Saraswathi bought an 8 gram gold chain for Rs.40,000 on 1st November 2020. After INE 10 months, she sold the chain and bought a new 10 grams gold chain by paying extra Rs.10,000. Suppose the rate of the gold (in thousands) per gram, denoted by G(t), is a polynomial function of the time t. That is, G(t) = 0.07t3-1.4t²+7t+5. (consider t = 0 to be the time when Saraswathi bought her first gold chain and t denotes the number of months after she bought her first gold chain). Let G(t) is the polynomial of the rate for both used and new gold.

1. What is the rate of the gold per gram when she sold her first gold chain?

2. If she had sold the first gold chain after 6 months, how much extra would she have paid for buying the 10 grams gold chain?

To solve this problem, we'll use the polynomial function \( G(t) \) provided to find the rate of gold per gram at different times and answer the given questions.

### Given Data:
1. **First Purchase:**
   - Date: **1 November 2020**
   - Weight: **8 grams**
   - Cost: **Rs. 40,000**
   - Price per gram: \( \frac{40,000}{8} = Rs. 5,000 \) (or \( 5 \) thousands)

2. **Second Purchase:**
   - After **10 months**, she sold the first chain.
   - Weight of new chain: **10 grams**
   - Extra amount paid: **Rs. 10,000**

3. **Gold Rate Function:**
   - \( G(t) = 0.07t^3 - 1.4t^2 + 7t + 5 \)
   - \( t \) is the number of months after **1 November 2020**.

### Questions:

1. **Rate of gold per gram when she sold her first gold chain:**
   - Sold after **10 months**: \( t = 10 \).

2. **Extra amount if sold after 6 months:**
   - Sold after **6 months**: \( t = 6 \).

### Solution:

#### 1. Rate of Gold per Gram When She Sold Her First Gold Chain (After 10 Months):

Calculate \( G(10) \):
\[ G(10) = 0.07(10)^3 - 1.4(10)^2 + 7(10) + 5 \]
\[ G(10) = 0.07(1000) - 1.4(100) + 70 + 5 \]
\[ G(10) = 70 - 140 + 70 + 5 \]
\[ G(10) = 5 \]

So, the rate of gold per gram after 10 months is **Rs. 5,000**.

#### 2. Extra Amount If Sold After 6 Months:

Calculate \( G(6) \):
\[ G(6) = 0.07(6)^3 - 1.4(6)^2 + 7(6) + 5 \]
\[ G(6) = 0.07(216) - 1.4(36) + 42 + 5 \]
\[ G(6) = 15.12 - 50.4 + 42 + 5 \]
\[ G(6) = 11.72 \]

So, the rate of gold per gram after 6 months is **Rs. 11,720**.

#### Cost Calculations:

- **Initial 8 grams purchase at Rs. 5000 per gram:**
  \[ 8 \times 5000 = Rs. 40,000 \]

- **New 10 grams purchase after 10 months at Rs. 5000 per gram:**
  \[ 10 \times 5000 = Rs. 50,000 \]
  \[ Extra amount paid = Rs. 50,000 - Rs. 40,000 = Rs. 10,000 \]

- **New 10 grams purchase after 6 months at Rs. 11,720 per gram:**
  \[ 10 \times 11,720 = Rs. 117,200 \]
  \[ Extra amount paid = Rs. 117,200 - Rs. 40,000 = Rs. 77,200 \]

So, if she had sold the first gold chain after 6 months, she would have paid an extra **Rs. 77,200** for buying the new 10 grams gold chain.

### Summary:
1. The rate of gold per gram when she sold her first gold chain (after 10 months) was **Rs. 5,000**.
2. If she had sold the first gold chain after 6 months, she would have paid an extra **Rs. 77,200** for buying the 10 grams gold chain.

Would you like to explore more details or have any questions?

### Additional Questions:
1. How is the polynomial function \( G(t) \) derived from real-world data?
2. What is the significance of each term in the polynomial \( G(t) \)?
3. How would the cost calculation change if gold prices fluctuated non-polynomially?
4. How does the rate of gold change if she waited longer than 10 months to sell the first chain?
5. Can the polynomial \( G(t) \) predict future trends in gold prices?
6. What are other factors affecting gold prices that are not considered in \( G(t) \)?
7. How would different rates of gold price increase impact her overall expenditure?
8. What is the derivative of \( G(t) \) and what does it signify in terms of rate of change of gold prices?

### Tip:
Always cross-verify polynomial functions with real-world data for accurate predictions and better financial decisions.

Explore the rate of gold per gram in Saraswathi's purchase and sale scenario. Calculate the cost implications of buying new gold chains based on a polynomial function describing gold rates over time. Suitable for advanced high school students and anyone interested in financial calculations involving gold prices.

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