setup an equation or expression to model the situation below:
in one year, sales for a company increase at a constant rate. after two months, the company sells 12,500 products, and after 8 months, the company sells 30,000 products. If the constant rate of increase continues, how many products do you project the company to sell in 4 years?
To model the situation, let's assume that the rate of increase in sales is linear and remains constant over time. We can set up an equation to determine the number of products sold based on the number of months.
Let's designate:
- x as the number of months
- y as the number of products sold
We know that after two months, the company sells 12,500 products, so we have the point (2, 12,500). Similarly, after eight months, the company sells 30,000 products, giving us the point (8, 30,000).
Using these two points, we can find the equation for the line that represents the sales growth:
First, determine the slope (m):
m = (y2 - y1) / (x2 - x1)
= (30,000 - 12,500) / (8 - 2)
= 17,500 / 6
= 2,916.67
Now, we have the slope (m) and can determine the equation using the point-slope form:
y - y1 = m(x - x1)
y - 12,500 = 2,916.67(x - 2)
Expanding the equation:
y - 12,500 = 2,916.67x - 5,833.33
y = 2,916.67x - 5,833.33 + 12,500
y = 2,916.67x + 6,666.67
This equation represents the linear growth of sales for the company.
To project the number of products the company will sell in four years (48 months), we substitute x = 48 into the equation:
y = 2,916.67(48) + 6,666.67
y = 140,000
Therefore, we project that the company will sell 140,000 products in four years.