Question

The cost of each of the components in the production of each size of pizza, and the selling price, is
given in the table below:
Cost of Production(cents)
Selling Size :Base Topping Cooking Labour Selling price($)
Small 32 110 8 200 9.50
Medium 50 175 10 240 11.50
Large 72 250 12 280 13.50
Family 98 345 15 320 17.50

a) Use matrix methods to find the total cost of production for each size of the pizzas. Give your
answers in cents?

b) If Profit = Selling price – Cost of Production, write a matrix equation that will enable you to find
the profit and find the profit, in dollars correct to the nearest cent, made on each size of pizza?

Answer 1

Let's write the matrix which will contain data about cost of each of the component:

$A=\begin{pmatrix} 32 &110 &8 &200 \\ 50 &175 &10& 240\\ 72 &250 &12 &280\\ 98 &345 &15 &320 \end{pmatrix}$

and matrix with selling prices:

$SP=\begin{pmatrix} 9.50 \\ 11.50\\ 13.50\\ 17.50 \end{pmatrix}$

a)  To find the total cost of production for each size of the pizzas we should use multiplication.

Just multiply our matrix by another matrix which has one column of 1's:

$B=\begin{pmatrix} 1 \\ 1\\ 1\\ 1 \end{pmatrix}$

$AB=\begin{pmatrix} 32 &110 &8 &200 \\ 50 &175 &10& 240\\ 72 &250 &12 &280\\ 98 &345 &15 &320 \end{pmatrix}x\begin{pmatrix} 1 \\ 1\\ 1\\ 1 \end{pmatrix}$ $= \begin{pmatrix} AB_{11} \\ AB_{21} \\ AB_{31} \\ AB_{41} \end{pmatrix}$

AB₁₁ = 32 * 1 + 110 * 1 + 8 * 1 + 200 * 1 = 350

AB₂₁ = 50 * 1 + 175 * 1 + 10 * 1 + 240 * 1 = 475

AB₃₁ = 72 * 1 + 250 * 1 + 12 * 1 + 280 * 1 = 614

AB₄₁ = 98 x 1 + 345 x 1 + 15 x 1 + 320 x 1 = 778

$AB= \begin{pmatrix} 350 \\ 475 \\ 614 \\ 778 \end{pmatrix}$

So we get costs of all pizzas: small - 350cents, medium - 475cents, large - 614cents, family - 778cents.

b) x = A*B, where x is AB

$P=SP-\dfrac{1}{100}*AB$

$P=\begin{pmatrix} 9.50 \\ 11.50\\ 13.50\\ 17.50 \end{pmatrix}-\dfrac{1}{100}*\begin{pmatrix} 350 \\ 475 \\ 614 \\ 778 \end{pmatrix}=$

$=\begin{pmatrix} 9.50-3.50 \\ 11.50-4.75 \\ 13.50-6.14 \\ 17.50-7.78 \end{pmatrix}=\begin{pmatrix} 6.00 \\ 6.75 \\ 7.36 \\ 9.72 \end{pmatrix}$

So our profit will be:

from small: $\approx6.0$ dollars

from middle: $\approx6.8$ dollars

from large: $\approx7.4$ dollars

from family: $\approx9.7$ dollars