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7. You wish to have $290,250 by investing P dollars at 11.4% interest compounded monthly for t years. A. Using only the compound interest formulas (from 4.2) in class, set up an equation relating P and
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7. You wish to have $290,250 by investing P dollars at 11.4% interest compounded monthly for t years. A. Using only the compound interest formulas (from 4.2) in class, set up an equation relating P and t. (The equation will have a P and a t, everything else should be numbers).B. Solve the equation you found in part A) for P. (It should have numbers and a t). Show all your work. C. Solve the equation you found in part A) for t. (It should have numbers and a P). Show all your work. D. Find P when t=17. Round to the nearest cent.E. Find t when P= $135000. Round to the nearest hundredth of a year.

User Emiliano Zilocchi
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\begin{gathered} \text{ Given} \\ A=290250 \\ r=0.114\text{ converted from percentage} \\ n=12,\text{ since it is compounded monthly (12 months in a year)} \\ A=P(1+(r)/(n))^(nt)\rightarrow\text{Compound Interest Formula} \\ \text{ A.) Substitute all the following values, that should leave us with} \\ \text{ P and t as variables} \\ A=P(1+(r)/(n))^(nt) \\ 290250=P(1+(0.114)/(12))^(12t)\rightarrow290250=P(1.0095)^(12t) \\ \text{ B.) Solve the equation for P} \\ \text{ We isolate P on the left hand side and we get} \\ 290250=P(1.0095)^(12t) \\ P(1.0095)^(12t)=290250 \\ \frac{P\cancel{(1.0095)^(12t)}}{\cancel{(1.0095)^(12t)}}=(290250)/((1.0095)^(12t)) \\ P=(290250)/((1.0095)^(12t)) \\ \text{ C.) Solve for t} \\ 290250=P(1.0095)^(12t) \\ P(1.0095)^(12t)=290250 \\ \frac{\cancel{P}(1.0095)^(12t)}{\cancel{P}}=(290250)/(P) \\ 1.0095^(12t)=(290250)/(P)\text{ get the natural logarithm of both sides} \\ \ln (1.0095^(12t))=\ln (290250)/(P) \\ 12t\cdot\ln 1.0095=\ln (290250)/(P),\text{ after this divide both sides by 12 and ln (1.0095)} \\ \text{ to isolate }t \\ \frac{\cancel{12}t\cdot\cancel{\ln (1.0095)}}{\cancel{12\cdot\ln(1.0095)}}=(\ln(290250)/(P))/(12\cdot\ln(1.0095)) \\ t=(\ln(290250)/(P))/(12\cdot\ln(1.0095)) \\ \text{ D.) Using the equation that we have on B, just substitute the values} \\ \text{ with }t=17 \\ P=(290250)/((1.0095)^(12t)) \\ P=(290250)/((1.0095)^(12(17))) \\ \text{ Input this in a calculator and we get} \\ P=\$42177.64 \\ \text{ E.) Using the equation that have on C, substitute the values with} \\ P=135000 \\ t=(\ln(290250)/(P))/(12\cdot\ln(1.0095)) \\ t=(\ln (290250)/(135000))/(12\cdot\ln (1.0095)) \\ \text{ Input this in a calculator and we get} \\ t=6.75\text{ years},\text{ rounded to a hundredth} \end{gathered}

User Bill Michell
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