Given the equation:
[tex]y=-0.26x^2-0.55x+91.81[/tex]Where x represents the number of years after 2000.
Let's solve for the following:
a.) Calculate the number of deaths per 100,000 for 2015 and 2017.
• For 2015, we have:
Number of years between 2015 and 2000 = 2015 - 2000 = 15
Substitute 15 for x and solve for y:
[tex]\begin{gathered} y=-0.26(15)^2-0.55(15)+91.81 \\ \\ y=-0.26(225)-8.25+91.81 \\ \\ y=-58.5-8.25+91.81 \\ \\ y=25.06\approx25 \end{gathered}[/tex]The number of deaths per 100,000 for 2015 is 25.
• For 2017:
Number of years between 2017 and 2000 = 2017 - 2000 = 17 years
Subustitute 17 for x and solve for y:
[tex]\begin{gathered} y=-0.25(17)^2-0.55(17)+91.81 \\ \\ y=7.32\approx7 \end{gathered}[/tex]The number of deaths oer 100,000 for 2017 is 7.
• b.) Let's solve for x when y = 50 using the quadratic formula.
Apply the quadratic formula:
[tex]x=\frac{-b\pm\sqrt[]{(b^2-4ac)}}{2a}[/tex]Now, subsitute 50 for y and equate to zero:
[tex]50=-0.26x^2-0.55x+91.81[/tex]Subtract 50 from both sides:
[tex]\begin{gathered} 50-50=-0.26x^2-0.55x+91.81-50 \\ \\ 0=-0.26x^2-0.55+41.81 \\ \\ -0.26x^2-0.55+41.81=0 \end{gathered}[/tex]Apply the general quadractic equation to get the values of a, b and c:
[tex]\begin{gathered} ax^2+bx+c=0 \\ \\ -0.26x^2-0.55+41.81=0 \end{gathered}[/tex]Hence, we have:
a = -0.26
b = -0.55
c = 41.81
Thus, we have:
[tex]\begin{gathered} x=\frac{-(-0.55)\pm\sqrt[]{-0.55^2-4(-0.26\ast41.81)}}{2(-0.26)} \\ \\ x=\frac{0.55\pm\sqrt[]{0.3025+43.4824}}{-0.52} \\ \\ x=\frac{0.55\pm6.617}{-0.52} \\ \\ x=-13.78,\text{ 11.}67 \end{gathered}[/tex]Since the number of years cannot be a negative value, let's take the positive value 11.67
Therefore, the value of x is 11.67 when y = 50.
