Q1.11. Threatened monarch butterflies are monitored by measuring the area occupied by overwintering butterflies in Mexico. According to data collected by World Wildlife Fund Mexico, monarch butterflies occupied 0.67 hectares one recent winter, and 1.13 hectares the following winter (one year later)1. If the population is growing geometrically and grows at the same rate, how much area should the overwintering butterflies occupy one year after that

The correct answer is 1.91 hectares.

Explanation:

Based on the given data, the monarch butterflies occupied 0.67 hectares of area one winter, and in the following winter the species occupied 1.13 hectares of the area. It is given that the population of the butterflies is growing geometrically, therefore, the area the butterflies will cover in the coming year will be:  

The formula that is used to calculate the geometric growth rate is Nt = No (1+r)^t,  

Here Nt is 1.13, t is 1 and No is 0.67,  

Therefore, 1.13 = 0.67 (1+r)^1

= 1.13 = 0.67 + 0.67r

= 1.13 - 0.67 = 0.67r

= 0.46 = 0.67r

= r = 0.46/0.67

= r = 0.686

For the coming year, the No will be 1.13, t is 1 and r is 0.686. Now the Nt will be,  

Nt = No (1+r)^t

Nt = 1.13 * (1+0.686)^1

Nt = 1.13 * 1.686

Nt = 1.91 hectares.  

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