The charges for hiring a car from three different companies, based on the number of days for which a car is hired and/or the number of kilometres for which the car is driven are as follows:
Company A charges $660 per day
Company B charges 48 cents on per kilometre driven
Company C charges $300 per day per day and $150 on per kilometre driven
John needs to hire a car for 4 days to drive 560 kilometres
i) Write down two matrices only such that the elements of their product under matrix multiplication give the charge of hiring a car from three different companies
ii) Find this product. Hence state the company that John should hire the car from in order to save cost.
Please help, I don't know where to start... What variables should I use to represent the kilometre driven, etc?
A | 660 0| d
B =| 0 , .48 | k
C | 300, 150| (I bet that 150 is a typo)
A = 660 d + 0 k
B = 0 d + .48 k
C = 300 d + 150 k
put in d = 4 and k = 560
(after proofreading your input)
i) To represent the charges for hiring a car from three different companies, we can use the following variables:
- Let "d" represent the number of days for which a car is hired.
- Let "k" represent the number of kilometers driven.
We can then create two matrices:
Matrix A:
| 660d 0 0 |
| 0 0.48k 0 |
| 0 300d 150k|
Matrix B:
| 1 |
| 1 |
| 1 |
ii) To find the product of Matrix A and Matrix B, we simply multiply them together:
Matrix A x Matrix B:
| 660d + 0 + 0 |
| 0 + 0.48k + 0 |
| 0 + 300d + 150k|
Simplifying the matrix multiplication gives us the following charges:
Company A: 660d
Company B: 0.48k
Company C: 300d + 150k
To find the company John should hire the car from in order to save cost, we need to calculate the charges for each company using the given values:
John needs to hire a car for 4 days (d = 4) and drive 560 kilometers (k = 560).
From the charges above, we have:
Company A charge: 660d = 660 * 4 = $2640
Company B charge: 0.48k = 0.48 * 560 = $268.80
Company C charge: 300d + 150k = (300 * 4) + (150 * 560) = $300 + $84,000 = $84,300
Based on these calculations, John should hire the car from Company B to save cost, as it has the lowest charge.
To solve this problem, let's define the following variables:
- d: The number of days for which the car is hired
- k: The number of kilometers driven
i) The first matrix will represent the charges for hiring a car based on the number of days:
Matrix A = [660d, 0, 0]
The second matrix will represent the charges for hiring a car based on the number of kilometers driven:
Matrix B = [0, 0.48k, 150k]
ii) To find the product of Matrix A and Matrix B, we multiply each element in Matrix A by the corresponding element in Matrix B and sum them up:
Matrix A x Matrix B = [660d * 0 + 0 * 0 + 0 * 150k, 660d * 0.48k + 0 * 0.48k + 0 * 150k, 660d * 150k + 0 * 0 + 0 * 150k]
Simplifying this results in:
Matrix A x Matrix B = [0, 316.8dk, 99,000dk]
Now, we substitute the given values:
Matrix A x Matrix B = [0, 316.8 * 4 * 560, 99,000 * 4 * 560]
Matrix A x Matrix B = [0, 712,320, 111,360,000]
From the product of the matrices, we can see that the first element represents the cost based on the number of days, the second element represents the cost based on the number of kilometers driven, and the third element represents the combined cost.
In this case, John needs to hire a car for 4 days and drive 560 kilometers. Therefore, John should hire the car from Company B, as it offers the lowest cost based on the number of kilometers driven.
Hence, John should hire the car from Company B in order to save cost.
To solve this problem, let's assign variables to represent the number of days a car is hired and the number of kilometers driven.
Let:
x = number of days a car is hired
y = number of kilometers driven
Now, we can determine the charges for each company based on these variables:
Company A charges $660 per day, so the charge for Company A is simply 660x.
Company B charges 48 cents for each kilometer driven, so the charge for Company B is 0.48y.
Company C charges $300 per day and $150 per kilometer driven. Therefore, the charge for Company C is 300x + 150y.
Now, we can represent the charges of each company as matrices:
Matrix A = [660x]
Matrix B = [0.48y]
Matrix C = [300x]
[150y]
To find the product of these matrices and determine the total charges, we can multiply Matrix A by Matrix B, and then multiply the result by Matrix C:
Matrix D = Matrix A * Matrix B
Matrix E = Matrix D * Matrix C
Matrix E will give us the charge of hiring a car from each company.
Let's calculate the product:
Matrix D = [660x] * [0.48y] = [660x * 0.48y]
Matrix E = [660x * 0.48y] * [300x]
[150y]
Now, we can simplify Matrix E:
Matrix E = [316.8xy]
[150xy]
The element in the top-left of Matrix E represents the charge for Company A, the element in the bottom-left represents the charge for Company B, and the element in the top-right represents the charge for Company C.
In this case, John needs to hire a car for 4 days and drive 560 kilometers, so we substitute the values into Matrix E:
Matrix E = [316.8 * 4 * 560]
[150 * 4 * 560]
Simplifying this, we get:
Matrix E = [893,760]
[336,000]
From this calculation, we can see that the charge for Company A is 893,760 and the charge for Company B is 336,000.
Therefore, John should hire the car from Company B in order to save costs, as it has the lowest charge.