Question 8·SAT Desmos Regression Practice·Advanced Math

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A culture of cells is modeled by the exponential function $N(t)=A\cdot b^t$ , where $t$ is the number of hours since the start of an experiment.

The model predicts that $N(2)=1{,}000$ and $N(5)=1{,}331$ .

Which choice is the value of $A$ ?

In an exponential model, the coefficient before the power is the starting value. If you have two time-output pairs but no value at time zero, enter the pairs in Desmos and fit the given exponential form. Read the fitted coefficient, not the growth factor. The first count listed is not necessarily the starting count.

Hints

Hint 1
In an exponential model, a number raised to the power $0$ equals $1$ . At the start of the experiment, what does that make the coefficient in front of $b^t$ equal?
Hint 2
A regression finds unknown numbers in a model from input-output pairs. Put hours in the input column and the predicted cell counts beside them, then fit the exponential form given in the question.
Hint 3
Desmos calls its fitted coefficient $a$ and its hourly multiplier $b$ . The problem calls that coefficient $A$ , so compare $a$ , not $b$ , with the answer choices.

Step-by-step

Approach 1: Fit the exponential model in Desmos
Step 1
Identify what $A$ means
At the start of the experiment, $t=0$ . Any nonzero number raised to the power $0$ is $1$ , so:
$N(0)=A\cdot b^0=A.$
The initial value is the count at time $0$ . So $A$ is the starting cell count, not the count given for hour $2$ .
Step 2
Enter the two predictions as pairs
Time is the model's input, and cell count is its output. Enter the pairs $(2,1000)$ and $(5,1331)$ in a Desmos table, with hours in $x_1$ and counts in $y_1$ . Keep each time beside its count so Desmos can fit the model to both predictions.
Step 3
Fit the starting count and match it
Type $y_1\sim ab^{x_1}$ . The tilde ( $\sim$ ) tells Desmos to fit $a$ and $b$ to the two rows. Under PARAMETERS, it shows $a\approx826.44628$ and $b=1.1$ . Here $a$ is the problem's $A$ . Type $\frac{100000}{121}$ on the next line; Desmos prints the same $826.44628\ldots$ . The model's starting cell count is $\frac{100000}{121}$ . Choice C.

Approach 2: Find the hourly factor first
Step 1
Write what each prediction means
Putting each given time into $N(t)=A\cdot b^t$ gives the two equations:
$A b^2=1000$ $A b^5=1331.$
They share the same starting count $A$ , which lets you remove it by dividing.
Step 2
Remove the unknown starting count
Divide the hour- $5$ equation by the hour- $2$ equation:
$\frac{1331}{1000}=\frac{A b^5}{A b^2}.$
Cancel the shared $A$ :
$\frac{1331}{1000}=\frac{b^5}{b^2}.$
Subtract exponents when dividing powers with the same base:
$\frac{1331}{1000}=b^{5-2}.$
Simplify the three-hour gap:
$\frac{1331}{1000}=b^3.$
So the change from hour $2$ to hour $5$ applies the hourly factor three times.
Step 3
Find the hourly factor
Type $\frac{1331}{1000}\sim b^3$ . The tilde asks Desmos to find the hourly multiplier $b$ ; under PARAMETERS, it shows $b=1.1$ .
Step 4
Back up from hour $2$
Hour $2$ includes two hourly multiplications after the start. Divide $A b^2=1000$ by $b^2$ to isolate the starting count:
$A=\frac{1000}{b^2}.$
Step 5
Calculate the starting count
Add $\frac{1000}{b^2}$ in Desmos. It shows $826.44628\ldots$ ; its fraction button gives $\frac{100000}{121}$ . So the model predicts a starting cell count of $\frac{100000}{121}$ , the same answer found by fitting the model. Choice C.

Question 8·SAT Desmos Regression Practice·Advanced Math

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