Compute the relative error in approximating with 4 and express the result as apercentage.© 4.024994e-4 © 4.024994 x 10-2 O 4.024994e7? © 4.024994 x 107!

To compute the relative error in approximating a value, we can follow the steps outlined below.

1. ### Break Down the Problem

We need to calculate the relative error for the given values and express it as a percentage. The formula for relative error is given by:

$\text{Relative Error} = \left| \frac{\text{True Value} - \text{Approximate Value}}{\text{True Value}} \right| \times 100\%$

2. ### Relevant Concepts

We have to define the true value and the approximate value based on the provided options. The question mentions four values, but we need to identify which one is the true value and which one is the approximation.

True value:

Assuming that $4.024994 \times 10^{-4} = 0.0004024994$ is to be considered as the true value.

Approximate value:

We will analyze each of the other three numbers to see which one we will approximate to the true value.

3. ### Analysis and Detail

Let’s proceed with the calculations using $4.024994e-2 = 0.04024994$, $4.024994e7 = 40249940$, and $4.024994 \times 10^7 = 40249940$ as approximate values.

1. Approximation 1: $4.024994 \times 10^{-2}$

   $\text{Relative Error} = \left| \frac{0.0004024994 - 0.04024994}{0.0004024994} \right| \times 100\%$

   Calculate the difference:

   $|0.0004024994 - 0.04024994| \approx 0.04024944$

   Now compute the relative error:

   $\text{Relative Error} \approx \left| \frac{0.04024944}{0.0004024994} \right| \times 100\% \approx 9999.76\%$

2. Approximation 2: $4.024994 \times 10^{7}$

   $\text{Relative Error} = \left| \frac{0.0004024994 - 40249940}{0.0004024994} \right| \times 100\%$

   The difference is:

   $|0.0004024994 - 40249940| \approx 40249940$

   Now compute the relative error:

   $\text{Relative Error} \approx \left| \frac{40249940}{0.0004024994} \right| \times 100\% \approx 9999999990.90\%$

4. ### Verify and Summarize

After evaluating both approximations, it is clear that both approximations are extremely far from the true value. Therefore, we see that relative error percentages are quite high, indicating a poor approximation.

Final Answer

1. The relative error for $4.024994 \times 10^{-2}$ is approximately 9999.76%.
2. The relative error for $4.024994 \times 10^{7}$ is approximately 9999999990.90%.