The mass of a substance, which follows a continuous exponential growth model, is being studied in a lab. The doubling time for this substance was observed to be 17 minutes. There were 8.7 mg of the substance present at the beginning of the study.

(a) Let [tex]$t$[/tex] be the time (in minutes) since the beginning of the study, and let [tex]$y$[/tex] be the amount of the substance at time [tex]$t$[/tex]. Write a formula relating [tex]$y$[/tex] to [tex]$t$[/tex]. Use exact expressions to fill in the missing parts of the formula. Do not use approximations.
[tex]$
y = 8.7 e^{\left( \frac{\ln(2)}{17} \right) t}
$[/tex]

(b) How much will be present in 11 minutes? Do not round any intermediate computations, and round your answer to the nearest tenth.
[tex]$ \square \text{ mg} $[/tex]



Answer :

Certainly! Let's proceed step by step:

### Part (a)
We start by establishing the formula for the exponential growth of the substance:

1. Exponential Growth Model:
The basic form of the exponential growth model is given by:
[tex]\[ y = y_0 e^{kt} \][/tex]
where:
- [tex]\( y \)[/tex] is the amount of the substance at time [tex]\( t \)[/tex],
- [tex]\( y_0 \)[/tex] is the initial amount of the substance,
- [tex]\( k \)[/tex] is the growth rate constant,
- [tex]\( t \)[/tex] is the time.

2. Doubling Time:
The doubling time is the time it takes for the substance to double in quantity. Given that the doubling time is 17 minutes, we can use this information to find [tex]\( k \)[/tex]. We set up the equation based on the doubling condition:
[tex]\[ 2 = e^{k \cdot 17} \][/tex]

3. Solving for [tex]\( k \)[/tex]:
To solve for [tex]\( k \)[/tex], take the natural logarithm (ln) on both sides of the equation:
[tex]\[ \ln(2) = k \cdot 17 \][/tex]
[tex]\[ k = \frac{\ln(2)}{17} \][/tex]
Substitute this back into the exponential growth formula:
[tex]\[ y = y_0 e^{\left(\frac{\ln(2)}{17}\right) t} \][/tex]

4. Initial Amount:
The initial amount [tex]\( y_0 \)[/tex] is 8.7 mg. Substituting [tex]\( y_0 \)[/tex] into the formula, we get:
[tex]\[ y = 8.7 e^{\left(\frac{\ln(2)}{17}\right) t} \][/tex]

So, the complete formula is:
[tex]\[ y = 8.7 e^{\left(\frac{\ln(2)}{17}\right) t} \][/tex]

### Part (b)
Next, we need to find the amount of the substance present at [tex]\( t = 11 \)[/tex] minutes:

1. Using the Formula:
Substitute [tex]\( t = 11 \)[/tex] minutes into the formula:
[tex]\[ y = 8.7 e^{\left(\frac{\ln(2)}{17}\right) \cdot 11} \][/tex]

2. Calculation:
Compute the exponent:
[tex]\[ \frac{\ln(2)}{17} \cdot 11 \approx \frac{0.6931471805599453}{17} \cdot 11 \approx 0.448210 \][/tex]

Substituting this value back into the formula, we get:
[tex]\[ y = 8.7 \cdot e^{0.448210} \][/tex]

Using [tex]\( e^{0.448210} \approx 1.565 \)[/tex]:
[tex]\[ y \approx 8.7 \cdot 1.565 \approx 13.6 \][/tex]

So, the amount of the substance present at 11 minutes is approximately:
[tex]\[ 13.6 \text{ mg} \][/tex]

In conclusion:
- (a) The formula relating [tex]\( y \)[/tex] to [tex]\( t \)[/tex] is [tex]\( y = 8.7 e^{\left(\frac{\ln(2)}{17}\right) t} \)[/tex].
- (b) The amount present in 11 minutes is approximately [tex]\( 13.6 \)[/tex] mg.