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Sagot :
Answer:
a) The initial height of the rock is of 48 feet.
b) It takes 1 seconds for the rock to reach maximum height.
c) The rock's maximum height is 50 feet.
d) It takes 6 seconds for the rock to land on the ground.
Step-by-step explanation:
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}, f(x_{v})[/tex]
In which
[tex]x_{v} = -\frac{b}{2a}[/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]f(x_{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{\bigtriangleup}}{2*a}[/tex]
[tex]x_{2} = \frac{-b - \sqrt{\bigtriangleup}}{2*a}[/tex]
[tex]\bigtriangleup = b^{2} - 4ac[/tex]
In this question:
The height of the ball after t seconds, in feet, is given by:
[tex]h(t) = -2t^2 + 4t + 48[/tex]
Which is a quadratic equation with [tex]a = -2, b = 4, c = 48[/tex]
a) What is the initial height?
This is [tex]h(0) = 48[/tex]. So the initial height of the rock is of 48 feet.
b) How many seconds does it take to reach the maximum height?
This is the value of t at the vertex. So
[tex]t_{v} = -\frac{b}{2a} = -\frac{4}{2(-2)} = 1[/tex]
It takes 1 seconds for the rock to reach maximum height.
c) What is the rock's maximum height?
This is the height of the ball after 1 second. So
[tex]h(1) = -2(1)^2 + 4(1) + 48 = 50[/tex]
The rock's maximum height is 50 feet.
d) How many seconds does it take to for the rock to land on the ground?
This is t for which [tex]h(t) = 0[/tex], so we solve the quadratic equation.
[tex]\bigtriangleup = (4)^2-4(-2)(48) = 400[/tex]
[tex]t_{1} = \frac{-4 + \sqrt{400}}{2*(-2)} = -4[/tex]
[tex]t_{2} = \frac{-4 - \sqrt{400}}{2*(-2)} = 6[/tex]
Time is a positive measures, so it takes 6 seconds for the rock to land on the ground.
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