An accountant for a corporation forgot to pay the firm's income tax of $$ \$ 725,974.06 $$ on time. The government charged a penalty of $$ 7.8 \% $$ interest for the 35 days the money was late. Find the total amount (tax and penalty) that was paid. Assume 365 days in a year.

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We first compute the penalty interest using the formula

$$
\text{Penalty} = \text{Principal} \times \text{Annual Rate} \times \frac{\text{Days Late}}{365}.
$$

Here, the principal (the unpaid tax) is $$\$725{,}974.06$$, the annual rate is $$7.8\%$$ (or $$0.078$$), and the money was late for $$35$$ days. Thus,

$$
\text{Penalty} = 725974.06 \times 0.078 \times \frac{35}{365}.
$$

Notice that

$$
0.078 \times \frac{35}{365} = \frac{0.078 \times 35}{365}.
$$

Since $$0.078 \times 35 = 2.73$$, we have

$$
\text{Penalty} = 725974.06 \times \frac{2.73}{365}.
$$

Now, calculate the fraction:

$$
\frac{2.73}{365} \approx 0.00748082.
$$

Then, the penalty becomes

$$
\text{Penalty} \approx 725974.06 \times 0.00748082 \approx 5431.59.
$$

The total amount paid (tax plus penalty) is

$$
\text{Total Payment} = 725974.06 + 5431.59 \approx 731405.65.
$$

Thus, the corporation paid approximately $$\$731{,}405.65$$ in total.
The total amount paid, including the penalty, is approximately \$731,405.65.

An accountant for a corporation forgot to pay the firm's income tax of \( \$ 725,974.06 \) on time. The government charged a penalty of \( 7.8 \% \) interest for the 35 days the money was late. Find the total amount (tax and penalty) that was paid. Assume 365 days in a year.

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