**Suppose you're going to make yourself a peanut butter sandwich for dinner,
and you've decided you need 300 calories of energy and 10g of protein.**

**How much peanut butter (in tbsps) and how much bread (in slices) should you use?**

peanut butter = [188 cal, 8g protein]/2 per tbsp = [94,4]

white bread = [100 cal, 3g protein] per slice = [100,3]

Want x1 [94,4] + x2 [100,3] = [300,10]

Ans: 2.2 slices of bread with 0.8 tbsp peanut butter.

b.
**What if you wanted the same number of calories, but 13g of protein instead of 10g?**
[Note to self: make it 14g next time.]

Want x1 [94,4] + x2 [100,3] = [300,13]

Mathematically, this has a solution: 3.4 tbsp of PB
and a **negative amount** of bread: -0.2 slices.

But physically, this is nonsense: we cannot consume negative amounts of bread, so we cannot obtain this desired combination of nutrients with PB and bread.

c.
**Draw a cartoon (i.e. not necessarily to scale) of the (calories,protein)-plane that shows schematically what's going on in parts a and b.**

Although the peanut butter vector and the white bread vector together span the entire nutrition plane, only the shaded region can be reached with non-negative amounts of these two foods. Thus getting the nutrition in 'a' is possible, but in 'b' it is not.

d.
**What if in addition to the 300 calories and 10g of protein, you also wanted 10% of
your Daily Value of calcium?**

Adding an extra row for the calcium constraint (0.5% per tbsp PB, 4% for slice of bread, 10% desired):

from fractions import Fraction as frac

we see from the row echelon form that the system is inconsistent (0=1). Therefore it is impossible to satisfy this 3rd constraint with only the 2 food amounts to adjust.

**Instructions: turn in paper that is a printout of your Jupyter Notebook,
showing the calculations you did, and your comments and conclusions - either handwritten
after printing, or typed in "markdown" cells of the notebook. Be sure to write your
name on the paper.
You will drop this in a box on the counter of the Math Main Office, Room 244 Math Bldg,
before 3:50pm Friday. The office closes then, and no late homework is accepted.**

Here is nutritional info I found by googling. Be careful: the PB numbers are given per 2 tbsp.