A cereal company finds that the number of people who will buy one of its products in the first month that it is introduced is linearly related to the amount of money it spends on advertising. If it spends $40,000 on advertising then 100,000 boxes of cereal will be sold, and if it spends $60,00 then 200,00 boxes will be sold.
a. Write and equation describing the relation between the amount A spent on advertising and the number x of boxes sold.
b. How much advertising is needed to sell 300.000 boxes of cereal?
c. Interpret slope
x = number sold
y = advertising cost
Do the whole problem in thousands, then multiply by 1000 at the end.
Points on graph
(X1,Y1) = (100, 40)
(X2,Y2) = (200, 60)
slope = (60-40)/(200-100)= (y - 40)/(x-100)
15000
a. To find the equation describing the relation between the amount A spent on advertising and the number x of boxes sold, we can use the given data points to determine the slope of the linear relationship.
Let A be the amount spent on advertising (in thousands of dollars) and x be the number of boxes sold (in thousands of boxes).
From the given data:
When A = 40, x = 100
When A = 60, x = 200
We can use the formula for slope, which is given by:
slope = (change in y) / (change in x)
slope = (200 - 100) / (60 - 40)
slope = 100 / 20
slope = 5
Therefore, the equation describing the relation between A and x is:
x = 5A
b. To determine how much advertising is needed to sell 300,000 boxes of cereal, we can substitute x = 300 into the equation:
300 = 5A
Solving for A:
A = 300 / 5
A = 60
Therefore, $60,000 needs to be spent on advertising to sell 300,000 boxes of cereal.
c. The slope of the equation (5 in this case) represents the rate of change or the increase in boxes sold for every $1,000 increase in advertising. In other words, for every additional $1,000 spent on advertising, the company can expect to sell an additional 5,000 boxes of cereal.
To solve this problem, we will use the given data points to create a linear equation that describes the relationship between the amount spent on advertising (A) and the number of boxes sold (x).
a. Let's start by writing the equation using the slope-intercept form, y = mx + b, where y represents the number of boxes sold and x represents the amount spent on advertising.
We are given two data points:
When A = $40,000, x = 100,000
When A = $60,000, x = 200,000
To find the slope (m), we can use the formula:
m = (change in y) / (change in x) = (200,000 - 100,000) / ($60,000 - $40,000) = 100,000 / $20,000 = 5
Now, let's plug in one of the data points into the equation and solve for the y-intercept (b). We'll use the point (40,000, 100,000):
100,000 = 5 * 40,000 + b
100,000 = 200,000 + b
b = 100,000 - 200,000
b = -100,000
Therefore, the equation is:
y = 5x - 100,000
b. To find out how much advertising is needed to sell 300,000 boxes of cereal (x), we substitute y = 300,000 into the equation and solve for x:
300,000 = 5x - 100,000
400,000 = 5x
x = 400,000 / 5
x = 80,000
Hence, $80,000 of advertising is needed to sell 300,000 boxes of cereal.
c. The slope of the equation, which is 5 in this case, represents the rate of change. It indicates that for every increase of $1,000 spent on advertising, the number of boxes sold will increase by 5,000. In other words, the slope shows the "marginal effect" of advertising on the number of boxes sold.