Solve for Time: Expanding (40a+30)(4a+5)-746 in Planetary Hours

To solve this problem, we'll determine the number of hours in a day on this planet:

• Step 1: Calculate the total number of minutes in a day given as $(40a + 30)(4a + 5) - 746$.
• Step 2: The total number of minutes calculated per the day's duration is $(20a - 4) \times (8a + 3)$ since there are $20a - 4$ hours each lasting $8a + 3$ minutes.
• Step 3: Equate the two expressions.

The given expression for the total minutes is:

Calculate the expression without subtraction:

Subtract 746:

The total minutes also correspond to:

Now equate the expressions:

Subtract $160a^2$ from both sides:

Subtract $28a$ from both sides and add 596 to both sides:

Solve for $a$:

Knowing $a = 2$, calculate hours per day:

Therefore, the number of hours in a day is $2$ hours.

The correct answer is: 2 hours.