e) The number of bacteria cells $$ (N) $$ in a culture, $$ t $$ hours after the start of a scientific experiment, is given by the formula $$ N=a \cdot 3^{k t} $$. The number of bacteria in the culture triples every 4 hours. Exactly 12 hours after the experiment started, there are 81000 bacteria cells in the culture. (1) Determine the value of $$ k $$. (2) If $$ k=\frac{1}{4} $$, determine the value of $$ a $$. It is now given that $$ a=3000 $$ and $$ k=\frac{1}{4} $$. (3) How many bacteria cells were present in the culture at the beginning of the (4) How many bacteria cells will be present in the culture 24 hours after the start of the (5) How long after the start of the experiment will there be 10000000 bacteria cells in (f) The pH of a solution is given by the formula pH $$ \mathrm{H}=-\log C $$, where $$ C $$ is the concentration of $$ \mathrm{H}^{+} $$ions in the solution (in moles per litre). (1) In a certain soft drink, the concentration of $$ \mathrm{H}^{+} $$ions is 0,000004 moles per litre. (2) The pH of white vinegar is 2,4 . What is the concentration of $$ \mathrm{H}^{+} $$ions in white (2)

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Let's break down the problem step by step.

### Part e

#### (1) Determine the value of $$ k $$.

We know that the number of bacteria triples every 4 hours. This means that if we let $$ t = 4 $$, then:

$$
N(4) = a \cdot 3^{k \cdot 4} = 3a
$$

This implies:

$$
3^{k \cdot 4} = 3 \implies k \cdot 4 = 1 \implies k = \frac{1}{4}
$$

#### (2) If $$ k = \frac{1}{4} $$, determine the value of $$ a $$.

We are given that after 12 hours, there are 81000 bacteria cells:

$$
N(12) = a \cdot 3^{\frac{1}{4} \cdot 12} = a \cdot 3^3 = a \cdot 27
$$

Setting this equal to 81000:

$$
a \cdot 27 = 81000 \implies a = \frac{81000}{27}
$$

Now, let's calculate $$ a $$:

$$
a = 3000
$$

#### (3) How many bacteria cells were present in the culture at the beginning of the experiment?

At the beginning of the experiment ($$ t = 0 $$):

$$
N(0) = a \cdot 3^{k \cdot 0} = a \cdot 1 = a
$$

Since we found $$ a = 3000 $$:

$$
N(0) = 3000
$$

#### (4) How many bacteria cells will be present in the culture 24 hours after the start of the experiment?

Using the formula:

$$
N(24) = a \cdot 3^{k \cdot 24} = 3000 \cdot 3^{\frac{1}{4} \cdot 24} = 3000 \cdot 3^6
$$

Now, let's calculate $$ N(24) $$:

$$
N(24) = 3000 \cdot 729
$$

Calculating this:

$$
N(24) = 2187000
$$

#### (5) How long after the start of the experiment will there be 10000000 bacteria cells in the culture?

We need to solve for $$ t $$ in the equation:

$$
10000000 = 3000 \cdot 3^{\frac{1}{4} t}
$$

Dividing both sides by 3000:

$$
\frac{10000000}{3000} = 3^{\frac{1}{4} t}
$$

Calculating the left side:

$$
\frac{10000000}{3000} = 3333.33
$$

Now we take the logarithm of both sides:

$$
\log(3333.33) = \frac{1}{4} t \cdot \log(3)
$$

Now, solving for $$ t $$:

$$
t = 4 \cdot \frac{\log(3333.33)}{\log(3)}
$$

Let's calculate $$ t $$:

$$
t \approx 4 \cdot \frac{3.5229}{0.4771} \approx 29.5
$$

### Part f

#### (1) In a certain soft drink, the concentration of $$ \mathrm{H}^{+} $$ ions is 0.000004 moles per litre.

Using the formula for pH:

$$
\text{pH} = -\log(0.000004)
$$

Calculating this:

$$
\text{pH} = -\log(4 \times 10^{-6}) = 6 - \log(4) \approx 6 - 0.602 = 5.398
$$

#### (2) The pH of white vinegar is 2.4. What is the concentration of $$ \mathrm{H}^{+} $$ ions in white vinegar?

Using the formula:

$$
\text{pH} = -\log C \implies C = 10^{-\text{pH}} = 10^{-2.4}
$$

Calculating this:

$$
C \approx 0.00398 \text{ moles per litre}
$$

### Summary of Results
1. $$ k = \frac{1}{4} $$
2. $$ a = 3000 $$
3. Initial bacteria count: $$ 3000 $$
4. Bacteria count after 24 hours: $$ 2187000 $$
5. Time to reach 10,000,000 bacteria: $$ \approx 29.5 $$ hours
6. pH of soft drink: $$ \approx 5.398 $$
7. Concentration of $$ \mathrm{H}^{+} $$ ions in vinegar: $$ \approx 0.00398 $$ moles per litre
1. $$ k = \frac{1}{4} $$ 2. $$ a = 3000 $$ 3. Initial bacteria count: 3000 4. Bacteria count after 24 hours: 2,187,000 5. Time to reach 10,000,000 bacteria: approximately 29.5 hours 6. pH of soft drink: approximately 5.398 7. Concentration of $$ \mathrm{H}^{+} $$ ions in vinegar: approximately 0.00398 moles per litre

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