Respuesta :
Answer:
6x2 + 16x = 672
Reorder the terms:
16x + 16x2 = 672
Solving
16x + 16x2 = 672
Solving for variable 'x'.
Reorder the terms:
-672 + 16x + 16x2 = 672 + -672
Combine like terms: 672 + -672 = 0
-672 + 16x + 16x2 = 0
Factor out the Greatest Common Factor (GCF), '16'.
16(-42 + x + x2) = 0
Factor a trinomial.
16((-7 + -1x)(6 + -1x)) = 0
Ignore the factor 16.
Subproblem 1
Set the factor '(-7 + -1x)' equal to zero and attempt to solve:
Simplifying
-7 + -1x = 0
Solving
-7 + -1x = 0
Move all terms containing x to the left, all other terms to the right.
Add '7' to each side of the equation.
-7 + 7 + -1x = 0 + 7
Combine like terms: -7 + 7 = 0
0 + -1x = 0 + 7
-1x = 0 + 7
Combine like terms: 0 + 7 = 7
-1x = 7
Divide each side by '-1'.
x = -7
Simplifying
x = -7
Subproblem 2
Set the factor '(6 + -1x)' equal to zero and attempt to solve:
Simplifying
6 + -1x = 0
Solving
6 + -1x = 0
Move all terms containing x to the left, all other terms to the right.
Add '-6' to each side of the equation.
6 + -6 + -1x = 0 + -6
Combine like terms: 6 + -6 = 0
0 + -1x = 0 + -6
-1x = 0 + -6
Combine like terms: 0 + -6 = -6
-1x = -6
Divide each side by '-1'.
x = 6
Simplifying
x = 6
Solution
x = {-7, 6}
Step-by-step explanation:
The given quadratic function models the projectile of the object as it is
launched off the top of the building.
The interpretation of the function values are;
- The maximum height reached by the object is 676 feet
- The height of the building is 672 feet
- Time of flight of the object is 7 seconds
The appropriate domain is 0 ≤ x ≤ 7
Reasons:
The given function for the height of the object is f(x) = -16·x² + 16·x + 672
The domain is given by the values of x for which the value of y ≥ 0
Therefore, when -16·x² + 16·x + 672 = 0, we get;
-16·x² + 16·x + 672 = 0
16·(-x² + x + 42) = 0
-x² + x + 42 = 0
x² - x - 42 = 0
(x - 7)·(x + 6) = 0
x = 7, or x = -6
The minimum value of time, x is 0, which is the x-value at the top of the
building, and when x = 7, the object is on the ground.
Therefore;
- The appropriate domain is 0 ≤ x ≤ 7
The maximum value of f(x) = a·x² + b·x + c, is given at [tex]x = -\dfrac{b}{2 \cdot a}[/tex]
Therefore;
We have;
[tex]x = -\dfrac{16}{2 \times (-16)} = \dfrac{1}{2}[/tex]
Which gives;
[tex]f \left(\frac{1}{2} \right) = -16 \times \left(\dfrac{1}{2} \right)^2 + 16 \times \left(\dfrac{1}{2} \right)+ 672 = 676[/tex]
- The maximum height reached by the object, [tex]f\left(\frac{1}{2} \right)[/tex] = 676 feet
The height of the building is given when the time, x = 0, as follows;
Height of building, f(0) = -16 × 0² + 16 × 0 + 672 = 672
- The height of the building, f(0) = 672 feet
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