My problem:
From 1992 to 1996, the annual income for all the private golf courses in the United States can be approximated by the model: I=59t^2+2254, where I is the annual income in millions of dollars and t is the year, with t=0 corresponding to 1990.
1. In which year did the annual income increase to more than $3,000,000?
2. In which year did the annual income increase to more than $4,000,000?
3. Predict the year in which the annual income will increase to more than $5,500,000.
4. In which year was the annual income $2,254,000?
Thanks for helping! In class right not we are learning quadratic formulas so i think your supposed to use that. Please show as much work as you can. thank you :)
I think you either have part of the formula incoreent or some faulty data
e.g.
#1.
59t^2 + 2254 > 3000000
59t^2 > 2 997 746
t^2 > 50 809.25
t > 225 , It would take 225 years??
also in terms of 1990 , t=0, the income would be
2254 million, so how can you just ask for 3 million?
Once you have established the correct equation, just plug in the matching values of a, b, and c into the formula.
To solve these questions using the given model, we need to substitute the given values into the equation and solve for t. I will guide you through the steps to find the answers to each question.
1. In which year did the annual income increase to more than $3,000,000?
To find the year when the annual income exceeds $3,000,000, we need to solve the equation for t. Let's substitute I = 3,000 into the equation:
3,000 = 59t^2 + 2,254
Now we need to isolate t, so let's subtract 2,254 from both sides:
59t^2 = 3,000 - 2,254
59t^2 = 746
To isolate t, divide both sides of the equation by 59:
t^2 = 746 / 59
t^2 = 12.644
To solve for t, take the square root of both sides:
t = sqrt(12.644)
t ≈ 3.556
Since t represents years from 1990, we add 3.556 to 1990:
1990 + 3.556 ≈ 1993.556
Therefore, the annual income exceeds $3,000,000 in approximately the year 1993-1994.
2. In which year did the annual income increase to more than $4,000,000?
Using the same steps as question 1, we substitute I = 4,000 into the equation:
4,000 = 59t^2 + 2,254
To isolate t, subtract 2,254 from both sides:
59t^2 = 4,000 - 2,254
59t^2 = 1,746
Divide both sides by 59 to isolate t:
t^2 = 1,746 / 59
t^2 ≈ 29.610
Taking the square root of both sides:
t = sqrt(29.610)
t ≈ 5.439
Adding t to 1990:
1990 + 5.439 ≈ 1995.439
Therefore, the annual income exceeds $4,000,000 in approximately the year 1995-1996.
3. Predict the year in which the annual income will increase to more than $5,500,000.
Apply the same steps as before, substitute I = 5,500 into the equation:
5,500 = 59t^2 + 2,254
Isolate t by subtracting 2,254 from both sides:
59t^2 = 5,500 - 2,254
59t^2 = 3,246
Divide both sides by 59 to isolate t:
t^2 = 3,246 / 59
t^2 ≈ 55.051
Taking the square root of both sides:
t = sqrt(55.051)
t ≈ 7.422
Adding t to 1990:
1990 + 7.422 ≈ 1997.422
Therefore, the annual income will exceed $5,500,000 in approximately the year 1997-1998.
4. In which year was the annual income $2,254,000?
To find the year when the annual income equals $2,254,000, we need to substitute I = 2,254 into the equation:
2,254 = 59t^2 + 2,254
Subtract 2,254 from both sides to isolate 59t^2:
59t^2 = 0
Dividing both sides by 59:
t^2 = 0 / 59
t^2 = 0
Taking the square root of both sides:
t = sqrt(0)
t = 0
Therefore, the annual income was $2,254,000 in the year 1990.
I hope this explanation helps you understand how to use the given model to solve these questions. If you have any further questions, feel free to ask!