Answer: [tex]k=\frac{M^t(v-P)}{P}[/tex]
Step-by-step explanation:
Given the following equation provided in the exercise:
[tex]P=\frac{v}{1+kM^{-t}}[/tex]
You can follow these steps in order to solve for "k":
1. You must multiply both sides of the equation by [tex]1+kM^{-t}[/tex]:
[tex](P)(1+kM^{-t})=(\frac{v}{1+kM^{-t}})(1+kM^{-t})\\\\P+PkM^{-t}=v[/tex]
2. The next step is to subtract "P" from both sides of the equation:
[tex]P+PkM^{-t}-(P)=v-(P)\\\\PkM^{-t}=v-P[/tex]
3. Now apply the Negative Exponent Rule, which states that:
[tex]a^{-n}=\frac{1}{a^n}[/tex]
Then:
[tex]\frac{Pk}{M^{t}}=v-P[/tex]
4. Multiply both sides of the equation by [tex]M^t[/tex]:
[tex](M^t)(\frac{Pk}{M^{t}})=(M^t)(v-P)\\\\Pk=M^t(v-P)[/tex]
5. And finally, you must divide both sides of the equation by "P":
[tex]\frac{Pk}P=\frac{M^t(v-P)}{P}\\\\k=\frac{M^t(v-P)}{P}[/tex]
