Question 143·200 Super-Hard SAT Math Questions·Advanced Math

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A video channel had 12,000 subscribers 3 days ago. The number of subscribers grows exponentially and triples every 40 hours.

Which choice gives a function $S$ that models the number of subscribers $w$ weeks from now, written in the form $S(w)=k\cdot 9^{mw}$ , where $k$ and $m$ are constants?

In an exponential model, the exponent counts how many growth intervals have passed. If the given count is from the past but the input measures time from now, first put both stretches of time on one clock. Then write the model using the given multiplier and rewrite it in the requested base. Changing the base also changes the exponent.

Hints

Hint 1
The count was measured before now, so the time from that measurement to $w$ weeks from now includes both stretches. How many hours have passed by the time the input is $w$ ?
Hint 2
An exponent counts applications of the multiplier. If subscribers triple every $40$ hours, divide the total hours since the old count by $40$ to find the exponent on $3$ .
Hint 3
Since $9=3^2$ , one multiplication by $9$ equals two multiplications by $3$ . To keep the same weekly growth while changing the base to $9$ , what must happen to the weekly exponent?

Step-by-step

Build the timeline, then change the base
Step 1
Count hours from the old measurement
The $3$ days ago are before $w=0$ , so add them to the future time. Each day has $24$ hours, and each week has $7$ days. The hours since the $12{,}000$ -subscriber count are
$t=3(24)+w(7)(24)=72+168w.$
Step 2
Turn the growth rule into a model
The count triples every $40$ hours, so the number of tripling intervals is the elapsed hours divided by $40$ . Starting from the old count gives
$S(w)=12{,}000\cdot3^{\frac{72+168w}{40}}.$
Step 3
Check what the model gives now
In Desmos, use $x$ for the number of weeks: type $f(x)=12{,}000\cdot3^{(72+168x)/40}$ , then $f(0)$ . Desmos prints about $86{,}696$ subscribers now. So $12{,}000$ is the count from three days ago, not the value at $w=0$ .
Step 4
Separate past growth from future growth
To see the starting factor and the weekly growth separately, split the fraction across the sum:
$\frac{72+168w}{40}=\frac{72}{40}+\frac{168w}{40}.$
Reduce both fractions:
$\frac{72}{40}+\frac{168w}{40}=\frac95+\frac{21}{5}w.$
Step 5
Pull out the growth before now
When powers with the same base multiply, their exponents add. Read that rule backward to make the growth before now a separate factor:
$S(w)=12{,}000\cdot3^{9/5}\cdot3^{\frac{21}{5}w}.$
Step 6
Rewrite the weekly growth using base 9
Because $9=3^2$ , halve the exponent when changing a power of $3$ to a power of $9$ :
$3^{\frac{21}{5}w}=9^{\frac{21}{10}w}.$
So the requested model is $S(w)=12{,}000\cdot3^{9/5}\cdot9^{\frac{21}{10}w}$ . Choice A.

Question 143·200 Super-Hard SAT Math Questions·Advanced Math

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