# In an expedition to seize his enemy's elephants, a king marched 2 yojanas the first day. Say,...

## Question:

In an expedition to seize his enemy's elephants, a king marched 2 yojanas the first day. Say, intelligent calculator, with what increasing rate of daily march did he proceed, since he reached his foe's city, a distance of 80 yojanas, in a week?

## Exponential Growth:

The exponential function is a function where the value of the function is dependent on an independent variable which is an exponent of a constant. In the exponential growth function, the constant in the function is greater than one, and the percent by which it is greater is the rate of growth of the function.

$$\text{Final value = Initial value}\times \left(1 +\dfrac{Rate}{100} \right)^{Time}$$

According to the question, the initial distance traveled is 20 yojanas and the army needs to cover 80 yojanas by the end of the 7th day. Using the exponential growth model:

$$\text{Final value = Initial value}\times \left(1 +\dfrac{Rate}{100} \right)^{Time}$$

$$80=20\left( 1 + \dfrac{R}{100}\right)^7$$

Dividing both sides by 20

$$4=\left( 1 + \dfrac{R}{100}\right)^7$$

$$\text{Taking log on both sides}$$

$$log\ 4=7\times log\left( 1 + \dfrac{R}{100}\right)$$

$$0.6020=7\times log\left( 1 + \dfrac{R}{100}\right)$$

$$\text{Dividing the equation by }7$$

$$0.0860=log\left( 1 + \dfrac{R}{100}\right)$$

$$\text{Taking log inverse on both sides}$$

$$1.2189=1 + \dfrac{R}{100}$$

$$\dfrac{R}{100} = 0.2189$$

Multiplying both sides by 100

$$R= 21.89\%$$

The increasing rate of daily march is {eq}21.89\%. {/eq} 