For 1983 through 1995, the amount of private donations D (in millions) allocated to education can be modeled by
D= 1.78t^3-6.02t^2+ 752t+ 6701
where t is the number of years since 1983. In what year was 14.3 billion of private donations allocated to education?
(My main problems were location the real zeros on the Ti-84 graphing calculator of which I could only find one line on the graph. Also I'm not sure if when I subtract 6701 from 14.3 billion if I am suppose to move the decimal back one for remaining numbers. Ex. being 1.78t^3 to 0.178t^3.)
Thanks in advance.
To find the year when 14.3 billion of private donations was allocated to education, we need to solve the equation D = 14.3 billion for t.
First, let's convert 14.3 billion to millions. 1 billion is equal to 1,000 million, so 14.3 billion is equivalent to 14,300 million.
Now we can substitute D = 14,300 million into the equation:
14,300 = 1.78t^3 - 6.02t^2 + 752t + 6701
To solve this equation, you can follow these steps:
1. Rearrange the equation to have all the terms on one side:
1.78t^3 - 6.02t^2 + 752t + 6701 - 14,300 = 0
2. Simplify the equation:
1.78t^3 - 6.02t^2 + 752t - 7628 = 0
3. At this point, you can use a graphing calculator, like the TI-84, to find the real zeros of the equation. Here's how to do it:
- Enter the equation: Y1 = 1.78X^3 - 6.02X^2 + 752X - 7628
- Use the graphing function and find where the curve intersects the x-axis. These are the real zeros of the equation.
- The calculator will give you the values of t (or X) where the equation equals zero. Make sure to note the sign of the zeros since we're looking for positive values of t.
Once you've found the real zeros, you can identify the value of t that corresponds to the year by adding it to 1983. For example, if t = 2, then the corresponding year would be 1983 + 2 = 1985.
Regarding your question about whether to move the decimal point, there is no need to do so. The equation is already in terms of millions, so there's no need to change the scale.
I hope the explanation and steps provided help you solve the problem. Feel free to ask if you have any further questions!