# 3.1: Pablo's Problem

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

Let us begin with an example. Consider again Pablo the nutritionist of problem 5, chapter 1. The Conundrum City school board has employed Pablo to design their school lunch program. Unfortunately for Pablo, their requirements are rather tricky:

Example 33: Pablo's Problem

The Conundrum City school board is heavily influenced by the local fruit grower’s association. They have stipulated that children eat at least 7 oranges and 5 apples per week. Parents and teachers have agreed that eating at least 15 pieces of fruit per week is a good thing, but school janitors argue that too much fruit makes a terrible mess, so that children should eat no more than 25 pieces of fruit per week

.

Finally Pablo knows that oranges have twice as much sugar as apples and that apples have 5 grams of sugar each. Too much sugar is unhealthy, so Pablo wants to keep the children's sugar intake as low as possible. How many oranges and apples should Pablo suggest that the school board put on the menu?

This is a rather gnarly word problem. Our first step is to restate it as mathematics, stripping away all the extraneous information:

Example 34: Pablo's problem restated

Let \(x\) be the number of apples and \(y\) be the number of oranges. These must obey

$$x\geq5\, \quad\mbox{and}\quad y\geq7\, ,$$

to fulfill the school board's politically motivated wishes. The teacher's and parent's fruit requirement means that

$$x+y\geq 15\, ,$$

but to keep the canteen tidy

$$x+y\leq 25\, .$$

Now let

$$s=5x+10y\, .$$

This linear function of \((x,y)\) represents the grams of sugar in \(x\) apples and \(y\) oranges. The problem is asking us to minimize \(s\) subject to the four linear inequalities listed above.

### Contributor

David Cherney, Tom Denton, and Andrew Waldron (UC Davis)