Homework 1.3.X1 due Friday Feb 9

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]

peanut_butter_sandwich_a.png

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]

peanut_butter_sandwich_b.png

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.

peanut_butter_sandwich_annotated_with_arrows.svg.png

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
peanut_butter_sandwich_d.png

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.

Nutritional info

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

peanut_butter.png white_bread.png