How many grams of mold were there in my fridge at 5:00 AM this morning, given that it grew from 0.4 grams at noon to 1.6 grams by 9:00 PM? Please round your answer to three decimal places and enter only the number without units or additional text. If the number is less than 1, include a leading 0.

Mathematics Questions

How many grams of mold were there in my fridge at 5:00 AM this morning, given that it grew from 0.4 grams at noon to 1.6 grams by 9:00 PM? Please round your answer to three decimal places and enter only the number without units or additional text. If the number is less than 1, include a leading 0.

Short Answer

To solve an exponential growth problem, use the formula M(t) = M‚ÄöCA e^(kt), where M‚ÄöCA is the initial amount and k is the growth constant. By calculating k using given values from the problem, you can estimate the amount of a substance at different times, concluding that the amount of mould at 5 AM is approximately 0.136 g.

Step-by-Step Solution

Step 1: Understand the Exponential Growth Formula

In problems involving exponential growth, the amount of a substance increases over time based on its current amount. The general formula used is:

M(t) = M₀ e^kt
M(t) represents the amount at time t.
M₀ is the initial amount.
k is the growth constant and t is time elapsed.

Step 2: Calculate the Growth Constant (k)

Using the information provided, we can calculate the growth constant k by substituting known values into the formula. From the problem:

At noon, M₀ = 0.4 g.
At 9 PM (t = 9), M(9) = 1.6 g.

This gives the equation:

1.6 = 0.4 e^9k
Solving leads to: k = ln(4)/9.

Step 3: Find the Amount of Mould at 5 AM (M(-7))

To find the amount of mould at 5 AM (which corresponds to t = -7), we substitute k back into the formula:

M(-7) = 0.4 e^-7k.
Replace k with ln(4)/9 to get:
M(-7) = 0.4 e^{-7 * (ln(4)/9)}.

This results in:

M(-7) = 0.136079 g.
Thus, the amount of mould at 5:00 AM was approximately 0.136 g.

How many grams of mold were there in my fridge …