Answer:
The given function is a periodic function with period $T = 16$ and the Fourier series representation is given by:
$$f(t) = \sum_{n=-\infty}^{\infty} a_n \cos\left(\frac{2n\pi}{T}t\right)$$
where $a_n$ are the Fourier coefficients. To find the Fourier coefficient $a_n$, we need to compute the definite integral:
$$a_n = \frac{1}{T} \int_{0}^{T} f(t) \cos\left(\frac{2n\pi}{T}t\right) dt$$
Since the function is periodic with period $T$, we can simplify the integral as:
$$a_n = \frac{1}{T} \int_{0}^{T} f(t) \cos\left(\frac{2n\pi}{T}t\right) dt = \frac{1}{T} \int_{0}^{T} 2.75 \cos\left(\frac{2n\pi}{T}t\right) dt$$
Now, we can evaluate the integral using the formula for the definite integral of a cosine function:
$$\int_{0}^{T} \cos\left(\frac{2n\pi}{T}t\right) dt = \frac{2}{T} \sin\left(\frac{2n\pi}{T}\right)$$
Therefore, we have:
$$a_n = \frac{1}{T} \int_{0}^{T} f(t) \cos\left(\frac{2n\pi}{T}t\right) dt = \frac{2}{T} \sin\left(\frac{2n\pi}{T}\right)$$
To find the value of $a_n$ for $n = 0$, we need to compute the integral:
$$a_0 = \frac{1}{T} \int_{0}^{T} f(t) dt = \frac{1}{T} \int_{0}^{T} 2.75 dt = \frac{2.75}{T}$$
Since $T = 16$, we have:
$$a_0 = \frac{2.75}{16} = 0.175$$
Therefore, the Fourier coefficient $a_0$ is $0.175$.
To find the value of $a_n$ for $n = 1$, we need to compute the integral:
$$a_1 = \frac{1}{T} \int_{0}^{T} f(t) \cos\left(\frac{2\pi}{T}t\right) dt = \frac{1}{T} \int_{0}^{T} 2.75 \cos\left(\frac{2\pi}{T}t\right) dt$$
Now, we can evaluate the integral using the formula for the definite integral of a cosine function:
$$\int_{0}^{T} \cos\left(\frac{2\pi}{T}t\right) dt = \frac{2}{T} \sin\left(\frac{2\pi}{T}\right)$$
Therefore, we have:
$$a_1 = \frac{1}{T} \int_{0}^{T} f(t) \cos\left(\frac{2\pi}{T}t\right) dt = \frac{2}{T} \sin\left(\frac{2\pi}{T}\right)$$
Since $T = 16$, we have:
$$a_1 = \frac{2}{16} \sin\left(\frac{2\pi}{16}\right) = \frac{2}{16} \sin\left(0.5\right) = 0.5$$
Therefore, the Fourier coefficient $a_1$ is $0.5$.
Similarly, we can find the values of $a_n$ for other values of $n$ by computing the corresponding integrals. However, since we are asked to provide the answer to at least four places of decimals, we will only provide the values of $a_0$ and $a_1$ to four places of decimals.
Therefore, the Fourier coefficients $a_0$ and $a_1$ are:
$$a_0 = 0.175$$
$$a_1 = 0.500$$
Step-by-step explanation: