a) Calculation:

Write the equation to calculate the cost of the capital (COC).

\(COC=[(\frac{Equity}{Equity+Debt}) \times ROE]+[(\frac{Debt}{Equity+Debt}) \times ROD \times (1-t)]\) .....(I)

Here, the amount that the partner has invested is Equity, the amount that firm has borrowed is Debt, the return on equity is ROE, return on debt is ROD, and tax rate is t.

Substitute $75000 for Equity, $125000 for Debt, \(\displaystyle{12}\%\) for ROE, \(\displaystyle{8}\%\) for ROD, and \(\displaystyle{0}\%\) for t in Equation (I).

\(\displaystyle{C}{O}{C}={\left[{\left({\frac{{\${75000}}}{{\${75000}+\${125000}}}}\right)}\times{0.12}\right]}+{\left[{\left({\frac{{\${125000}}}{{\${75000}+\${125000}}}}\right)}\times{0.08}\times{\left({1}-{0}\right)}\right]}\)

\(\displaystyle={0.045}+{0.05}\)

\(\displaystyle={0.095}\)

\(\displaystyle={9.5}\%\)

Conclusion:

Therefore, the firms cost of capital before tax is \(\displaystyle{9.5}\%\),

b) Calculation:

Substitute $75000 for Equity, $125000 for Debt, \(\displaystyle{12}\%\) for ROE, \(\displaystyle{8}\%\) for ROD, and \(\displaystyle{30}\%\) for t in Equation (I).

\(\displaystyle{C}{O}{C}={\left[{\left({\frac{{\${75000}}}{{\${75000}+\${125000}}}}\right)}\times{0.12}\right]}+{\left[{\left({\frac{{\${125000}}}{{\${75000}+\${125000}}}}\right)}\times{0.08}\times{\left({1}-{0.30}\right)}\right]}\)

\(\displaystyle={0.045}+{0.035}\)

\(\displaystyle={0.08}\)

\(\displaystyle={8}\%\)

Conclusion:

Therefore, the firms cost of capital after tax is, \(\displaystyle{8}\%\).