Chapter 3.6, Problem 45PPS

(a)

To determine

Total cost of each package using matrices.

Answer to Problem 45PPS

Total cost for each package:

Package A costs $421.

Package B costs $274.

Package C costs $150.

Package D costs $68.

Explanation of Solution

Given information:

Kangy Studio packages for senior portraits:

  

The total amount of each package can be obtained by multiplying the price by number in each package.

For Package A:

Represent the matrices:

  $\left[\begin{array}{c}7\\ 10\\ 14\\ 45\\ 95\\ 13\end{array}\right]\left[\begin{array}{c}10\\ 4\\ 2\\ 1\\ 1\\ 11\end{array}\right]$

Multiply for the total amount:

  $\left[\begin{array}{c}7\left(10\right)\\ 10\left(4\right)\\ 14\left(2\right)\\ 45\left(1\right)\\ 95\left(1\right)\\ 13\left(11\right)\end{array}\right]$

Simplify:

  $\left[\begin{array}{c}70\\ 40\\ 28\\ 45\\ 95\\ 143\end{array}\right]$

For the total amount of package A, add up all the elements in the matrices.

  $70+40+28+45+95+143=\$421$

Thus,

The total amount for package A is $421.

For Package B:

Represent the matrices:

  $\left[\begin{array}{c}7\\ 10\\ 14\\ 45\\ 95\\ 13\end{array}\right]\left[\begin{array}{c}10\\ 4\\ 2\\ 1\\ 0\\ 7\end{array}\right]$

Multiply for the total amount:

  $\left[\begin{array}{c}7\left(10\right)\\ 10\left(4\right)\\ 14\left(2\right)\\ 45\left(1\right)\\ 95\left(0\right)\\ 13\left(7\right)\end{array}\right]$

Simplify:

  $\left[\begin{array}{c}70\\ 40\\ 28\\ 45\\ 0\\ 91\end{array}\right]$

For the total amount of package B, add up all the elements in the matrices.

  $70+40+28+45+0+91=\$274$

Thus,

The total amount for package B is $274.

For Package C:

Represent the Matrices:

  $\left[\begin{array}{c}7\\ 10\\ 14\\ 45\\ 95\\ 13\end{array}\right]\left[\begin{array}{c}8\\ 4\\ 2\\ 0\\ 0\\ 2\end{array}\right]$

Multiply for the total amount:

  $\left[\begin{array}{c}7\left(8\right)\\ 10\left(4\right)\\ 14\left(2\right)\\ 45\left(0\right)\\ 95\left(0\right)\\ 13\left(2\right)\end{array}\right]$

Simplify:

  $\left[\begin{array}{c}56\\ 40\\ 28\\ 0\\ 0\\ 26\end{array}\right]$

For the total amount of package C, add up all the elements in the matrices.

  $56+40+28+0+0+26=\$150$

Thus,

The total amount for package C is $150.

For Package D:

Represent the matrices:

  $\left[\begin{array}{c}7\\ 10\\ 14\\ 45\\ 95\\ 13\end{array}\right]\left[\begin{array}{c}0\\ 4\\ 2\\ 0\\ 0\\ 0\end{array}\right]$

Multiply for the total amount:

  $\left[\begin{array}{c}7\left(0\right)\\ 10\left(4\right)\\ 14\left(2\right)\\ 45\left(0\right)\\ 95\left(0\right)\\ 13\left(0\right)\end{array}\right]$

Simplify:

  $\left[\begin{array}{c}0\\ 40\\ 28\\ 0\\ 0\\ 0\end{array}\right]$

For the total amount of package D, add up all the elements in the matrices.

  $0+40+28+0+0+0=\$68$

Thus,

The total amount for package D is $68.

(b)

To determine

Early bird price for each package.

Answer to Problem 45PPS

After 15% off,

Early bird price:

Package A costs $357.85.

Package B costs $232.90.

Package C costs $127.50.

Package D costs $57.80.

Explanation of Solution

Given information:

Kangy Studio packages for senior portraits:

  

According to the statement,

There is an early bird discount of 15% off.

That means

The prices are reduced by 15% and we need to multiply the prices obtained in Part (a) by 0.85:

  $0.85\left[\begin{array}{c}421\\ 274\\ 150\\ 68\end{array}\right]$

Distribute the scalar:

  $\left[\begin{array}{c}0.85\left(421\right)\\ 0.85\left(274\right)\\ 0.85\left(150\right)\\ 0.85\left(68\right)\end{array}\right]$

For each element, perform the multiplication:

  $\left[\begin{array}{c}357.85\\ 232.90\\ 127.50\\ 57.80\end{array}\right]$

Thus,

The prices with early bird discount:

Package A costs $357.85.

Package B costs $232.90.

Package C costs $127.50.

Package D costs $57.80.