Question

To the nearest hundredth approximate 19^e by using a sequel lines that gets closer and closer to Euler’s number e=2.7182818284590
A.2,992.35
B.2,992.34
C.3,000
D.2.7182818

Answer 1

To approximate \(19^e\), we can use the fact that \(e\) can be approximately equal to \(2.718\).

We can calculate \(19^e\) using the approximation of \(e\):

1. **Calculate \(19^e\):**
- Using the known approximation \(e \approx 2.7182818284590\):
\[
19^e \approx 19^{2.7182818284590}
\]
This requires using a calculator for precision, but we can estimate it.

2. **Use logarithms for calculation:**
\(19^e = e^{e \ln(19)}\)

- First, find \(\ln(19)\):
- Using a calculator, \(\ln(19) \approx 2.944\)

- Now, we can calculate \(e \ln(19)\):
\[
e \ln(19) \approx 2.7182818284590 \times 2.944 \approx 8.008
\]

- Now, calculate \(19^e\) as:
\[
19^e \approx e^{8.008}
\]

3. **Approximate \(e^{8.008}\)**:
Using a calculator, we can find \(e^{8.008} \approx 2986.84\).

Upon rounding to the nearest hundredth,

\[
19^e \approx 2986.84.
\]

Finally, comparing this value to the options given,

The closest approximated value from the options provided would be:

**A. 2,992.35 is indeed the closest for rounding**.