1. A photocopy shop charges P1.00 per photocopy for orders of 20 or fewer photocopies, P0.85 per photocopy for orders of 50 or fewer but more than 20 photocopies, and P0.75 per photocopy for orders of more than 50 photocopies.

a. Write a function representing the cost f in pesos for x number of photocopies.

b. How much will you pay if you order 34 photocopies?

c. How much will you pay if you order 55 photocopies?

2. In 1995, Bangkok's population was about 13,000,000. If the city's population has grown about 1.7% each year, what is the population of Bangkok in year 1999? Create a function representing the population of Bangkok after t years.

3. You have inherited land that was purchased for P680,000 in 1960. The value of the land increased by approximately 5% per year. What is the approximate value of the land in the year 2011? Write a function that represents the amount of land after t years.

### 1. **Photocopy Pricing Problem**

#### a. Function for Cost \( f(x) \)

To define the cost function \( f(x) \), we can break it down into three intervals based on the number of photocopies \( x \) as follows:

\[
f(x) = 
\begin{cases} 
1.00x & \text{if } x \leq 20 \\
0.85x & \text{if } 20 < x \leq 50 \\
0.75x & \text{if } x > 50
\end{cases}
\]

#### b. How much will you pay for 34 photocopies?

Since \( 34 \) photocopies falls in the interval \( 20 < x \leq 50 \), we apply the rate of P0.85 per photocopy.

\[
f(34) = 0.85 \times 34 = 28.90 \, \text{pesos}
\]

#### c. How much will you pay for 55 photocopies?

Since \( 55 \) photocopies falls in the interval \( x > 50 \), we apply the rate of P0.75 per photocopy.

\[
f(55) = 0.75 \times 55 = 41.25 \, \text{pesos}
\]

---

### 2. **Bangkok Population Growth Problem**

The population growth can be modeled by exponential growth, which is given by:

\[
P(t) = P_0 (1 + r)^t
\]

Where:
- \( P_0 = 13,000,000 \) (initial population in 1995),
- \( r = 0.017 \) (growth rate of 1.7% per year),
- \( t \) is the number of years after 1995.

So, the function is:

\[
P(t) = 13,000,000 \times (1.017)^t
\]

#### Population in 1999 (after 4 years):
For \( t = 4 \):

\[
P(4) = 13,000,000 \times (1.017)^4
\]

Let’s calculate this:

\[
P(4) \approx 13,000,000 \times 1.07083 = 13,920,800
\]

Thus, the population in 1999 is approximately **13,920,800**.

---

### 3. **Land Value Growth Problem**

The value of the land increases by 5% each year, so we can also model this using exponential growth:

\[
V(t) = V_0 (1 + r)^t
\]

Where:
- \( V_0 = 680,000 \) (initial value of land in 1960),
- \( r = 0.05 \) (growth rate of 5% per year),
- \( t \) is the number of years after 1960.

The function is:

\[
V(t) = 680,000 \times (1.05)^t
\]

#### Value in 2011:
To find the value in 2011, we calculate the number of years between 1960 and 2011, which is \( t = 2011 - 1960 = 51 \).

Now, let's calculate the value:

\[
V(51) = 680,000 \times (1.05)^{51}
\]

Calculating this gives:

\[
V(51) \approx 680,000 \times 12.119 = 8,241,020
\]

Thus, the approximate value of the land in 2011 is **P8,241,020**.

---

### Follow-up Questions:

1. What is the total cost if you order 100 photocopies?
2. How many photocopies can you get if you have P200 to spend?
3. What would be the population of Bangkok in 2005 assuming the same growth rate?
4. How does the exponential growth formula change if the growth rate changes from 1.7% to 2%?
5. What would the land value be in 2050 if the growth rate remains at 5%?

**Tip:** Exponential growth models are powerful in financial and population contexts because they account for compounded growth over time.

1. A photocopy shop charges P1.00 per photocopy for orders of 20 or fewer photocopies, P0.85 per photocopy for orders of 50 or fewer but more than 20 photocopies, and P0.75 per photocopy for orders of more than 50 photocopies.

 a. Write a function representing the cost f in pesos for x number of photocopies.

 b. How much will you pay if you order 34 photocopies?

 c. How much will you pay if you order 55 photocopies?

 2. In 1995, Bangkok's population was about 13,000,000. If the city's population has grown about 1.7% each year, what is the population of Bangkok in year 1999? Create a function representing the population of Bangkok after t years.

 3. You have inherited land that was purchased for P680,000 in 1960. The value of the land increased by approximately 5% per year. What is the approximate value of the land in the year 2011? Write a function that represents the amount of land after t years.

Solve real-world problems involving photocopy costs, population growth in Bangkok, and land value increase. Learn how to use piecewise functions and exponential growth models to determine the total cost, population, and land value over time. Suitable for students in Grades 9-12.

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