The function [tex]y = -0.03(x - 14)^2 + 6[/tex] models the jump of a red kangaroo where [tex]x[/tex] is the horizontal distance in meters and [tex]y[/tex] is the vertical distance in meters for the height of the jump.
What is the kangaroo's maximum height?
How long is the kangaroo's jump?
Answer (3)
Kangaroo's maximum height is 6 m and the kangaroo's jump is 28 m long ;
From the vertex of the quadratic equation, we find that: The kangaroos maximum height is of 6 meters.
From the roots of the equation, we find that: The kangaroo's jump is 28.14 meters long .
Vertex of a quadratic function:
Suppose we have a quadratic function in the following format:
f ( x ) = a x 2 + b x + c
It's vertex is the point ( x v , y v )
In which
x v = − 2 a b
y v = − 4 a Δ
Where
Δ = b 2 − 4 a c
If a<0, the vertex is a maximum point , that is, the maximum value happens at x v , and it's value is y v .
Solving a quadratic equation:
Given a second order polynomial expressed by the following equation:
a x 2 + b x + c , a = 0 .
This polynomial has roots x 1 , x 2 such that a x 2 + b x + c = a ( x − x 1 ) ∗ ( x − x 2 ) , given by the following formulas:
x 1 = 2 ∗ a − b + Δ
x 2 = 2 ∗ a − b − Δ
Δ = b 2 − 4 a c
The quadratic equation is:
y = − 0.03 ( x − 14 ) 2 + 6
Placing in standard form:
y = − 0.03 ( x 2 − 28 x + 196 ) + 6
y = − 0.03 x 2 + 0.84 x + 0.12
Thus, it has coefficients a = − 0.03 , b = 0.84 , c = 0.12
The kangaroo's maximum height is the y-value of the vertex , thus:
Δ = b 2 − 4 a c = ( 0.84 ) 2 − 4 ( − 0.03 ) ( 0.12 ) = 0.72
y v = − 4 a Δ = − 4 ( − 0.03 ) 0.72 = 6
The kangaroos maximum height is of 6 meters.
The length of the kangaroo's jump is the positive root. The roots are found at the values of x for which y = 0, thus, the solutions of the quadratic equation.
x 1 = 2 ( − 0.03 ) − 0.84 + 0.72 = − 0.14
x 2 = 2 ( − 0.03 ) − 0.84 − 0.72 = 28.14
The kangaroo's jump is 28.14 meters long .
A similar question is given at https://brainly.com/question/16858635
The kangaroo's maximum height is 6 meters, and the length of its jump is approximately 28.14 meters.
;
