As per bartleby guidelines for the more than 3 subparts only first three are to be answered please upload the others separately.

(a)

Consider the two points from table (a) as (200 53.00) and (250 52.50).

Then, the equation is,

\(\displaystyle{p}-{p}_{{1}}=\frac{{{p}_{{2}}-{p}_{{1}}}}{{{N}_{{2}}-{N}_{{1}}}}{\left({N}-{N}_{{1}}\right)}\)

\(\displaystyle{p}-{53.00}=\frac{{{52.50}-{53.00}}}{{{250}-{200}}}{\left({N}-{200}\right)}\)

\(\displaystyle{p}-{53}=\frac{{-{0.50}}}{{50}}{\left({N}-{200}\right)}\)

\(\displaystyle{p}-{53}=-{0.01}{N}+{2}\)

\(\displaystyle{p}=-{0.01}{N}+{55}\)

Thus, the equation \(\displaystyle{p}=-{0.01}{N}+{55}\)

(b)

Consider the formula to express the total monthly revenue R, in dollars, of this manufacturer in a month as a function of the number N of widgets produced in a month.

\(\displaystyle{R}={\left(-{0.01}{N}+{55}\right)}{N}\)

\(\displaystyle=-{0.01}{N}^{2}+{55}{N}\)

Thus, the formula is \(\displaystyle{R}=-{0.01}{N}^{2}+{55}{N}.\)

Here, R is not the linear function of N.

(c)

The formula for the monthly profit is,

\(\displaystyle{P}=-{0.01}{N}^{2}+{55}{N}-{35}{N}-{900}\)

\(\displaystyle=-{0.01}{N}^{2}+{20}{N}-{900}\)

Here, P is not the linear function of N.

(a)

Consider the two points from table (a) as (200 53.00) and (250 52.50).

Then, the equation is,

\(\displaystyle{p}-{p}_{{1}}=\frac{{{p}_{{2}}-{p}_{{1}}}}{{{N}_{{2}}-{N}_{{1}}}}{\left({N}-{N}_{{1}}\right)}\)

\(\displaystyle{p}-{53.00}=\frac{{{52.50}-{53.00}}}{{{250}-{200}}}{\left({N}-{200}\right)}\)

\(\displaystyle{p}-{53}=\frac{{-{0.50}}}{{50}}{\left({N}-{200}\right)}\)

\(\displaystyle{p}-{53}=-{0.01}{N}+{2}\)

\(\displaystyle{p}=-{0.01}{N}+{55}\)

Thus, the equation \(\displaystyle{p}=-{0.01}{N}+{55}\)

(b)

Consider the formula to express the total monthly revenue R, in dollars, of this manufacturer in a month as a function of the number N of widgets produced in a month.

\(\displaystyle{R}={\left(-{0.01}{N}+{55}\right)}{N}\)

\(\displaystyle=-{0.01}{N}^{2}+{55}{N}\)

Thus, the formula is \(\displaystyle{R}=-{0.01}{N}^{2}+{55}{N}.\)

Here, R is not the linear function of N.

(c)

The formula for the monthly profit is,

\(\displaystyle{P}=-{0.01}{N}^{2}+{55}{N}-{35}{N}-{900}\)

\(\displaystyle=-{0.01}{N}^{2}+{20}{N}-{900}\)

Here, P is not the linear function of N.