Eyjafjallajökull is a volcano in ice land. during a recent eruption, the volcano, spewed out copious amounts of ash. one small was ejected from the volcano with an initial velocity of 336ft/sec. the height h, in feet, of our ash projectile is given by the equation h= -16t2 +336t. where t is the time is seconds, assuming the volcano has no height. h=0 t=0
1. when does the ash projectile reach its maximum height?
2. what is its maximum height?
3. when does the ash projectile return to the ground?

answer: 12

step-by-step explanation:

1. when "a" is a root of a polynomial, (x -a) is a factor of it. for the three roots given, the factors of the desired polynomial are (x +2)(x -1/2)(x -3).

in order to make the leading coefficient be negative, we need to multiply this product by a negative number. any negative number will do, but we choose a small (magnitude) value that will eliminate the fraction: -2.

then

f(x) = -2(x +2)(x -1/2)(x -3) = -(x +2)(2x -1)(x -3)

= -(2x² +3x -2)(x -3)

= -(2x³ -3x² -11x +6)

f(x) = -2x³ +3x² +11x -6

2. a graph created by the desmos on-line graphing calculator is shown, and the zeros are highlighted.

3. as indicated in part 1, the multiplier of this equation can be anything and the zeros will remain the same. you want a negative leading coefficient, so the "anything" is restricted to any of the infinite number of numbers that will make that be the case.