To solve this problem, we need to determine how much the number of packages distributed per hour must increase so that all 9135 packages are distributed in 35 hours. Here’s a breakdown of the solution:

Problem Summary:

• Total packages: $9135$
• Total hours: $35$
• Packages distributed in the first hour: $40$
• The number of packages distributed increases by a constant value $x$ every hour.
  In other words:
  • Hour 1: $40$
  • Hour 2: $40 + x$
  • Hour 3: $40 + 2x$
  • ...
  • Hour $n$: $40 + (n-1)x$
• After $35$ hours, the sum of all distributed packages must equal $9135$.

We need to find the value of $x$.

Mathematical Solution:

The total number of packages distributed over $35$ hours follows an arithmetic sequence where:

• First term $a = 40$
• Common difference $d = x$
• Number of terms $n = 35$

The sum of an arithmetic sequence is given by: $S_n = \frac{n}{2} \cdot \left( 2a + (n-1)d \right)$

Substitute the known values: $S_{35} = \frac{35}{2} \cdot \left( 2(40) + (35-1)x \right)$

Set this equal to the total number of packages: $9135 = \frac{35}{2} \cdot \left( 80 + 34x \right)$

Step-by-Step Calculation:

1. Multiply through by $2$ to eliminate the fraction: $2 \cdot 9135 = 35 \cdot (80 + 34x)$ $18270 = 35 \cdot (80 + 34x)$

2. Divide through by $35$: $\frac{18270}{35} = 80 + 34x$ $522 = 80 + 34x$

3. Subtract $80$ from both sides: $442 = 34x$

4. Solve for $x$: $x = \frac{442}{34}$ $x = 13$

Final Answer:

The number of packages distributed per hour must increase by 13 packages per hour.

Verification:

• Hour 1: $40$
• Hour 2: $53$ (40 + 13)
• Hour 3: $66$ (40 + 2 \cdot 13)
  ...
• Hour 35: $40 + 34 \cdot 13 = 482$

The total sum of packages: $S_{35} = \frac{35}{2} \cdot \left( 2(40) + 34(13) \right)$ $S_{35} = \frac{35}{2} \cdot \left( 80 + 442 \right) = \frac{35}{2} \cdot 522 = 9135$

The solution is correct.

Do you need further clarification or step-by-step details?

Related Questions:

1. How do you derive the formula for the sum of an arithmetic sequence?
2. What would happen if the increase $x$ were halved?
3. How would the result change if the total number of hours increased to $40$?
4. Can we solve this problem graphically to understand the distribution pattern?
5. How do you determine the general term of an arithmetic sequence?

Tip:

When dealing with increasing or decreasing sequences, the arithmetic sequence formula is an efficient tool for summing terms systematically.