Imagine that the amount of bacteria in a petri dish grows exponentially through time, doubling every day until the dish is completely covered by bacteria on day 100. On what day would the petri dish be 50% covered by bacteria?

Correct answer: 50% will be on day 99

Step-by-step explanation:

Given:

- amount of bacteria is doubling every day

- on day 100 amount is 100%

- on day n = ? will be 50%

We define the initial amount of bacteria with A

First day - A₁ = A · 2

Second day - A₂ = (A · 2) · 2 = A · 2²

Third day - A₃ = A₂ · 2 = A · 2³

...

n-th day Aₙ = A · 2ⁿ

on day 100 A₁₀₀ = A · 2¹⁰⁰ - 100%

on day n Aₙ = A · 2ⁿ - 50%

A₁₀₀ : Aₙ = A · 2¹⁰⁰ : A · 2ⁿ = 100% : 50% = 2 : 1

(A · 2¹⁰⁰) : (A · 2ⁿ) = 2 : 1 ⇒ A · 2ⁿ = (A · 2¹⁰⁰) / 2 ⇒

2ⁿ = 2¹⁰⁰ / 2¹ = 2⁹⁹ ⇒ 2ⁿ = 2⁹⁹ ⇒ n = 99

n = 99

This problem could have been solved more simply. If we know that the quantity increases double every day, it is logical if 99 days is 50% that on the 100th day it will be double or 100%

God is with you!

Day 99 because if it doubles everyday it would go from 50% to 100% after one day since 50•2=100