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.

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}(14)/(9)&(-2)/(9)&(-2)/(9)\\\\(14)/(3)&(-2)/(3)&(1)/(3)\\\\(-47)/(9)&(8)/(9)&(-1)/(9)\end{array}\right]`

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

`x=\left[\begin{array}{ccc}(14)/(9)&(-2)/(9)&(-2)/(9)\\\\(14)/(3)&(-2)/(3)&(1)/(3)\\\\(-47)/(9)&(8)/(9)&(-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