can someone please explain why the differential equation of this prolem ( An object is placed in 68 F room. write a differentiatal equation for H , the temperature of the object at time t.) as dH/dt=k(68−H) and not as dH/dt=−k(H−68)
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Both of them are actually the same since we never say explicitly whether the k must be positive or negative.
If k is positive then yours is correct.
does the positive or negative affect the problem
Since in this problem to temperature is going to decrease, then it is understood implicitly that \(dH/dt\) is negative. So if the equation is written as \9dH/dt=k(68-H)\) it is understood that \(k<0\). In general it won't affect the problem because usually to solve the problem we need to solve for \(k\) first. If the equation written in the first form, you will get negative k, if it is written as the second form you will get positive k. But if we substitute these k's into the equation you will exactly get the same equation (imagine that you plug in k=-2 to the first or k=2 to the second, you'll get the same thing).