1984-1994 the equation the number of aids cases can be modeled by the equation
C(x)=3034x^2+14,018x+6400
X represents years after 1984 estimate the year when 200,000 aids cases had been diagnosed.
X=1990
3034x^2+14,018x+6400 = 200000
3034x^2+14,018x - 193600 = 0
use the formula, you are just dealing with large numbers, I got
x = 6.005 or a negative
so adding 6 to 184 --->1990
To estimate the year when 200,000 AIDS cases had been diagnosed, we can set up the equation C(x) = 200,000 and solve for x.
C(x) = 3034x^2 + 14,018x + 6400
200,000 = 3034x^2 + 14,018x + 6400
Rearranging the equation:
3034x^2 + 14,018x + 6400 - 200,000 = 0
3034x^2 + 14,018x - 193,600 = 0
Now, we can use the quadratic formula to solve for x:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 3034, b = 14,018, and c = -193,600.
x = (-14,018 ± √((14,018)^2 - 4 * 3034 * (-193,600))) / (2 * 3034)
Calculating this using a calculator, we get two possible values for x:
x ≈ -1.192 or x ≈ 12.425
Since x represents years after 1984, we discard the negative value because it does not make sense in this context.
x ≈ 12.425
Therefore, it can be estimated that around 12.425 years after 1984, or approximately in the year 1996, 200,000 AIDS cases had been diagnosed.
To estimate the year when 200,000 AIDS cases had been diagnosed, we need to solve the equation C(x) = 200,000.
The equation C(x) = 3034x^2 + 14,018x + 6400 represents the number of AIDS cases (C(x)) in a given year (x) after 1984.
To solve C(x) = 200,000, we can rearrange the equation:
3034x^2 + 14,018x + 6400 = 200,000
Simplify the equation:
3034x^2 + 14,018x - 193,600 = 0
Now, we can solve this quadratic equation using various methods such as factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula:
The quadratic formula states that for an equation in the form ax^2 + bx + c = 0, the solutions for x can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation: 3034x^2 + 14,018x - 193,600 = 0, we have:
a = 3034
b = 14,018
c = -193,600
Plugging these values into the quadratic formula, we get:
x = (-14,018 ± √(14,018^2 - 4 * 3034 * -193,600)) / (2 * 3034)
Calculating this expression will give us two possible values for x, one positive and one negative. Since we are looking for years after 1984, the negative value is not relevant in this case.
By solving this equation using the quadratic formula, you will find the year when approximately 200,000 AIDS cases had been diagnosed.