Question 254211
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We are looking for a function


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ h(d)]


such that


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ h(0)\ =\ 0]


and


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ h(300)\ =\ 0]


We are given that this is a quadratic function, so:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ h(d)\ =\ ad^2\ +\ bd\ +\ c]


The graph is a parabola with the vertex on the axis of symmetry, namely half-way between *[tex \Large d\ =\ 0] and *[tex \Large d\ =\ 300], to wit:  *[tex \Large d\ =\ 150]  and we are given that


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ h(150)\ =\ 15]


So from


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ h(0)\ =\ 0]


we can deduce that *[tex \Large c\ =\ 0] in


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ h(d)\ =\ ad^2\ +\ bd\ +\ c]


and then


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ h(d)\ =\ ad^2\ +\ bd]


remains.  Then, from


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ h(300)\ =\ 0]


we can write:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ h(300)\ =\ 90000a\ +\ 300b\ =\ 0]


and from


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ h(150)\ =\ 15]


we can write:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ h(150)\ =\ 22500a\ +\ 150b\ =\ 15]


Giving us a system of two linear equations in two variables.  I recommend that you solve them by the elimination method.  Once you have the coefficients for the quadratic, set the quadratic expression equal to 10 and solve for the two roots.



John
*[tex \LARGE e^{i\pi} + 1 = 0]
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