Question 546·Medium·Math

0 / 2000

A microbiologist observes that a bacterial culture contains $2,400$ cells at time $t = 0$ hours. Eight hours later, the culture contains $9,600$ cells. The growth of the culture is modeled by an exponential function $N$ , where $N(t)$ is the number of cells $t$ hours after the initial observation.

Which equation could approximately define $N$ , with the hourly growth factor rounded to the nearest hundredth?

In an exponential model, the count at time zero is the starting coefficient, and the base is the factor for one hour. Use the later observation to write an equation for that factor, then find it with a Desmos regression and round. Don't confuse a growth percent with the multiplier applied to the whole count.

Hints

Hint 1
In $N(t)=ab^t$ , the starting value is $a$ : at time zero, $b^0=1$ . Which observed cell count should go in front of the power?
Hint 2
An hourly growth factor multiplies the count once per hour. Over eight hours, it acts eight times. What power of that factor connects the starting count to the later count?
Hint 3
A regression lets Desmos find an unknown in an equation. Replace $=$ with $\sim$ and require a positive factor; an eighth-power equation can also have a negative solution.

Step-by-step

Find the hourly factor
Step 1
Write the model from the starting count
Let $b$ be the hourly growth factor, the number the count is multiplied by each hour. An exponential model has the form $N(t)=ab^t$ . At $t=0$ , $b^0=1$ , so the starting count gives $a=2{,}400$ and $N(t)=2{,}400b^t$ .
Step 2
Connect the two observations
After $8$ hours, the count has been multiplied by $b$ eight times. The hourly factor's eighth power must produce the full eight-hour change. Since the later count is $9{,}600$ , write: $2{,}400b^8=9{,}600$ .
Step 3
Find the positive factor in Desmos
Type $2{,}400b^8\sim9{,}600\{b>0\}$ . The $\sim$ tells Desmos to fit $b$ ; the restriction keeps the positive factor needed for a cell count. Under PARAMETERS, Desmos shows $b\approx1.189207$ .
Step 4
Round and choose the model
Round $b\approx1.189207$ to the nearest hundredth to get $1.19$ . So the approximate model is $N(t)=2{,}400(1.19)^t$ . Choice D.

Question 546·Medium·Math

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this