a systematic equation?

Question

A dietitian works in a hospital and prepares meals under the guidance of a physician. For a particular patient the physician prescribes a meal to have 550 calories, 40 grams of protein and 320 milligrams of vitamin c. The dietitian decides to prepare a meal using steak ( 3 oz serving contains 300 calories, 20 grams of protein, and no vitamin c), baked potato (one baked potato contains 100 calories, 5 grams of protein, and 20 milligrams of vitamin c) and brussel sprouts (one 156 gram serving contains 50 calories, 5 grams of protein and 100 milligrams of vitamin c). How many servings of each are required to satisfy the physicians requirements?

Answer 1

You do use systems of equations! Let's start by writing down the essential information from the problem.

Needed:

550 cal

40g protein

320mg vitC

1 steak: 300cal, 20g protein, 0mg vitC

1 baked potato:100cal, 5g protein, 20mg vitC

1 serving sprouts: 50cal, 5g protein, 100mg vitC

As we can see from the info above, no one item will give the patient the right nutrients. We can't get any vitamin C from the steak, if we get enough protein from just potatoes we have too many calories, and if we get enough protein from brussels sprouts we don't get enough calories. So the meal should contain at least two of the items. Let's make some equations. I'll make calories "C", protein "P", vitamin C "V.

C = 550 = 300(steak) + 100(potato) + 50(sprouts)

P = 40 = 20(steak) + 5(potato) + 5(sprouts)

V = 320 = 20(potato) + 100(sprouts)

V has only 2 items (since the steak has no vitamin C), so it will be the simplest to work out first.

320 = 20potato + 100sprouts

20potato = 320 - 100sprouts

potato = 16 - 5 sprouts

Now we can substitute potato where it shows up in the calories equation.

So 550 = 300steak + 100(16 - 5sprouts) + 50sprouts

550 = 300steak + 1600 - 500sprouts + 50sprouts

-1050 = 300steak + -450sprouts

300steak = 450sprouts - 1050

steak = 1.5sprouts - 3.5

Now we can substitute both steak and potato in the protein equation.

40 = 20(1.5sprouts - 3.5) + 5(16 - 5sprouts) + 5sprouts

40 = 30sprouts - 70 + 80 - 25 sprouts + 5sprouts

10sprouts = 30

sprouts = 3

Yay! We have part of the answer! We need 3 servings of brussels sprouts

Now we use the equations we made earlier to find steak and potato servings:

steak = 1.5(3) - 3.5

steak = 1

potato = 16-5(3)

potato = 1

To summarize, we need 3 servings of brussels sprouts, 1 serving of steak, and 1 serving of potato. Before moving on, let's make sure this is correct.

C = 550 = 300(1) + 100(1) + 50(3)

= 300 + 100 + 150 = 550

Looks good!

P = 40 = 20(1) + 5(1) + 5(3)

= 20 + 5 + 15 = 40

Also looks good!

V = 320 = 20(1) + 100(3)

= 20 + 300 = 320

Everything checks out! Our answer is correct.