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The path that Gloria follows when she jumped is a path of parabola.
The equation of the parabola that describes the path of her jump is [tex]\mathbf{y = -\frac{5}{49}(x - 14)^2 + 20}[/tex]
The given parameters are:
[tex]\mathbf{Height = 20}[/tex]
[tex]\mathbf{Length = 28}[/tex]
Assume she starts from the origin (0,0)
The midpoint would be:
[tex]\mathbf{Mid = \frac 12 \times Length}[/tex]
[tex]\mathbf{Mid = \frac 12 \times 28}[/tex]
[tex]\mathbf{Mid = 14}[/tex]
So, the vertex of the parabola is:
[tex]\mathbf{Vertex = (Mid,Height)}[/tex]
Express properly as:
[tex]\mathbf{(h,k) = (14,20)}[/tex]
A point on the graph would be:
[tex]\mathbf{(x,y) = (28,0)}[/tex]
The equation of a parabola is calculated using:
[tex]\mathbf{y = a(x - h)^2 + k}[/tex]
Substitute [tex]\mathbf{(h,k) = (14,20)}[/tex] in [tex]\mathbf{y = a(x - h)^2 + k}[/tex]
[tex]\mathbf{y = a(x - 14)^2 + 20}[/tex]
Substitute [tex]\mathbf{(x,y) = (28,0)}[/tex] in [tex]\mathbf{y = a(x - 14)^2 + 20}[/tex]
[tex]\mathbf{0 = a(28 - 14)^2 + 20}[/tex]
[tex]\mathbf{0 = a(14)^2 + 20}[/tex]
Collect like terms
[tex]\mathbf{a(14)^2 =- 20}[/tex]
Solve for a
[tex]\mathbf{a =- \frac{20}{14^2}}[/tex]
[tex]\mathbf{a =- \frac{20}{196}}[/tex]
Simplify
[tex]\mathbf{a =- \frac{5}{49}}[/tex]
Substitute [tex]\mathbf{a =- \frac{5}{49}}[/tex] in [tex]\mathbf{y = a(x - 14)^2 + 20}[/tex]
[tex]\mathbf{y = -\frac{5}{49}(x - 14)^2 + 20}[/tex]
Hence, the equation of the parabola that describes the path of her jump is [tex]\mathbf{y = -\frac{5}{49}(x - 14)^2 + 20}[/tex]
See attachment for the graph
Read more about equations of parabola at:
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