Brennan, a farmer, orders a large box of ladybugs to control pests in his greenhouse. To make sure that he was shipped as many ladybugs as he ordered, he decides to figure out approximately how many ladybugs there are in the box. He catches 30 ladybugs, marks them with special paint, and returns them to the box. Then a little while later, he catches 390 ladybugs and counts 13 marked ladybugs among them. To the nearest whole number, what is the best estimate for the ladybug population? ____

Question

Brennan, a farmer, orders a large box of ladybugs to control pests in his greenhouse. To make sure that he was shipped as many ladybugs as he ordered, he decides to figure out approximately how many ladybugs there are in the box. He catches 30 ladybugs, marks them with special paint, and returns them to the box. Then a little while later, he catches 390 ladybugs and counts 13 marked ladybugs among them. To the nearest whole number, what is the best estimate for the ladybug population?   ____

Bytelearn · Accepted Answer

Set up method:  Step 1: Set up the capture-recapture method to estimate the total population of ladybugs. Brennan marked 30 ladybugs and later found 13 of them in a sample of 390. Calculate proportion:  Step 2: Use the proportion of marked ladybugs in the second sample to estimate the total population. The proportion is calculated as the number of marked ladybugs found (13) divided by the total number of ladybugs in the second sample (390), which should equal the number of marked ladybugs initially released (30) divided by the total population (N). Equation: $ \frac{13}{390} = \frac{30}{N} $ Cross-multiply for N:  Step 3: Cross-multiply to solve for N. $ 13 \times N = 30 \times 390 $ Continue solving for N:  Step 4: Continue solving for N. $ 13N = 11700 $ Isolate N:  Step 5: Divide both sides by 13 to isolate N. $ N = \frac{11700}{13} $ Calculate final population:  Step 6: Calculate N. $ N = 900 $

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