Finding the time given an exponential function with base e that... The number of bacteria $$ P(h) $$ in a certain population increases according to the following function, where time $$ h $$ is measured in hours. $$ P(h)=1700 e^{0.19 h} $$ How many hours will it take for the number of bacteria to reach 2400 ? Round your answer to the nearest tenth, and do not round any intermediate computations. $$ \square $$ hours

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To find the time $$ h $$ it takes for the number of bacteria $$ P(h) $$ to reach 2400, we start with the given exponential function:

$$
P(h) = 1700 e^{0.19 h}
$$

We need to set $$ P(h) $$ equal to 2400 and solve for $$ h $$:

$$
1700 e^{0.19 h} = 2400
$$

### Step 1: Isolate the exponential term

First, we divide both sides by 1700:

$$
e^{0.19 h} = \frac{2400}{1700}
$$

Calculating the right side:

$$
\frac{2400}{1700} = \frac{24}{17} \approx 1.411764705882353
$$

### Step 2: Take the natural logarithm of both sides

Next, we take the natural logarithm (ln) of both sides:

$$
\ln(e^{0.19 h}) = \ln\left(\frac{24}{17}\right)
$$

Using the property of logarithms that $$ \ln(e^x) = x $$:

$$
0.19 h = \ln\left(\frac{24}{17}\right)
$$

### Step 3: Solve for $$ h $$

Now, we can solve for $$ h $$:

$$
h = \frac{\ln\left(\frac{24}{17}\right)}{0.19}
$$

### Step 4: Calculate the value of $$ h $$

Now, we will calculate $$ h $$ using the natural logarithm:

$$
h = \frac{\ln\left(\frac{24}{17}\right)}{0.19}
$$

Let's compute this value.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\ln{\left(\frac{24}{17}\right)}}{0.19}$$
- step1: Rewrite the expression:
  $$\frac{\ln{\left(\frac{24}{17}\right)}}{\frac{19}{100}}$$
- step2: Multiply by the reciprocal:
  $$\ln{\left(\frac{24}{17}\right)}\times \frac{100}{19}$$
- step3: Multiply the terms:
  $$\frac{\ln{\left(\frac{24}{17}\right)}\times 100}{19}$$
- step4: Multiply the terms:
  $$\frac{100\ln{\left(\frac{24}{17}\right)}}{19}$$
Round the number $$ 1.81495 $$ to nearest tenth.
Round the number by following steps:
- step0: Round $$1.81495$$ to $$1$$ decimal places using standard rounding method:
  $$1.81495$$
- step1: Round $$1.81495$$ to $$1$$ decimal places using standard rounding method$$:$$
  $$1.8$$
After calculating, we find that:

$$
h \approx 1.81495
$$

Rounding this to the nearest tenth, we get:

$$
h \approx 1.8
$$

Thus, it will take approximately $$ \boxed{1.8} $$ hours for the number of bacteria to reach 2400.
It will take approximately 1.8 hours for the number of bacteria to reach 2400.

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