https://file.aiquickdraw.com/mathbot/file/17283969406126o5u1let.jpg

We are tasked with determining how many days the Department of Health's fund will last, given their budget and spending pattern.

### Problem Breakdown:
- **Budget**: ₱3,312,000
- **Spending pattern**: 
  - Day 1: ₱72,000
  - Day 2: ₱90,000
  - Day 3: ₱108,000
  - and so on, increasing by ₱18,000 daily.

This spending pattern represents an **arithmetic sequence**:
- First term (\(a_1\)) = ₱72,000
- Common difference (\(d\)) = ₱18,000

The goal is to find the number of days (\(n\)) before the budget is exhausted.

### Step 1: Sum of the arithmetic sequence
The total spending after \(n\) days can be expressed using the sum of an arithmetic sequence formula:
\[
S_n = \frac{n}{2} \times (2a_1 + (n - 1)d)
\]
Where:
- \(S_n\) = total budget (₱3,312,000)
- \(a_1 = 72,000\) 
- \(d = 18,000\)

### Step 2: Plugging in the known values
We substitute these values into the equation:
\[
3,312,000 = \frac{n}{2} \times (2(72,000) + (n - 1)(18,000))
\]
Simplifying:
\[
3,312,000 = \frac{n}{2} \times (144,000 + 18,000n - 18,000)
\]
\[
3,312,000 = \frac{n}{2} \times (126,000 + 18,000n)
\]
\[
3,312,000 = n(63,000 + 9,000n)
\]

### Step 3: Solve the quadratic equation
Expand and rearrange the equation:
\[
9,000n^2 + 63,000n - 3,312,000 = 0
\]
Divide by 1,000 to simplify:
\[
9n^2 + 63n - 3,312 = 0
\]
Solve this quadratic equation using the quadratic formula:
\[
n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Where:
- \(a = 9\)
- \(b = 63\)
- \(c = -3,312\)

Let's calculate this next.The quadratic equation gives us two solutions: \( n_1 = 16 \) and \( n_2 = -23 \).

Since the number of days (\(n\)) cannot be negative, the valid solution is \( n = 16 \).

### Conclusion:
The Department of Health's fund will last for **16 days**.

Would you like more details or have any questions? Here are five related questions you could explore:

1. How would the solution change if the common difference was ₱20,000 instead of ₱18,000?
2. What if the initial budget was reduced to ₱2,500,000?
3. How can we derive the total spent after exactly 10 days?
4. Can we solve this problem using a different method, such as graphing?
5. What happens if the spending decreases instead of increases each day?

**Tip**: When dealing with arithmetic sequences, the sum formula is a powerful tool for solving problems related to cumulative growth or decline.

The Department of Health has a budget of ₱3,312,000. They spend ₱72,000 on day 1, ₱90,000 on day 2, ₱108,000 on day 3, and so on, increasing by ₱18,000 daily. In how many days will the budget last?

Learn how to solve an arithmetic sequence problem where a budget of ₱3,312,000 is spent over increasing daily amounts. Includes a step-by-step breakdown of how to apply the sum of arithmetic sequences and the quadratic formula to find the number of days the budget will last. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation