#### Answer

(a) $C(x)=90x+1800$, where $C(x)$ represents the cost (in dollars) of making $x$ bicycles in a day.
(b) See the image given.
(c) Cost of manufacturing $14$ bicycles a day: $C(14)=\$3060$.
(d) $22$ bicycles can be manufactured for $\$3780$.

#### Work Step by Step

Step-1: It is given that a linear cost function is of the form $$C(x) = mx+b$$
According to the question, the manufacturer has a fixed daily cost of $\$1800$, and it costs $\$90$ to manufacture a $single$ bicycle. Suppose the manufacturer made $4$ bicycles in a day. His cost of manufacturing will be $(90\times 4)+1800=\$2160$. Thus, replacing the bicycle quantity $4$ with a more general number, $x$, for the number of bicycles, we obtain the required cost function:
$$C(x)=90x+1800$$
Step-2: Let us take values from $x=0$ to $x=4$ bicycles, to obtain the desired data points for the graph.
For $C(x=0)=\$1800$
For $C(x=1)=\$1890$
For $C(x=2)=\$1980$
For $C(x=3)=\$2070$
For $C(x=4)=\$2160$
The obtained graph is given below.
Step-3: $C(x=14)=(90 \times 14) + 1800 = \$3060$
Step-4: Suppose $x$ bicycles can be manufactured for $\$3780$ a day. Thus,
$$3780=90x+1800$$
$$\implies 90x=3780-1800$$
$$\implies x=\frac{1980}{90}=22$$