Question

Donnie is considering two different DVD rental stores, both of which have a one-time membership fee and a fee per rented DVD.

The cost for renting DVDs at the first store is shown below:
Number of DVDs rented Cost ($)
1 5.50
2 7.00
3 8.50

The cost for renting DVDs at the second store is modeled by the linear function C = 5 + 2d, where d is the number of DVDs rented and C is the cost.

Compare the rates of change for each function and explain what this means in terms of the context.

Answer 1

Answer:

⇒Store i charges $4 for one-time membership fee and a fee of $1.50 per rented DVD

⇒Store ii charges $5 for one-time membership fee and a fee of $2 per rented DVD

⇒ Renting a DVD in shop {i} is cheaper than in shop {ii}

Step-by-step explanation:

In the first store, the equation for cost can be calculated as;

Find slope using the data set

Number of DVDs rented               Cost {$}

1                                                        5.50

2                                                        7.0

3                                                         8.50

m= Δy/Δx

m= 8.50 - 5.50 / 3-1

m= 3/2 = 1.5

Write the equation

m= Δy/Δx

1.5 = y-7/ x-2

1.5 { x-2 } = y-7

1.5 x - 3.0 = y-7

1.5x -3.0 + 7 = y

1.5 x + 4= y

C= 4 + 1.5 d  ------where d is the number of DVDs rented

Comparing the two equations that model the cost of renting DVDs

i) C= 4 + 1.5d

ii) C=5 + 2d

⇒Store i charges $4 for one-time membership fee and a fee of $1.50 per rented DVD

⇒Store ii charges $5 for one-time membership fee and a fee of $2 per rented DVD

For example 2 DVDs are rented in both store, you can find the store which is cheaper as;

i) C= 4 + 1.5d  = 4+ 1.5*2 = 4 + 3 = $7

ii) C=5 + 2d = 5 + 2 * 2 = 5 + 4 = $9

⇒ Renting a DVD in shop {i} is cheaper than in shop {ii}

Answer 2

   Renting a DVD from store (2) is costlier than renting from store (2).

Linear equation:

•  A linear equation represents the relation between two variables 'x' and 'y',

         y = mx + b

         Here, m = slope of the line

         b = y-intercept

For store (1),

Let the linear equation representing the cost for renting the DVDs is,

C = md + b

Here, m = slope of the line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$

m = $\frac{y_2-y_1}{x_2-x_1}$

And 'd' = Number of DVDs

Let the two points lying on the line are (1, 5.5) and (2, 7),

So the slope will be,

$m=\frac{7-5.5}{2-1}=1.5$

Equation of the line will be → C = 1.5d + b

Since, this line passes through a third point (3, 8.5),

Substitute the values in the equation of the line,

8.5 = 1.5(3)+ b

8.5 - 4.5 = b

b = 4

Hence, equation of the line for the cost of renting a DVD from store (1) will be,

C = 1.5d + 4 ---------(1)

For store (2),

Equation of the line has been given as,

C = 2d + 5 ------(2)

Equation (1) represents a fixed cost of $4 and $1.5 for renting each DVD.

Equation (2) represents a fixed cost of $5 and $2 for renting each DVD.

If Donnie rents a DVD from both the stores,

Cost for renting DVDs at store (1) → C = 1.5(1) + 4 = $5.5

Cost for renting one DVD at store (2) → C = 2(1) + 5 = $7

      Therefore, renting a DVD from store (2) is costlier than renting from store (2).

Learn more about the linear equations here,

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