consider the model y=4/9x + 7/3 for the relation between price and quality of chewing gum. a) what does the slope mean? b) use this equation to predict the likely rating of gum which costs 8 cent a stick

c) according to this equation what would a chewing gum with a rating of 3 be likely to cost
d) what is the largest possible domain for which the model can hold?

a) The slope is the coefficient of x, which in this case is 4/9. I am assuming that y is price (in cents) and x is quality. That should have been made clear.

The slope is the ratio of the change in y to the change in x.

b) Substitute y = 8 in the equation and solve for x (the quality)

c) Substitute x = 3 and solve for y

d) The lowest price is x = zero. The quality must exceed 7/3

a) The slope of the equation, 4/9, represents the rate of change of the quality of the chewing gum with respect to its price. It indicates that for every 1 unit increase in the price of the gum, the quality increases by 4/9 units.

b) To predict the likely rating of gum that costs 8 cents a stick, substitute the given price (8) into the equation:

y = (4/9)x + 7/3
y = (4/9)(8) + 7/3
y β‰ˆ 3.56

Therefore, the predicted likely rating for gum that costs 8 cents a stick is approximately 3.56.

c) According to the equation, to find the likely cost for a gum with a rating of 3, substitute the given rating (3) into the equation and solve for x (price):

3 = (4/9)x + 7/3
(4/9)x = 3 - 7/3
(4/9)x β‰ˆ 2.33
x β‰ˆ 5.25

Therefore, a chewing gum with a rating of 3 is likely to cost approximately 5.25 cents.

d) The largest possible domain for which the model can hold depends on the context of the problem. Since the equation is linear, it implies a linear relationship between price and quality. Thus, technically the model can hold for any real number for price (x). However, it is important to note that the model assumes that the relationship remains linear and holds true across all possible price values. In practical terms, the model's validity may be limited by factors such as market conditions, consumer preferences, or other external factors.

a) The slope of the equation represents the rate of change between the price (x) and the quality rating (y) of chewing gum. In this case, the slope is 4/9, which means that for every unit increase in price, the quality rating of the gum is expected to increase by 4/9 units.

b) To predict the likely rating of gum that costs 8 cents a stick, substitute x = 8 into the equation:

y = (4/9)(8) + 7/3
= 32/9 + 7/3
= (32 + 21) / 9
= 53/9

Therefore, the likely rating of the gum would be 53/9.

c) According to the equation, to determine the likely cost of a chewing gum with a rating of 3, substitute y = 3 into the equation:

3 = (4/9)x + 7/3

Now, solve for x:

(4/9)x = 3 - 7/3
= 9/3 - 7/3
= 2/3

Multiply both sides by (9/4) to isolate x:

x = (2/3) * (9/4)
= (2 * 9) / (3 * 4)
= 18/12
= 3/2

Therefore, a chewing gum with a rating of 3 would likely cost 3/2.

d) The largest possible domain for which the model can hold depends on the context of the problem. However, since the equation is a linear model representing the relationship between price and quality rating, mathematically, the model can hold for any real values of x (price) and y (quality rating). In practical terms, the domain may be limited by other factors such as feasibility or the range of prices and ratings being considered in the context of chewing gum.