At Westonci.ca, we connect you with the answers you need, thanks to our active and informed community. Our Q&A platform provides quick and trustworthy answers to your questions from experienced professionals in different areas of expertise. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
Answer:
Kangaroo's maximum height is 6 m and the kangaroo's jump is 28 m long
Step-by-step explanation:
Given :[tex]y = -0.03(x - 14)^2 + 6[/tex]
To Find : What is the kangaroo's maximum height? How long is the kangaroo's jump?
Solution:
[tex]y = -0.03(x - 14)^2 + 6[/tex]
x is the horizontal distance in meters
y is the vertical distance in meters for the height of the jump.
Substitute y = 0
[tex]0 = -0.03(x - 14)^2 + 6[/tex]
[tex]0.03(x - 14)^2 = 6[/tex]
[tex](x - 14)^2 = \frac{6}{0.03}[/tex]
[tex](x - 14)^2 =200[/tex]
[tex](x - 14) =\sqrt{200}[/tex]
[tex](x - 14) =14.142[/tex]
[tex]x =14.142+14[/tex]
[tex]x =28.142[/tex]
x≈ 28 m
Now the maximum height will be attained at mid point i.e. [tex]\frac{28}{2} =14[/tex]
Now substitute x= 14
[tex]y = -0.03(14 - 14)^2 + 6[/tex]
[tex]y = 6[/tex]
So, 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:
[tex]f(x) = ax^{2} + bx + c[/tex]
It's vertex is the point [tex](x_{v}, y_{v})[/tex]
In which
[tex]x_{v} = -\frac{b}{2a}[/tex]
[tex]y_{v} = -\frac{\Delta}{4a}[/tex]
Where
[tex]\Delta = b^2-4ac[/tex]
If a<0, the vertex is a maximum point, that is, the maximum value happens at [tex]x_{v}[/tex], and it's value is [tex]y_{v}[/tex].
----------------------------
Solving a quadratic equation:
Given a second order polynomial expressed by the following equation:
[tex]ax^{2} + bx + c, a\neq0[/tex].
This polynomial has roots [tex]x_{1}, x_{2}[/tex] such that [tex]ax^{2} + bx + c = a(x - x_{1})*(x - x_{2})[/tex], given by the following formulas:
[tex]x_{1} = \frac{-b + \sqrt{\Delta}}{2*a}[/tex]
[tex]x_{2} = \frac{-b - \sqrt{\Delta}}{2*a}[/tex]
[tex]\Delta = b^{2} - 4ac[/tex]
----------------------------
The quadratic equation is:
[tex]y = -0.03(x - 14)^2 + 6[/tex]
Placing in standard form:
[tex]y = -0.03(x^2 - 28x + 196) + 6[/tex]
[tex]y = -0.03x^2 + 0.84x + 0.12[/tex]
Thus, it has coefficients [tex]a = -0.03, b = 0.84, c = 0.12[/tex]
----------------------------
The kangaroo's maximum height is the y-value of the vertex, thus:
[tex]\Delta = b^2 - 4ac = (0.84)^2 - 4(-0.03)(0.12) = 0.72[/tex]
[tex]y_{v} = -\frac{\Delta}{4a} = -\frac{0.72}{4(-0.03)} = 6[/tex]
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.
[tex]x_{1} = \frac{-0.84 + \sqrt{0.72}}{2(-0.03)} = -0.14[/tex]
[tex]x_{2} = \frac{-0.84 - \sqrt{0.72}}{2(-0.03)} = 28.14[/tex]
The kangaroo's jump is 28.14 meters long.
A similar question is given at https://brainly.com/question/16858635
Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.
