09SpringHwk03_Soln - Math 3301 Homework Set 3 Solutions 10...

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Math 3301 Homework Set 3 – Solutions 10 Points 1. (3p ts) Here’s the IVP’s we need for this problem. ( ) ( ) ( ) ( ) ( ) 0 50 3 150 7 4 9 4 8 0 50 P rP P P P rP rP P ¢ = = = ¢ = + - = - = Solving the first gives, ( ) ( ) 3 1 3 50 150 50 ln 3 r t r P t r = = = e e The second IVP is now, ( ) ( ) 1 3 ln 3 8 0 50 P P P ¢ = - = The solution is, ( ) ln3 3 24 ln3 28.1542 t P t = + e From this we can see that they will survive to take over the world! 2. (3 pts) Here’s the IVP’s for this case. ( ) ( ) 10 2 1 1 1 1 75 15 30 2 2 2 2 2 75 5 9.8 9.8 0 2 9.8 9.8 30 c v v v v v v v v t ¢ = - = - = ¢ = - = - = The solution to the first one is, ( ) 2 15 1 73.5 71.5 t v t - = - e and we’ll need to set this equal to 30 and solve to determine the time when the second parachute opens. 2 2 15 15 30 73.5 71.5 0.6084 3.7269 t t t - - = - fi = ° = e e Eventually we’ll need the “height” function for this so let’s get that and the “height” of the sky diver at this time. ( ) ( ) ( ) 2 2 15 15 1 1 1 73.5 71.5 0 0 73.5 536.25 536.25 t t s t dt s s t t - - = - = ° = + - ± e e The sky diver is now ( ) 3.7269 63.9325 s m = below the plane. The second IVP is now, ( ) 2 2 2 2 5 9.8 3.7269 30 v v v ¢ = - = The solution to this one is, ( ) 2 5 2 24.5 24.4226 t v t - = + e Next we’ll need the “height” function.
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