Question

a rock is thrown from the top of a tall building. the distance, in feet, between the rock and the ground x seconds after it is thrown is given by $f(x) = -16x^2 - 4x + 382$. how long after the rock is thrown is it 340 feet from the ground? your answer

Explanation:

Step1: Set f(x) equal to 340

$340 = -16x^2 - 4x + 382$

Step2: Rearrange to standard quadratic form

$16x^2 + 4x - 42 = 0$

Step3: Simplify the quadratic equation

Divide all terms by 2: $8x^2 + 2x - 21 = 0$

Step4: Solve via quadratic formula

For $ax^2+bx+c=0$, $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Here $a=8$, $b=2$, $c=-21$:

$$ x=\frac{-2\pm\sqrt{2^2-4(8)(-21)}}{2(8)} $$

Calculate discriminant: $\sqrt{4 + 672}=\sqrt{676}=26$

$$ x=\frac{-2\pm26}{16} $$

Step5: Select positive solution

$x=\frac{-2+26}{16}=\frac{24}{16}=1.5$; discard negative $x=\frac{-2-26}{16}=-1.75$

Answer:

1.5 seconds