Math  /  Algebra

Question4 A bird population has decreased at a rate of 2.05\% every month for the last two years. There are 533 birds remaining, how many birds were there originally two years ago. (A) 558 (B) 1073 (C) 290 (D) 876

Studdy Solution

STEP 1

Assumptions
1. The current bird population is 533.
2. The population decreases at a rate of 2.05% every month.
3. The time period for the decrease is 2 years.
4. We need to find the original bird population 2 years ago.

STEP 2

First, we need to determine the total number of months over which the population has been decreasing. Since the time period is 2 years, we calculate:
Totalmonths=2years×12months/year=24monthsTotal\, months = 2 \, years \times 12 \, months/year = 24 \, months

STEP 3

The population decreases at a rate of 2.05% per month. To find the monthly decay factor, we convert the percentage to a decimal and subtract it from 1.
Decayfactor=10.0205=0.9795Decay\, factor = 1 - 0.0205 = 0.9795

STEP 4

The relationship between the original population P0 P_0 and the current population P P after 24 months can be expressed using the exponential decay formula:
P=P0×(Decayfactor)TotalmonthsP = P_0 \times (Decay\, factor)^{Total\, months}

STEP 5

Substitute the known values into the equation. We know P=533 P = 533 , Decayfactor=0.9795 Decay\, factor = 0.9795 , and Totalmonths=24 Total\, months = 24 .
533=P0×(0.9795)24533 = P_0 \times (0.9795)^{24}

STEP 6

To solve for P0 P_0 , divide both sides of the equation by (0.9795)24 (0.9795)^{24} .
P0=533(0.9795)24P_0 = \frac{533}{(0.9795)^{24}}

STEP 7

Calculate (0.9795)24 (0.9795)^{24} .
(0.9795)240.6176(0.9795)^{24} \approx 0.6176

STEP 8

Substitute the calculated value back into the equation to find P0 P_0 .
P0=5330.6176P_0 = \frac{533}{0.6176}

STEP 9

Calculate the original population P0 P_0 .
P05330.6176863.08P_0 \approx \frac{533}{0.6176} \approx 863.08

STEP 10

Since the population must be a whole number, we round 863.08 to the nearest whole number. The closest option is 876.
The original bird population two years ago was approximately 876.

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