Question

Insect larvae grow incredibly quickly over the first several days of their lives. The length of one particular species increases exponentially at a rate of 25% each day. A scientist studying this species of larva measures one specimen to be 3 mm in length at the beginning of his observation period. The scientist will transfer the larvae to a new environment once the larvae are over 9.16 mm long. If t represents the number of days since the scientist began his observations, write an inequality to represent the situation, and use it to determine after how many days the larvae are transferred to the new environment.

Answer 1

The exponent graph for this question would be: an= 3mm(1.25)^t
If the target length of the larvae is 9.16 mm, the time needed would be:an= 3mm(1.25)^t9.16 mm = 3mm(1.25)^t3mm*3.053=3mm(1.25)^t1.25^t= 3.0531.25^t= 1.25^5t=5

Answer 2

The equation that represents this situation: A = P*(1+r)^t

Number of days after the larvae are transferred to the new environment: 5

Explanation:

The exponential function that relates the length of the larvae (A, in mm) after t days is:

A = P*(1+r)^t

where P is the length at the beginning (3 mm) and r is the daily growth parameter (0.25)

The time needed to reach 9.16 mm long is:

9.16 = 3*(1+0.25)^t

9.16/3 = 1.25^t

ln(9.16/3) = t*ln(1.25)

t = ln(9.16/3)/ln(1.25)

t = 5 days