Question

A coach purchases 47 hats for his players and their families at a total cost of $302. The cost of a small hat is $5.50. A medium hat costs $6.00. A large hat costs $7.00. He purchases three times as many medium hats as small hats. Using matrices, how many large hats did the coach purchase?
a. 6

b. 11

c. 18

d. 23

Answer 1

Answer: Option 'd' is correct.

Step-by-step explanation:

Let the number of small hat be 'x'

Let the number of medium hat be 'y'.

Let the number of large hat be 'x'.

Cost of each small hat = $5.50

Cost of each medium hat = $ 6.00

Cost of each large hat = $7.00

According to this, our equations would be

$x+y+z=47---(1)\\\\5.5x+6y+7z=\ 302------(2)$

And we have given that He purchases three times as many medium hats as small hats.

So, our equation would be

$y=3x\\\\-3x+y=0-------(3)$

so, the matrix would be

Ax=b

$\left[\begin{array}{ccc}1&1&1\\5.5&6&7\\-3&1&0\end{array}\right] \left[\begin{array}{ccc}x\\y\\z\end{array}\right] =\left[\begin{array}{ccc}47\\302\\0\end{array}\right]$

So, According to this, we can rewrite it as :

$x=A^{-1}B$

Here, Inverse of A would be

$A^{-1}=\left[\begin{array}{ccc}\dfrac{14}{9}&\dfrac{-2}{9}&\dfrac{-2}{9}\\\\\dfrac{14}{3}&\dfrac{-2}{3}&\dfrac{1}{3}\\\\\dfrac{-47}{9}&\dfrac{8}{9}&\dfrac{-1}{9}\end{array}\right]$

So, the value of 'X' would be written as

$x=\left[\begin{array}{ccc}\dfrac{14}{9}&\dfrac{-2}{9}&\dfrac{-2}{9}\\\\\dfrac{14}{3}&\dfrac{-2}{3}&\dfrac{1}{3}\\\\\dfrac{-47}{9}&\dfrac{8}{9}&\dfrac{-1}{9}\end{array}\right]\left[\begin{array}{ccc}47\\\\302\\\\0\end{array}\right] =\left[\begin{array}{ccc}6\\\\18\\\\23\end{array}\right]$

So, we using calculator, we get that

x = 6

y = 18

z = 23

So, Number of large hats purchased by the coach are 23.

Hence, Option 'd' is correct.

Answer 2

First, we define variables:
 x: small hat
 y: medium hat
 z: large hat
 We now write the system of equations:
 x + y + z = 47
 5.50x + 6y + 7z = 302
 -3x + y = 0
 We can write the system in matrix form as:
 Ax = b
 Where,
 A = [1  1  1; 5.50  6  7; -3  1  0]
 b = [47;  302;  0]
 x = [x;  y ;  z]
 Solving the system we have:
 x = 6
 y = 18
 z = 23
 Answer:
 the coach did purchase 23 large hats
 d. 23