christine obrien, who is self - employed, wants to invest $70,000 in a pension plan. one investment offers…

Explanation:

Step1: Calculate future - value for 7% compounded quarterly

The compound - interest formula is $A = P(1+\frac{r}{n})^{nt}$, where $P=$70000$, $r = 0.07$, $n = 4$ (quarterly compounding), and $t = 4$. $A_1=70000(1 +\frac{0.07}{4})^{4\times4}=70000(1 + 0.0175)^{16}$ $A_1=70000\times1.0175^{16}\approx70000\times1.32129096\approx$92490.37$ Interest earned $I_1=A_1 - P=92490.37−70000=$22490.37$

Step2: Calculate future - value for 6.75% compounded continuously

The continuous - compounding formula is $A = Pe^{rt}$, where $P = 70000$, $r=0.0675$, and $t = 4$. $A_2=70000e^{0.0675\times4}=70000e^{0.27}$ $A_2\approx70000\times1.31001677\approx$91701.17$ Interest earned $I_2=A_2 - P=91701.17−70000=$21701.17$ Since $I_1>I_2$, the 7% compounded quarterly investment earns more interest.

Step3: Calculate the difference in interest

The difference in interest is $I_1 - I_2=22490.37−21701.17=$789.20$

Step4: Calculate the effective rate for 7% compounded quarterly

The effective - rate formula is $r_{eff}=(1+\frac{r}{n})^{n}-1$. For $r = 0.07$ and $n = 4$, $r_{eff1}=(1+\frac{0.07}{4})^{4}-1=(1 + 0.0175)^{4}-1$ $r_{eff1}=1.0175^{4}-1\approx1.071859 - 1=0.071859\approx7.19%$

Step5: Calculate the effective rate for 6.75% compounded continuously

The effective - rate formula for continuous compounding is $r_{eff}=e^{r}-1$. For $r = 0.0675$, $r_{eff2}=e^{0.0675}-1\approx1.07008 - 1=0.07008\approx7.01%$

Step6: Find the time for continuous compounding

We use the formula $A = Pe^{rt}$. Given $P = 70000$, $A = 80000$, and $r = 0.0675$. $80000=70000e^{0.0675t}$ $\frac{80000}{70000}=e^{0.0675t}$ $\frac{8}{7}=e^{0.0675t}$ Take the natural logarithm of both sides: $\ln(\frac{8}{7})=\ln(e^{0.0675t})$ Since $\ln(e^{x})=x$, we have $\ln(\frac{8}{7}) = 0.0675t$ $t=\frac{\ln(\frac{8}{7})}{0.0675}\approx\frac{0.13353}{0.0675}\approx1.98\approx2$ years

Step7: Find the time for quarterly compounding

We use the formula $A = P(1+\frac{r}{n})^{nt}$. Given $P = 70000$, $A\geq80000$, $r = 0.07$, and $n = 4$. $80000\leq70000(1+\frac{0.07}{4})^{4t}$ $\frac{80000}{70000}\leq(1 + 0.0175)^{4t}$ $\frac{8}{7}\leq1.0175^{4t}$ Take the natural logarithm of both sides: $\ln(\frac{8}{7})\leq\ln(1.0175^{4t})$ Since $\ln(a^{b})=b\ln(a)$, we have $\ln(\frac{8}{7})\leq4t\ln(1.0175)$ $t\geq\frac{\ln(\frac{8}{7})}{4\ln(1.0175)}$ $\ln(\frac{8}{7})\approx0.13353$, $\ln(1.0175)\approx0.01735$ $4\ln(1.0175)\approx0.0694$ $t\geq\frac{0.13353}{0.0694}\approx1.92$ years. Since interest is added quarterly, we need to round up to the next - quarter. So $t = 2$ years (8 quarters).

Answer:

a. The investment with 7% compounded quarterly. b. $$789.20$ c. 7.19% for 7% compounded quarterly; 7.01% for 6.75% compounded continuously. d. Approximately 2 years. e. 2 years (8 quarters).