Question

# The graph shows the subscription cost to a certain club in town. Write an equation of the

line. Explain what the slope and y-intercept mean in this situation. Then find the total

cost for 2 years.

## The correct answer is: cost of functioning for 2 years = $900.

### Hint:-

1. When we have 2 points that lie on a given line then we can find the equation of the said line by using the 2-point formula-

(y-y1) = (y2-y1) * (x-x1)

(x2-x1)

2. The slope of a line can be defined as the change in y coordinates of any 2 points on that line corresponding to the change in the x coordinates of those 2 points. This is generally referred to as the rise to run ratio of the given line i.e. how much did the y-coordinates rise vis-a-vis how long a distance was covered by the x-coordinates.

3. y-intercept for any line refers to the point at which the given line intersects with the Y-axis. At this point the x-coordinate is 0.

Step-by-step solution:-

From the given graph, we can observe that-

monthly cost of subscription in Dollars has been given and a line representing such cost has been plotted on the graph.

We observe that the given line passes through 2 points (0,100) & (6,300).

Hence, x1 = 0, y1 = 100, x2 = 6 & y2 = 300.

Coordinates of any point that lie on a line, when substituted in the equation of the given line, satisfies the equation.

Hence, we can find out the equation of the given lie using 2 point formula-

(y-y1) = (y2-y1) * (x-x1)

(x2-x1)

∴ (y-100) = (300 -100) * (x-0)

(6-0)

∴ (y-100) = 200 * x

6

∴ 6 * (y-100) = 200x ................................................................................. (Multiplying both sides by 6)

∴ 6y - 600 = 200x

∴ 6y = 200x + 600

∴ y = 200/6 x + 600/6 ............................................................... (Dividing both sides by 6)

∴ y = 100/3 x + 100 ..................................................................... (Equation i)

Comparing Equation i with the slope intercept form of a line i.e. y = mx + b, we get-

Slope = m = 100/3

& Y-intercept = b = 100

The slope of a line can be defined as the change in y coordinates of any 2 points on that line corresponding to the change in the x coordinates of those 2 points. This is generally referred to as the rise to run ratio of the given line i.e. how much did the y-coordinates rise vis-a-vis how long a distance was covered by the x-coordinates.

In the given situation, slope of the given equation refers to the movement in Subscription cost to the club for every increase in the months that it is functioning.

since the slope = 100/3 ≈ 33.33, we can say that for every month that the club functions (From the first month), its subscription cost increases by $33.33.

Also, y-intercept for any line refers to the point at which the given line intersects with the Y-axis.

At this point the x-coordinate is 0.

This means that when the value of x-coordinate is 0, the value of y-coordinate is $100.

For the given situation, this means that the club has to incur a subscription cost of $100, irrespective of whether it functions for its customers or not. i.e. even before starting its operations for earning revenue, it has to incur $100 towards subscription cost.

Now, Since the given equatuion is represented in terms of months, to find the total cost for 2 years, we first need to convert this time period (2 years) into months. i.e. 2 * 12 months = 24 months.

∴ We need to find the total subscription cost to the club for 24 months.

∴ We substitute x = 24 in Equation i, to find the total cost = y

∴ y = 100/3 x + 100

∴ y = 100/3 * 24 + 100

∴ y = 100 * 8 + 100

∴ y = 800 + 100

∴ y = 900.

Final Answer:-

∴ The given line can be written as- y = 100/3 x + 100. Its slope (i.e. change in cost for a given change in time period) is 100/3 or 33.33. Its Y-intercept (i.e. cost at 0 months or before starting operations) is $100. Also, cost of functioning for 2 years = $900.

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**Note:**

There is a shorter way of doing this problem. There is a simple rule followed while increasing or decreasing a value by a certain percentage. When we decrease a quantity by x%, we multiply the quantity by (1-0.x). Whereas when we increase a quantity by x%, we multiply it with (1+0.x).

Carrie, a packaging engineer, is designing a container to hold 12 drinking glasses shaped as regular octagonal prisms. Her initial sketch of the top view of the base of the container is shown above.

Carrie redesigned the container because the initial sketch did not account for cushioning material between the glasses. The area of the base of the newly designed container is greater than the area of the base in the initial sketch. What is the area, in square inches, of the base of the newly designed container?

**Note:**

There is a shorter way of doing this problem. There is a simple rule followed while increasing or decreasing a value by a certain percentage. When we decrease a quantity by x%, we multiply the quantity by (1-0.x). Whereas when we increase a quantity by x%, we multiply it with (1+0.x).

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Slope exists on every line. Because it shows how quickly our line is rising or falling, the slope of a line reveals how steep a line is. Mathematically, the slope of a line is known as the ratio of change in the line's y-value to the change in its x-value.

¶A line's slope can be determined using its two points (x_{1}, y_{1}) and (x_{2}, y_{2}). The formula (y_{2} - y_{1}) / is used to find the change in y and divided by the change in x. (x_{2} - x_{1}).

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**Note:**

A few simple ideas are used in solving this problem, like, area of a rectangle is given by the product of its length and breadth and the basic idea of division.

Carrie, a packaging engineer, is designing a container to hold 12 drinking glasses shaped as regular octagonal prisms. Her initial sketch of the top view of the base of the container is shown above.

If the length and width of the container base in the initial sketch were doubled, at most how many more glasses could the new container hold?

**Note:**

A few simple ideas are used in solving this problem, like, area of a rectangle is given by the product of its length and breadth and the basic idea of division.