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Answer :
- Given the vertex, the model is given by:
[tex]y = -2x^2 + 8x + 3[/tex]
- Using the model above, it is found that it lands on the ground after 4.35 seconds.
The equation of a quadratic function of vertex (h,k) is given by:
[tex]y = a(x - h)^2 + k[/tex]
The vertex is the maximum point, which is (2,11), hence [tex]h = 2, k = 11[/tex]. Then:
[tex]y = a(x - 2)^2 + 11[/tex]
The initial height is of 3 feet, then when [tex]x = 0, y = 3[/tex], and this is used to find a.
[tex]y = a(x - 2)^2 + 11[/tex]
[tex]3 = 4a + 11[/tex]
[tex]4a = -8[/tex]
[tex]a = -\frac{8}{4}[/tex]
[tex]a = -2[/tex]
Then:
[tex]y = -2(x - 2)^2 + 11[/tex]
In standard format, the model is:
[tex]y = -2x^2 + 8x - 8 + 11[/tex]
[tex]y = -2x^2 + 8x + 3[/tex]
It hits the ground when [tex]y = 0[/tex], so:
[tex]-2x^2 + 8x + 3 = 0[/tex]
[tex]2x^2 - 8x - 3 = 0[/tex]
Which has coefficients [tex]a = 2, b = -8, c = -3[/tex]. So
[tex]\Delta = b^2 - 4ac = (-8)^2 - 4(2)(-3) = 88[/tex]
[tex]x_1 = \frac{-b + \sqrt{\Delta}}{2a} = \frac{8 + sqrt{88}}{4} = 4.35[/tex]
[tex]x_1 = \frac{-b - \sqrt{\Delta}}{2a} = \frac{8 - sqrt{88}}{4} = -0.35[/tex]
Time is positive, so it lands on the ground after 4.35 seconds.
A similar problem is given at https://brainly.com/question/17987697
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Final answer:
The motion of the toy rocket is modeled by the equation h = -4(t-2)^2 + 14. Solving this equation results in the rocket landing at t = 3 seconds after launch.
Explanation:
The subject area of this problem is mathematics, particularly about studying the motions in a vertical line or free fall. The problem involves a toy rocket that is launched vertically and then returns to the ground. We can model this motion using a quadratic equation which fits the key stages of the rocket's path.
Let's denote the height of the rocket as h (in feet) and time as t (in seconds). Since the rocket initially launches from a height of 3 feet and reaches a maximum height of 11 feet in 2 seconds, our equation will look like: h = -4(t-2)^2 + 11 + 3.
To find out when the toy rocket will land, we need to set h to 0 and solve for t: 0 = -4(t-2)^2 + 14. After simplifying and taking the positive root, we find t = 3 seconds.
Learn more about Mathematical Modeling here:
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