Answer:
The equation of the parabola that models the path of the long jumper through the air is [tex]y = -x^{2} +12[/tex]×[tex]x-27[/tex]
Step-by-step explanation:
Mathematically, we know that parabolas are second-order polynomials and every second-order polynomials, also known as quadratic functions, can be constructed by knowing three different points of the curve. The standard form of the parabola is:
[tex]y = a[/tex] ×[tex]x^{2} +b[/tex]×[tex]+ C[/tex]
Where:
x - Horizontal distance from the start line, measured in meters.
y - Height of the long jumper, measured in meters.
a,b,c - Polynomial constants, measured in [tex]\frac{1}{m}[/tex], dimensionless and meters, respectively.
If we know that[tex](x_{1},y_{1}) = (3\,m, 0\,m), (x_{2},y_{2}) = (8\,m, 0.05\,m)[/tex]and [tex](x_{3}, y_{3}) = (9\,m, 0\,m),[/tex] this system of linear equations is presented below:
[tex]9\cdot a + 3\cdot b + c = 0 (Eq. 1)81\cdot a + 9\cdot b + c = 0 (Eq. 2)64\cdot a + 8\cdot b + c = 0.05 (Eq. 3)[/tex]
The coefficients of the polynomial are, respectively:
[tex]a = -\frac{1}{100}, b = \frac{3}{25}, c = -\frac{27}{100}[/tex]
The equation of the parabola that models the path of the long jumper through the air is [tex]y' = -\frac{1}{100}\cdot x^{2}+\frac{3}{25}\cdot x -\frac{27}{100}.[/tex]
But we need y measured in centimeters, then, we use the following conversion:
[tex]y = 100\cdot y'[/tex]
Then, we get that:
[tex]y = -x^{2}+12\cdot x -27[/tex]
Where x and y are measured in meters and centimeters, respectively.
The equation of the parabola that models the path of the long jumper through the air is [tex]y = -x^{2}+12\cdot x -27.[/tex]
