Science:Math Exam Resources/Courses/MATH152/April 2022/Question B1 (a)

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MATH152 April 2022

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Question B1 (a)

A factory produces four types of cookies: A, B, C and D. Ingredients needed to make one box of cookies of each type are given in the table below:

	A	B	C	D
butter (lb.)	1	2	1	3
flour (cups)	3	6	3	9
eggs	1	4	3	7
sugar (cups)	2	1	3	8
chocolate (lb.)	0	5	1	2

The following amounts of ingredients were used during one hour of operation: 11 lb. of butter, 33 cups of flour, 21 eggs, 23 cups of sugar and an unknown amount $H$ lb. of chocolate.

(a) Find how much chocolate was used.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
A number of boxes of each type of cookie was produced. Use four variables for these numbers of boxes and write down a linear system describing the total amount of ingredients used as a function of these 4 variables. Can you express this system as an augmented matrix to solve for $H$ ?

Hint 2
Let $n_{A},\ n_{B},\ n_{C},\ n_{D}$ denote the numbers of boxes of each type of cookie that were produced. Then we have the following system of equations, where each row corresponds to the use of one of the ingredients: ${\begin{cases}n_{A}+2n_{B}+n_{C}+3n_{D}&=11\\3n_{A}+6n_{B}+3n_{C}+9n_{D}&=33\\n_{A}+4n_{B}+3n_{C}+7n_{D}&=21\\2n_{A}+n_{B}+3n_{C}+8n_{D}&=23\\5n_{B}+n_{C}+2n_{D}&=H\end{cases}}.$ Did you learn tools to reduce such a system to one that is as simple as possible?

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work.
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Solution
The linear system described in the hint is given by the following augmented matrix: $\left[{\begin{array}{cccc\|c}1&2&1&3&11\\3&6&3&9&33\\1&4&3&7&21\\2&1&3&8&23\\0&5&1&2&H\end{array}}\right]$ We can solve the linear system by applying row operations (we use the notation $L_{i,j}(m)$ for the operation "add $m$ times the $j^{\mathrm {th} }$ row to the $i^{\mathrm {th} }$ row" and $D_{i}(m)$ for "multiply the $i^{\mathrm {th} }$ row by $m$ "): ${\begin{aligned}\left[{\begin{array}{cccc\|c}1&2&1&3&11\\3&6&3&9&33\\1&4&3&7&21\\2&1&3&8&23\\0&5&1&2&H\end{array}}\right]\xrightarrow {L_{2,1}(-3),\ L_{3,1}(-1),\ L_{4,1}(-2)} &\left[{\begin{array}{cccc\|c}1&2&1&3&11\\0&0&0&0&0\\0&2&2&4&10\\0&-3&1&2&1\\0&5&1&2&H\end{array}}\right]\\\xrightarrow {D_{3}(1/2),\ L_{1,3}(-2),\ L_{4,3}(3),\ L_{5,3}(-5)} &\left[{\begin{array}{cccc\|c}1&0&-1&-1&1\\0&0&0&0&0\\0&1&1&2&5\\0&0&4&8&16\\0&0&-4&-8&H-25\end{array}}\right]\\\xrightarrow {D_{4}(1/4),\ L_{1,4}(1),\ L_{3,4}(-1),\ L_{5,4}(1)} &\left[{\begin{array}{cccc\|c}1&0&0&1&5\\0&0&0&0&0\\0&1&0&0&1\\0&0&1&2&4\\0&0&0&0&H-9\end{array}}\right]\end{aligned}}$ From the last row, we see that $H=9$ .