A fat cat that was 12 kg reduces its weight by 10% each month How long does it take for the cat to be at least 6 kg lighter than its original weight

Question

It is part of the Australian Math Syllabus. I am having trouble with this question. It is in the Indices and Surds chapter

Answer 1

in month 1, the loss is 1.2 kg so new weight is 10.8

in month 2, the loss is 1.08 kg so new weight is then 9.72

to simplify, just multiply above weight by 0.9 to get

8.748 kg, then 7.8732 kg, then 7.08588, then 6.377292 kg at end of months 3, 4, 5 & 6.

Then the next month's ending weight will be 5.74 kg, so it takes almost 7 months to get to 6 kg.

This agrees with the Exponential Decay solution; I present this because the student may not be at the Algebra II level yet.

Answer 2

Use exponential decay formula:

A = P (1-r)^t

A = amount decreased to

P = original amount

r = rate of decrease

t = time

 

6 ≥ 12 (1-0.10)^t      divide both sides by 12

0.5 ≥ (0.9)^t            take log of both sides

log (0.5) ≥ log (0.9)^t     power rule of logs lets the exponent t drop to front of log to become a factor

log (0.5) ≥ t log(0.9)     divide both sides by log(0.9) which is negative so reverse the inequality sign

log (0.5) / log (0.9) ≤ t     use calculator

6.57 ≤ t

 

It takes at least 7 months for the cat to be at least 6 kg lighter that original weight.

 

check:

12 (0.9)⁷≈ 5.73

12 (0.9)⁶ ≈ 6.37