A shipping company charges a base rate of Php 150.00 plus an additional Php 40.00 per kilogram of the packages weight. The function f(w) = 15+4w represents the cost based on weight w in kilograms.

a. If a customer wants to ship two packages with weighs w. and w2 kg respectively, find the total cost function.

b. Calculate the total cost to ship a 3kg and a 5kg package

Let's break down the problem:

### a. Finding the Total Cost Function

Given that the shipping cost function for a single package is \( f(w) = 150 + 40w \), where \( w \) is the weight in kilograms:

- For the first package with weight \( w_1 \), the cost is \( f(w_1) = 150 + 40w_1 \).
- For the second package with weight \( w_2 \), the cost is \( f(w_2) = 150 + 40w_2 \).

The total cost for shipping both packages is the sum of these two individual costs:

\[
\text{Total Cost} = f(w_1) + f(w_2)
\]

\[
\text{Total Cost} = (150 + 40w_1) + (150 + 40w_2)
\]

Combine like terms:

\[
\text{Total Cost} = 150 + 150 + 40w_1 + 40w_2
\]

\[
\text{Total Cost} = 300 + 40(w_1 + w_2)
\]

So, the total cost function for shipping two packages with weights \( w_1 \) and \( w_2 \) is:

\[
\text{Total Cost} = 300 + 40(w_1 + w_2)
\]

### b. Calculating the Total Cost for Specific Weights

Now, let's calculate the total cost to ship a 3 kg package and a 5 kg package.

Substitute \( w_1 = 3 \) and \( w_2 = 5 \) into the total cost function:

\[
\text{Total Cost} = 300 + 40(3 + 5)
\]

\[
\text{Total Cost} = 300 + 40(8)
\]

\[
\text{Total Cost} = 300 + 320
\]

\[
\text{Total Cost} = 620
\]

Thus, the total cost to ship a 3 kg and a 5 kg package is **Php 620.00**.

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Would you like any further details or explanations?

Here are some related questions you might find useful:

1. How would the total cost function change if the base rate increased?
2. What is the cost to ship a single 10 kg package?
3. How does the cost function change if the rate per kilogram is reduced to Php 35.00?
4. What would be the total cost to ship three packages weighing 2 kg, 4 kg, and 6 kg?
5. How can the function be used to calculate the cost of shipping multiple packages of the same weight?
6. What is the cost difference between shipping a 1 kg package and a 7 kg package?
7. How does the function behave as the weight of the packages increases?
8. How would you set up the function if there were an additional surcharge for packages over 10 kg?

**Tip:** Always check for any additional fees or discounts that might apply when using cost functions for real-world applications.

Learn how to calculate shipping costs for multiple packages using a linear cost function. This problem explores the cost calculation for two packages of different weights, with a step-by-step explanation. Calculate specific costs for shipping a 3 kg and a 5 kg package. 

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