yourself.
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    Senior Member nycmath's Avatar
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    Re: Find Angle Theta...Optimization

    I plugged this into an online derivative calculator and the answer involved partial derivatives which is a calculus 3 concept. I tried taking the first derivative on my own and got a crazy answer. The letters V and g threw me off.
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