Question 35·SAT Desmos Regression Practice·Advanced Math

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The function

P(t)=ab^{t/8}

models the population, in thousands, of a laboratory culture $t$ hours after observations begin, where $a$ and $b$ are positive constants and $b>1$ . Measurements show that $P(12)=100$ and $P(36)=2700$ .

According to the model, after how many hours will the population be 8,000% greater than it was at $t=0$ ?

Your answer

The form $P(t)=ab^{t/8}$ signals exponential growth: the population is multiplied by the same factor every $8$ hours. Fit both measurements to the given model with one Desmos table regression, then solve for the time when the population reaches the requested multiple of its starting value. Greater than includes the starting amount as well as the increase.

Hints

Hint 1
Each measurement is an input-output pair: hours are inputs, and populations in thousands are outputs. Put both pairs in one Desmos table so the regression fits the same $a$ and $b$ to both.
Hint 2
At $t=0$ , the exponent is $0$ , so the starting population is $a$ . The percent increase is measured from that value, not from the population recorded at hour $12$ .
Hint 3
An $8,000\%$ increase adds $80$ copies of the starting population to the one copy already there. What equation says the model has reached $81$ times its starting value?

Step-by-step

Approach 1: Fit the model, then solve for time
Step 1
Turn the measurements into model equations
Put each given hour into $t$ and match the model to its measured population: $ab^{12/8}=100,\qquad ab^{36/8}=2700$ . Both equations use the same starting amount and growth factor because they describe one culture.
Step 2
Find the eight-hour growth factor
Enter $(12,100)$ and $(36,2700)$ in a table, with $x_1$ for hours and $y_1$ for population in thousands. Type $y_1\sim ab^{x_1/8}$ to fit the given model to both rows. Under PARAMETERS, Desmos shows $b=3$ and $a\approx19.245$ . So the population triples every $8$ hours.
Step 3
Identify the starting population
At $t=0$ , the exponent is $0$ , and a positive number to the zeroth power is $1$ . So the starting population is $P(0)=ab^{0/8}=a$ .
Step 4
Convert the increase into a target
Type $1+8000/100$ ; Desmos shows $81$ . The $8,000\%$ increase adds $80$ starting populations to the original one, so the target is $P(t)=81a$ , not $80a$ .
Step 5
Match the model to the target
The population must equal $81a$ at the requested time, so set the model equal to that target: $ab^{t/8}=81a$ .
Step 6
Remove the starting amount
Because $a$ is positive, divide both sides by $a$ to leave only the growth factor: $b^{t/8}=81$ .
Step 7
Solve for the number of hours
Use $b=3$ from the fit. Type $3^{t/8}\sim81$ ; the tilde asks Desmos to solve for the unknown time $t$ . Under PARAMETERS, it shows $t=32$ . The culture reaches the requested increase after $32$ hours. Grid in 32.

Approach 2: Use the ratio to see why the factor is 3
Step 1
Divide the two model outputs
Divide their model outputs so the shared $a$ cancels: $\frac{P(36)}{P(12)}=\frac{ab^{36/8}}{ab^{12/8}}$ .
Step 2
Simplify the ratio
Subtract the exponents when dividing powers with the same base: $\frac{P(36)}{P(12)}=b^{(36-12)/8}$ .
Step 3
Count the growth intervals
The measurements are $36-12=24$ hours apart. Type $(36-12)/8$ to count the eight-hour growth intervals; Desmos shows $3$ . So the ratio is $\frac{P(36)}{P(12)}=b^3$ .
Step 4
Substitute the measurements
Substitute the measured populations: $b^3=\frac{2700}{100}$ .
Step 5
Find the ratio's value
Divide the measured populations: $b^3=27$ .
Step 6
Find the growth factor
Since $27=3^3$ and $b>1$ , the growth factor is $b=3$ .
Step 7
Match the model to the target
The target is $81$ times the starting population, so set the growth factor's power equal to $81$ : $3^{t/8}=81$ .
Step 8
Write both sides as powers of 3
Rewrite $81$ as a power of $3$ : $3^{t/8}=3^4$ .
Step 9
Match the exponents
Equal powers of the same base greater than $1$ have equal exponents: $\frac{t}{8}=4$ .
Step 10
Convert intervals to hours
Multiply by $8$ to get the number of hours: $t=32$ . So the increase occurs after $32$ hours. Grid in 32.

Question 35·SAT Desmos Regression Practice·Advanced Math

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