Question number four

Question

An icecream store reports sales of 350, 000 dollars in 2019 and 400, 000 in 2020. Assume a linear relationship exists which gives the sales of the icecream store S (in dollars) in terms of time t (in years).

(a) Find the linear function S(t) which models the information given in the

problem statement.

(b) Use S(t) to find when the company’s sales will total 600, 000

Answer 1

You could consider the given data as absolute years or as years after 2019. Let's choose the later since it produces a simpler equation. So we'll "clarify" that "t" is years after 2019. Consider the given data as points:

(0, 350,000) and (1, 400,000)

Use the point-slope form of a line:

y - y₁ = m(x - x₁)

First calculate the slope (m) using the slope equation:

m = (400,000 - 350,000)/(1 - 0) = 50,000

Plug in the slope into y - y₁ = m(x - x₁) to get y - y₁ = 50,000(x - x₁) then plug in one of the points. I'll choose (1, 400,000):

y - 400,000 = 50,000(x - 1)

y = 50,000x + 350,000

Converting to our correct variables:

S(t) = 50,000t + 350,000

To find when sales are 600,000, we plug in 600,000 for S(t):

600,000 = 50,000t + 350,000

250,000 = 50,000t

t = 5

So 5 years after 2019, which would be 2024

If you feel more comfortable using absolute years, make the data points (2019, 350,000) and (2020, 400,000) and repeat the process. You'll get a different function S(t) with larger numbers but the same year when sales are 600,000.