### Problem Statement**12.** Benny received 4 pennies on the first day of the month, and each day after that, he received quadruple the number of pennies that he received the day before. On what day of the month did Benny first receive over 1 million dollars on a single day?### ExplanationIn this problem, Benny's pennies grow exponentially, quadrupling each day. Here is the breakdown:- Day 1: 4 pennies- Day 2: 4 x 4 = 16 pennies- Day 3: 16 x 4 = 64 pennies- Day 4: 64 x 4 = 256 pennies- ...Let’s denote the number of pennies Benny receives on the $ n $th day as $ P(n) $. The relationship can be expressed mathematically as:\[ P(n) = 4 \times 4^{(n-1)} = 4^n \text{ pennies} \]To find out when Benny first receives over 1 million dollars, knowing that 1 dollar = 100 pennies, we set up the inequality:\[ 4^n > 100 \text{ (pennies needed for 1 dollar)} \times 1,000,000 \text{ (number of dollars)} \]\[ 4^n > 100,000,000 \]Taking the logarithm (base 10) of both sides, we get:\[ \log_{10}(4^n) > \log_{10}(100,000,000) \]\[ n \log_{10}(4) > 8 \]\[ n > \frac{8}{\log_{10}(4)} \]Using the value $ \log_{10}(4) \approx 0.602 $:\[ n > \frac{8}{0.602} \approx 13.29 \]Since $ n $ must be a whole number (representing days), Benny first receives over 1 million dollars on the 14th day of the month.### ConclusionBenny first receives over 1 million dollars on a single day on the 14th day of the month.

Given: Money received on Day 1 = 1 penny Money is being quadrupled every day. So, Money received on…

12. Benny received 4 pennies on the first day of the month, and each day after that, he received quadruple the number of pennies that he received the day before. On what day of the month did Benny first receive over 1 million dollars on a single day?

12. Benny received 4 pennies on the first day of the month, and each day after that, he received quadruple the number of pennies that he received the day before. On what day of the month did Benny first receive over 1 million dollars on a single day?

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

### Problem Statement

**12.** Benny received 4 pennies on the first day of the month, and each day after that, he received quadruple the number of pennies that he received the day before. On what day of the month did Benny first receive over 1 million dollars on a single day?

### Explanation

In this problem, Benny's pennies grow exponentially, quadrupling each day. Here is the breakdown:

- Day 1: 4 pennies
- Day 2: 4 x 4 = 16 pennies
- Day 3: 16 x 4 = 64 pennies
- Day 4: 64 x 4 = 256 pennies
- ...

Let’s denote the number of pennies Benny receives on the $ n $th day as $ P(n) $. The relationship can be expressed mathematically as:
\[ P(n) = 4 \times 4^{(n-1)} = 4^n \text{ pennies} \]

To find out when Benny first receives over 1 million dollars, knowing that 1 dollar = 100 pennies, we set up the inequality:

\[ 4^n > 100 \text{ (pennies needed for 1 dollar)} \times 1,000,000 \text{ (number of dollars)} \]
\[ 4^n > 100,000,000 \]

Taking the logarithm (base 10) of both sides, we get:
\[ \log_{10}(4^n) > \log_{10}(100,000,000) \]
\[ n \log_{10}(4) > 8 \]
\[ n > \frac{8}{\log_{10}(4)} \]

Using the value $ \log_{10}(4) \approx 0.602 $:
\[ n > \frac{8}{0.602} \approx 13.29 \]

Since $ n $ must be a whole number (representing days), Benny first receives over 1 million dollars on the 14th day of the month.

### Conclusion

Benny first receives over 1 million dollars on a single day on the 14th day of the month.

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