Question textA glass of lemonade is observed over time through photos and the number of bubbles is noted by taking photographs of the glass every two hours. The number of bubbles added over time diminishes, which means the rate of increase of cumulative bubbles diminishes as well.A saturation-growth model is fitted to the data, where y is the cumulative total number of bubbles and t is time in hours: $$ y(t)\ =\ \frac{1351t}{(t+7.75)} $$To two decimal places, when does the cumulative total of observed bubbles reach 1248? Answer 1 Question 1[input] hours

Question

BlackTom AI · Accepted Answer

First, restate the problem in my own words: we are given a saturation-growth model y(t) = 1351 t / (t + 7.75), where y is the cumulative number of bubbles and t is time in hours. We want the time t when y(t) reaches 1248, to two decimal places.

Step 1: Set up the equation for y = 1248.
1248 = 1351 t / (t + 7.75).
To solve for t, cross-multiply:
1248 (t + 7.75) = 1351 t.
This expands to 1248t + 1248 × 7.75 = 1351t.
Calculate 1248 × 7.75: 1248 × 7 = 8736 and 1248 × 0.75 = 936, so the product is 8736 + 936 = 9672.
So the equation is 1248t + 9672 = 1351t.

Step 2: Solve for t.
Subtract 1248t from both sides: 9672 = (1351 − 1248)t = 103t.
Therefore t = 9672 / 103.

Step 3: Compute the quotient to needed precision.
Perform the division: 9672 ÷ 103 ≈ 93.90 hours (more precisely ~93.902...), so to two decimal places, t ≈ 93.90 hours.

Step 4: Interpret the result and check plausibility.
The model is a saturating growth function that approaches a finite asymptote as t grows; reaching 1248 bubbles at about 93.90 hours is consistent with slow, diminishing increments over time. Since 1248 is less than the potential asymptotic total, a positive finite time is expected.

Final note on the answer format: the calculation yields t ≈ 93.90 hours when y(t) = 1248, which matches the provided numeric result.

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