Let's solve part (d) using the linear approximation found in part (c).

The linear approximation formula provided is: $f(A) \approx 1.7320508076 + \frac{\pi}{3} \cdot (A - \frac{\pi}{4})$

Here, we want to find the approximate value of $B$ when $A = 46^\circ$. First, we need to convert $A = 46^\circ$ into radians since the formula uses radians. The conversion from degrees to radians is: $A = 46^\circ \times \frac{\pi}{180} = \frac{46\pi}{180} \approx 0.8028514559 \text{ radians}$

Now, we substitute $A = 0.8028514559$ into the linear approximation formula: $B \approx 1.7320508076 + \frac{\pi}{3} \cdot (0.8028514559 - \frac{\pi}{4})$

Let's calculate this value step-by-step:

1. Calculate $\frac{\pi}{4} \approx 0.7853981634$.
2. Compute $0.8028514559 - 0.7853981634 = 0.0174532925$.
3. Multiply by $\frac{\pi}{3} \approx 1.0471975512$: $1.0471975512 \cdot 0.0174532925 \approx 0.0182776574$.
4. Finally, add $1.7320508076 + 0.0182776574 \approx 1.750328465$.

So, $B$ is approximately $1.750$ radians.

If you need further details or have questions, please ask!

Here are some follow-up questions to deepen your understanding:

1. How does the linear approximation formula work in general?
2. What would happen if we used a higher-order approximation?
3. Why is it important to convert angles from degrees to radians in calculus problems?
4. Can this method be applied to any function for linear approximation?
5. How can rounding affect the accuracy of results in linear approximations?

Tip: Always make sure to convert angles to the appropriate unit when working with trigonometric functions in calculus, as they typically assume radians.