Question 402·Medium·Math

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The number of bacteria in a culture $t$ hours after the culture is started is modeled by

P(t)=500\cdot 3^{t/4}

At what time, in hours, will the culture first contain exactly 13,500 bacteria?

An exponential model multiplies by the same factor over equal time intervals. To find when it reaches a target count, set the model equal to that count and use a Desmos regression to find the unknown time. Read the exponent before choosing: it counts intervals, not individual hours, so one multiplication need not happen after one hour.

Hints

Hint 1
In a target-value equation, the count you want is the output, $P(t)$ , while $t$ is the time you need. What equation says that the output equals the requested count?
Hint 2
The exponent $t/4$ counts four-hour stretches. Each time it increases by $1$ , the culture is multiplied by $3$ . Watch the four-hour interval rather than treating each hour as a full multiplication step.
Hint 3
In a Desmos regression, write the unknown time as $t_1$ and use $\sim$ instead of $=$ . Desmos lists the value it finds under PARAMETERS. Which value answers the question?

Step-by-step

Approach 1: Find the time with Desmos
Step 1
Turn the target count into an equation
The culture contains exactly 13,500 bacteria when the model's output equals 13,500, so write:
$500\cdot3^{t/4}=13{,}500.$
Step 2
Read the time interval in the exponent
The exponent $t/4$ increases by $1$ every $4$ hours, so the culture is multiplied by $3$ once every four hours, not every hour. Because $3>1$ , the population keeps growing and can reach the target only once.
Step 3
Solve for the time
Type $500\cdot3^{t_1/4}\sim13500$ in Desmos. This regression uses $t_1$ for the unknown time and $\sim$ to ask Desmos to find a match. Under PARAMETERS, Desmos shows $t_1=12$ . That's 12 hours, so the culture first contains exactly 13,500 bacteria after 12 hours. Choice C.

Approach 2: Match powers of 3
Step 1
Isolate the power
Divide both sides of the target equation by the starting count, $500$ :
$\frac{500\cdot3^{t/4}}{500}=\frac{13{,}500}{500}$
Simplify:
$3^{t/4}=27.$
Step 2
Give both sides the same base
Since $27=3^3$ , rewrite the right side as a power of $3$ . Now both sides have the same base:
$3^{t/4}=3^3.$
Step 3
Set the exponents equal
Powers of $3$ can be equal only if their exponents are equal: a greater exponent makes a greater power of $3$ . So set the exponents equal:
$\frac{t}{4}=3.$
Step 4
Convert intervals to hours
Multiply both sides by $4$ to turn the three four-hour intervals into hours:
$\frac{t}{4}\cdot4=3\cdot4$
Simplify:
$t=12.$
The culture first contains exactly 13,500 bacteria after 12 hours. Choice C.

Question 402·Medium·Math

Strategy

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