Final answer:
The complex number 464e¹.⁸⁵ᵗ is expressed in rectangular form by calculating the cosine and sine of 1.85 radians and multiplying them with the magnitude. The calculated result does not exactly match any of the provided options, suggesting there may be a typo in the original exponential form.
Step-by-step explanation:
To express the complex number 464e¹.⁸⁵ᵗ in rectangular form, we can use Euler's formula, which relates the exponential form of complex numbers to their rectangular form. Euler's formula states that eˣᵗ = cos(θ) + i*sin(θ), where θ is the angle in radians. In this case, the angle is 1.85 radians.
Calculate the cosine and sine of 1.85 radians:
- cos(1.85) ≈ -0.117
- sin(1.85) ≈ 0.993
Now, let's multiply these values by the magnitude, 464, to obtain the rectangular form:
The real part: 464 * (-0.117) ≈ -54.288
The imaginary part: 464 * 0.993 ≈ 460.512
Therefore, the rectangular form of the given complex number is approximately -54.288 + 460.512i. However, this does not match exactly with any of the provided options. Please double-check the given exponential form of the complex number, since it might contain a typo.