Respuesta :
Answer:
[tex]y = \frac{-2}{81}x^2 + 50[/tex]
Step-by-step explanation:
The parabola in Cartesian terms goes through the points:
(0, 50) [assuming center at 0]
(-45,0) and (45,0) (touches floor at both edges)
Since it is round
So, width = 90 cm = Diameter
So, Radius = [tex]\frac{Diameter}{2}=\frac{90}{2} = 45 cm[/tex]
So, take points (0, 50) and (45,0) and (-45,0)
Equation :[tex]y = ax^2 + bx + c[/tex] --A
since x= 0 gives 50
So, [tex]50 = a(0)^2 + b(0) + c[/tex]
So,c = 50
Since x= -45 and x = 45 gives the same result So, b has no effect i.e. b = 0
So, plugging in b and c for the value 45 for the expression
[tex]a\times 45^2 +0 \times 45 + 50 = 0[/tex]
[tex]2025a = -50[/tex]
[tex]a =\frac{ -50}{2025} [/tex]
[tex]a =\frac{ -2}{81} [/tex]
Now substitute values of a,b and c in A
[tex]y = \frac{-2}{81}x^2 + 50[/tex]
Hence the equation of parabola is [tex]y = \frac{-2}{81}x^2 + 50[/tex]
A parabola is simply an approximately U-shaped curve which is mirror-symmetrical
The equation of the parabola is: [tex]\mathbf{y = -\frac{2}{81}x^2 + 50}[/tex]
The equation of a parabola is represented as:
[tex]\mathbf{y = ax^2 + bx + c}[/tex]
The parabola passes through:
[tex]\mathbf{(x,y) = (0,50)}[/tex] --- the height
[tex]\mathbf{(x,y) = (-45,0), (45,0)}[/tex] --- the width
Substitute [tex]\mathbf{(x,y) = (0,50)}[/tex] in [tex]\mathbf{y = ax^2 + bx + c}[/tex]
[tex]\mathbf{50 = a(0)^2 + b(0) +c}[/tex]
[tex]\mathbf{50 = c}[/tex]
Rewrite as:
[tex]\mathbf{c = 50 }[/tex]
Substitute 50 for c in [tex]\mathbf{y = ax^2 + bx + c}[/tex]
[tex]\mathbf{y = ax^2 + bx + 50}[/tex]
Substitute [tex]\mathbf{(x,y) = (-45,0), (45,0)}[/tex] in [tex]\mathbf{y = ax^2 + bx + 50}[/tex]
[tex]\mathbf{0 = a(-45)^2 -45b + 50}[/tex]
[tex]\mathbf{0 = a(45)^2 +45b + 50}[/tex]
Add the above equations
[tex]\mathbf{0 + 0 = a(-45)^2 + a(45)^2 - 45b + 45b + 50 + 50}[/tex]
[tex]\mathbf{2025a + 2025a + 100 = 0}[/tex]
[tex]\mathbf{4050a + 100 = 0}[/tex]
Rewrite as:
[tex]\mathbf{4050a = -100}[/tex]
Divide both sides by 4050
[tex]\mathbf{a = -\frac{2}{81}}[/tex]
Substitute [tex]\mathbf{a = -\frac{2}{81}}[/tex] in [tex]\mathbf{0 = a(45)^2 +45b + 50}[/tex]
[tex]\mathbf{0 =-\frac{2}{81} \times 45^2 + 45b + 50}[/tex]
[tex]\mathbf{0 =-50 + 45b + 50}[/tex]
[tex]\mathbf{0 = 45b }[/tex]
Divide both sides by 45
[tex]\mathbf{b =0}[/tex]
Substitute values a, b and c in [tex]\mathbf{y = ax^2 + bx + c}[/tex]
[tex]\mathbf{y = -\frac{2}{81}x^2 + 50}[/tex]
Hence, the equation of the parabola is: [tex]\mathbf{y = -\frac{2}{81}x^2 + 50}[/tex]
Read more about parabolas at:
https://brainly.com/question/4074088
