Thge function D(x) models the cumulative number of deaths from a disease x years after 1984. Estimate the year when ther were 89,000 deaths. D(x) = 2386x^2 +5147x + 5398. there were approximately 89,000 deaths in the year?
To estimate the year when there were approximately 89,000 deaths, we need to solve the equation D(x) = 89,000.
The given function is D(x) = 2386x^2 + 5147x + 5398.
So, we need to solve the equation:
2386x^2 + 5147x + 5398 = 89,000.
Let's rearrange the equation to get it in standard quadratic form (ax^2 + bx + c = 0):
2386x^2 + 5147x + (5398 - 89000) = 0.
Simplifying, we have:
2386x^2 + 5147x - 83602 = 0.
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a).
For this equation, a = 2386, b = 5147, and c = -83602.
Plugging in the values, we get:
x = (-5147 ± √(5147^2 - 4*2386*(-83602))) / (2*2386).
Calculating this expression, we get two possible solutions for x, representing two different years when the estimated deaths were approximately 89,000.
x ≈ -3.893 or x ≈ 8.044.
Since the number of years cannot be negative in this context, we consider x ≈ 8.044 as the appropriate solution.
To find the year, we add 8.044 to 1984:
Year ≈ 1984 + 8.044 ≈ 1992.044.
Therefore, the estimated year when there were approximately 89,000 deaths is around 1992.
To estimate the year when there were approximately 89,000 deaths, we need to find the value of x (number of years) when the function D(x) equals 89,000.
Given that D(x) = 2386x^2 + 5147x + 5398, we can set it equal to 89,000 and solve for x:
2386x^2 + 5147x + 5398 = 89,000
Now, let's rearrange the equation:
2386x^2 + 5147x + 5398 - 89,000 = 0
By simplifying the equation, we get a quadratic equation:
2386x^2 + 5147x - 84,602 = 0
Since this is a quadratic equation, we can solve it using various methods like factoring, completing the square, or using the quadratic formula.
In this case, let's use the quadratic formula to solve for x:
The quadratic formula is:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 2386, b = 5147, and c = -84,602.
Substituting these values into the quadratic formula, we get:
x = (-5147 ± √(5147^2 - 4 * 2386 * -84,602))/(2 * 2386)
Now, we can solve this equation to find two possible values for x. The positive value of x represents the number of years after 1984 when there were approximately 89,000 deaths.
By evaluating the equation, we find that the positive value of x is approximately 5.8.
Therefore, there were approximately 89,000 deaths in the year 1984 + 5.8 = 1990.8.
Rounded to the nearest whole year, this would be the year 1991.
D(x)=2386x^2+5147x+5398=89000
2386x^2+5147x+5398-89000=0
2386x^2+5147x-83602=0
If you don't know solve that equation in google type:
quadratic equation online
When you see list of results click on:
Free Online Quadratic Equation Solver:Solve by Quadratic Formula
When page be open in rectangle type:
2386x^2+5147x-83602=0
and click option solve it.
You will see solution step by step.
4.94 approx = 5
Approximately the year=1984+5=1989