Question 688946
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If the curve *[tex \LARGE y\ =\ ab^x] contains the point *[tex \LARGE (3,13.5)] then


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ ab^3\ =\ 13.5]


which is to say


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ b^3\ =\ \frac{13.5}{a}]


Similarly, since the curve contains the point *[tex \LARGE (5,30.375)],


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ b^5\ =\ \frac{30.375}{a}]


But since *[tex \LARGE b^5\ =\ b^3\,\cdot\,b^2], we can write


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ b^2\ =\ \frac{\frac{30.375}{a}}{\frac{13.5}{a}}]


Solving for *[tex \LARGE b]:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ b^2\ =\ 2.25]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ b\ =\ 1.5]


From this we get


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ b^3\ =\ 3.375]


So


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ a\ =\ \frac{13.5}{3.375}\ =\ 4]


And finally:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ 4\,\cdot\,1.5^x]


John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
<font face="Math1" size="+2">Egw to Beta kai to Sigma</font>
My calculator said it, I believe it, that settles it
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