7. A Company finds that its sales since the company started in 2000 can be modelled by the function s(t)=20t^2+800t+300/8t^2+10t+100, where S is the total sales, in millions of dollars, and t is the number of years since 2000.
(a) Calculate the years when the sales are 9 million, algebraically. (3 marks)
(b) After many years, what does the model predict sales will be? (1 mark)
(a) To algebraically calculate the years when the sales are 9 million, we need to set the function s(t) equal to 9 million and solve for t.
The given function is: s(t) = (20t^2 + 800t + 300) / (8t^2 + 10t + 100)
Setting s(t) equal to 9 million:
9 million = (20t^2 + 800t + 300) / (8t^2 + 10t + 100)
To simplify the equation, let's multiply both sides by the denominator to eliminate the fraction:
9 million * (8t^2 + 10t + 100) = 20t^2 + 800t + 300
Expanding and rearranging the equation:
72 million t^2 + 90 million t + 900 million = 20t^2 + 800t + 300
Combining like terms:
52 million t^2 + 90 million t + 900 million = 0
Now, we have a quadratic equation that can be solved using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 52 million, b = 90 million, and c = 900 million.
Substituting the values into the quadratic formula:
t = (-(90 million) ± √((90 million)^2 - 4 * 52 million * 900 million)) / (2 * 52 million)
t = (-90 million ± √(8100 million^2 - 187200 million^2)) / (104 million)
Now, we can simplify the equation by using a calculator to compute the values within the square root:
t = (-90 million ± √(14580000 million^2)) / (104 million)
t = (-90 million ± 381591.7728 million) / (104 million)
Dividing the denominators:
t = (-90 ± 381591.7728) / 104
Now, we have two possible values for t:
t = (-90 + 381591.7728) / 104
t = 3,634.8857 years
t = (-90 - 381591.7728) / 104
t = -3,674.0296 years
Since t represents the number of years since 2000, we exclude negative values. Therefore, the years when sales are 9 million are approximately 3,635 years since 2000.
(b) To predict sales after many years, we need to evaluate the function s(t) as t approaches infinity.
As t approaches infinity, the higher order terms in the numerator and denominator will dominate the expression.
Looking at the terms in the function s(t), we see that the highest power of t in the numerator and denominator is t^2.
Therefore, as t approaches infinity, the terms with lower powers of t become insignificant.
So, we can simplify the function s(t) by considering only the highest power terms:
s(t) ≈ 20t^2 / 8t^2
Now, we can cancel out the common terms to get:
s(t) ≈ (20/8)
Simplifying the expression:
s(t) ≈ 2.5
Therefore, the model predicts that sales will be approximately 2.5 million dollars after many years.