Answer:
The equation of the parabola that models the path of the long jumper through the air is
×![x-27](https://tex.z-dn.net/?f=x-27)
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:
×
×![+ C](https://tex.z-dn.net/?f=%2B%20C)
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
, dimensionless and meters, respectively.
If we know that
and
this system of linear equations is presented below:
![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)](https://tex.z-dn.net/?f=9%5Ccdot%20a%20%2B%203%5Ccdot%20b%20%2B%20c%20%3D%200%20%28Eq.%201%2981%5Ccdot%20a%20%2B%209%5Ccdot%20b%20%2B%20c%20%3D%200%20%28Eq.%202%2964%5Ccdot%20a%20%2B%208%5Ccdot%20b%20%2B%20c%20%3D%200.05%20%28Eq.%203%29)
The coefficients of the polynomial are, respectively:
![a = -\frac{1}{100}, b = \frac{3}{25}, c = -\frac{27}{100}](https://tex.z-dn.net/?f=a%20%3D%20-%5Cfrac%7B1%7D%7B100%7D%2C%20b%20%3D%20%5Cfrac%7B3%7D%7B25%7D%2C%20c%20%3D%20-%5Cfrac%7B27%7D%7B100%7D)
The equation of the parabola that models the path of the long jumper through the air is ![y' = -\frac{1}{100}\cdot x^{2}+\frac{3}{25}\cdot x -\frac{27}{100}.](https://tex.z-dn.net/?f=y%27%20%3D%20-%5Cfrac%7B1%7D%7B100%7D%5Ccdot%20x%5E%7B2%7D%2B%5Cfrac%7B3%7D%7B25%7D%5Ccdot%20x%20-%5Cfrac%7B27%7D%7B100%7D.)
But we need y measured in centimeters, then, we use the following conversion:
![y = 100\cdot y'](https://tex.z-dn.net/?f=y%20%3D%20100%5Ccdot%20y%27)
Then, we get that:
![y = -x^{2}+12\cdot x -27](https://tex.z-dn.net/?f=y%20%3D%20-x%5E%7B2%7D%2B12%5Ccdot%20x%20-27)
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 ![y = -x^{2}+12\cdot x -27.](https://tex.z-dn.net/?f=y%20%3D%20-x%5E%7B2%7D%2B12%5Ccdot%20x%20-27.)