A chef is estimating the cost to prepare a fixed-price menu for x guests. The anticipated cost of the appetizers is
a(x) = 3.25x, the anticipated cost for the main course is m(x) = 5.25x, and the anticipated cost for the dessert course is d(x) = 1.25x.
If the functions are combined so that
c(x) = a(x) + m(x) + d(x), then the function c(x) includes the point (40,

To find the function $ c(x) $ that includes the point $(40, y)$, we first need to combine the given functions $ a(x) $, $ m(x) $, and $ d(x) $.

Given:
- $ a(x) = 3.25x $ (cost of appetizers)
- $ m(x) = 5.25x $ (cost of the main course)
- $ d(x) = 1.25x $ (cost of dessert)

The total cost function $ c(x) $ is the sum of these three functions:

\[
c(x) = a(x) + m(x) + d(x)
\]

Substituting the expressions for $ a(x) $, $ m(x) $, and $ d(x) $:

\[
c(x) = 3.25x + 5.25x + 1.25x
\]

Now, combine like terms:

\[
c(x) = (3.25 + 5.25 + 1.25)x = 9.75x
\]

So, the function $ c(x) $ is:

\[
c(x) = 9.75x
\]

We are given that the function $ c(x) $ includes the point $(40, y)$. To find $ y $, substitute $ x = 40 $ into the function:

\[
y = c(40) = 9.75 \times 40
\]

\[
y = 390
\]

Thus, the point is $(40, 390)$.

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

1. What would the cost be for 50 guests?
2. How would the cost function change if the price of the main course increased by $2 per guest?
3. What is the total cost for 100 guests?
4. How would you adjust the function if the dessert cost doubled?
5. What is the individual contribution of each course to the total cost for 40 guests?

**Tip:** Always check your function by substituting values to ensure the calculations are consistent with given points.

Learn how to calculate the cost function for preparing a fixed-price menu based on the number of guests. Includes step-by-step calculations for appetizers, main course, dessert, and total cost. Suitable for advanced high school students.

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