Question

How do you do this? Todd invests $ 7,000 $ dollars at an annual interest rate of $ 4.2\% $ compounded continuously. Determine how many years, to the nearest $ 10th $ of a year, it will take for Todd’s investment to triple.

Answer 1

Hint: Here we will use the concept of amount equals Principal times the mathematical constant (e) to the power of the rate of the interest (r) times the times in years (t) and will use the formula, $ A = P{e^{rt}} $

Complete step-by-step answer:
First of all convert the given rate of percentage in the form of decimal.
  $ r = \dfrac{{4.2}}{{100}} = 0.042 $
Now, use the formula $ A = P{e^{rt}} $
Place the given values in the above equation-
  $ 21000 = 7000{e^{0.042t}} $
The above equation can be written as –
  $ 7000{e^{0.042t}} = 21000 $
Term multiplicative on one side if moved to the opposite side then it goes to the denominator.
  $ {e^{0.042t}} = \dfrac{{21000}}{{7000}} $
Find the factors for the term on the numerator on the right hand side of the equation.
  $ {e^{0.042t}} = \dfrac{{7000 \times 3}}{{7000}} $
Common factors from the numerator and the denominator cancel each other.
  $ {e^{0.042t}} = 3 $
Take logarithm on both the sides of the equation.
  $ \ln ({e^{0.042t}}) = \ln 3 $
Apply the logarithmic properties and simplify the above equation-
  $ 0.042t = \ln (3) $
Term multiplicative on one side if moved to the opposite side then it goes to the denominator.
  $ \Rightarrow t = \dfrac{{\ln (3)}}{{0.042}} $
By reading the values from the logarithmic table
  $ \Rightarrow t = \dfrac{{1.098612}}{{0.042}} $
Simplify the above expression –
  $ \Rightarrow t = 26.2 $ years
This is the required solution.
So, the correct answer is “ $ t = 26.2 $ years”.

Note: Always double check for the page, and the function for which you are looking. Use scale to refer and go through the horizontal lines for the required values for the minutes. Since, a log book is a series of the numbers, observe and note the given number carefully and always cross-check it.