Respuesta :
INFORMATION:
If Q=5000e^0.18t we must find the effective annual yield and the continuous growth rate
STEP BY STEP EXPLANATION:
The annual yield is the return over 1 year:
- Initial investment: To find it, we need to replace t = 0 in the equation for Q
[tex]\begin{gathered} \text{ Initial investment}=5000\cdot e^{0.18t} \\ \text{ When t = 0,} \\ \text{ Initial investment}=5000\cdot e^{0.18\cdot0}=5000 \end{gathered}[/tex]So, the initial investment is 5000
- Value after 1 year: To find it, we need to replace t = 1 in the equation for Q
[tex]\begin{gathered} \text{ Value after 1 year}=5000\cdot e^{0.18t} \\ \text{When t=1,} \\ \text{In}\imaginaryI\text{t}\imaginaryI\text{al}\imaginaryI\text{nvestment}=5,000e^{0.18\times1}=5986.0868 \end{gathered}[/tex]Therefor the effective annual yield is
[tex]\text{ \lparen}\frac{5986.0868-5000}{5000}-1)\cdot100\text{ \%}=19.7217\text{ \%}[/tex]Finally, The continuous growth rate is the constant in the exponential so in this case is 0.18 or 18%
ANSWER:
(a) The continuous growth rate is 18%
(b) The effective annual rate is 19.7217%
