A synthetic culture model has Q(t)=30·2^(t/5), where t is elapsed days and Q is the modeled quantity. A technician records one value at the end of each whole day. When does the model first reach 100, and on which report does the technician first record at least 100? These are two different questions even when the same formula answers both.
Predict
At day 5 the quantity is 60 and at day 10 it is 120. Predict a threshold time between those days. Would rounding to the nearest day reliably produce the first report meeting a threshold? Consider a threshold at day 8.1: day 8 would still be too early. Also predict what changes if the inequality is strictly above rather than at least.
Experiment
Make the following decision table, retaining the exact exponential expressions until your final comparison. Use a calculator for the approximate column, then compare each result with 100.
Time in days
Exact modeled quantity
Approximate quantity
8
30·2^(8/5)
90.943
5 log₂(10/3)
100
100
9
30·2^(9/5)
104.466
The first row fails, the middle row is the continuous threshold, and the last row succeeds. For this increasing model, those neighboring checks justify the first whole-day report. They do not establish the model’s biological accuracy.
Understand
Divide the target equation by 30: 2^(t/5)=10/3. A logarithm answers the exponent question. By definition, log₂(10/3) is the exponent that makes base 2 equal 10/3. Therefore t=5 log₂(10/3)≈8.684828 days. A calculator can use ln(10/3)/ln(2); changing the logarithm base consistently does not change the answer.
More generally, a b^(ct)=d gives t=ln(d/a)/(c ln b), provided a is positive, d is positive, b is positive and different from 1, and c is nonzero. Here c has units inverse to the units of t, so the exponent ct is dimensionless. Check the resulting time against the allowed domain, usually t≥0. A negative algebraic answer can be outside the modeled time period.
For an increasing model and a target met at t₀, the first integer report at or above the target is the ceiling of t₀: the smallest integer at least t₀. However, a strictly-above requirement needs the next integer when t₀ itself is an integer. For decreasing models, solve the inequality carefully: dividing by a negative logarithm reverses its direction. A positive exponential approaches zero but never reaches it at a finite time, so taking the logarithm of zero is not a valid shortcut.
Apply
For a fresh increasing model, first solve the equality, then write the inequality and the reporting schedule. Check the proposed integer and its predecessor using the original formula. For decay, interpret “at most” as a falling threshold and again verify neighboring reports. Keep enough calculator precision to avoid accidentally moving a value across an integer boundary; an exact equality check is preferable when available.
Continue
Write a decision statement containing the model, time units, continuous crossing, discrete report, inequality sign and neighboring checks. A rounded number without this context is incomplete evidence. The following graph-construction lessons revisit how formulas, marked features and contextual domains constrain a whole graph, beyond one target time.
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
A model has Q(t)=40·2^(t/3), t in days. What is the first whole-number day n≥0 with Q(n)≥150?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A model has Q(t)=800·(1/2)^(t/4), t in hours. At what real time does Q(t)=100?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For Q(t)=2·10^(0.3t), solve Q(t)=9 and find the first integer time at or above 9. Supply neighboring checks.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A concentration is C(t)=100e^(−0.2t) for t≥0 hours. Solve C(t)≤30 and find the first whole-hour report satisfying it.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For Q(n)=10·2^n at integer n≥0, find the last n with Q(n)<80, the last n with Q(n)≤80, and the first n with Q(n)>80. Explain why blindly rounding fails.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A decaying model Q(t)=60·(0.8)^t has t≥0. Decide whether targets 90, 0 and 15 are reached at finite allowed times; write the valid target-time formula.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
