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10-21.

A certain type of bacteria is being grown on a Petri dish in the school’s biology lab. Inez does some measurements and determines that the area of the bacteria covering the Petri dish is doubling each day. She started the bacteria colony on February $9$ and predicts that it will cover the entire Petri dish by February $21$.

1. When will the Petri dish be half covered by the bacteria?

Since the bacteria doubles the area it covers each day, it will have covered half the dish the day before it has covered the whole dish.
February $20$ will be the day that $50$% of the dish will be covered.

2. Let a represent the unknown percentage covered by bacteria on the starting day. Write an equation for a geometric sequence that represents percentage covered by bacteria.

Review the Math Notes box in section 5.3.3

$t(n) = a · 2^n$

The general form for a geometric sequence is $t(n) = ab^n$ where a is the zeroth term, $b$ is the common ratio, and $n$ is the term number. In this case, $b = 2$, since the bacteria population is doubling in area each day.

3. If $100$% of the Petri dish is covered after $12$ days have passed, what percentage was covered on the starting day? Use your equation from part (b).

Substitute $100$ in for $t(n)$ and solve for $a$.

4. How much of the Petri dish was covered on February $14$?

• February $14$ is day $5$ relative to the zeroth day.
Evaluate the geometric sequence when $n = 5$.