Step-by-step explanation:
y = -40x² + 120x + 160
a) When the rocket lands, y = 0. So we need to write the equation in factored form.
y = -40 (x² − 3x − 4)
y = -40 (x + 1) (x − 4)
Setting y equal to 0:
0 = -40 (x + 1) (x − 4)
x = -1 or x = 4
x can't be negative, so the rocket lands after 4 seconds.
b) The rocket reaches its maximum height at the vertex of the parabola. So we need to write the equation in vertex form.
y = -40 (x² − 3x) + 160
y = -40 (x² − 3x + 9/4) + 40(9/4) + 160
y = -40 (x − 3/2)² + 250
The vertex is (3/2, 250). So the rocket reaches a maximum height of 250 feet after 1.5 seconds.
c) There are many possible answers. As long as one root is 4 and the other root is nonpositive, we can write an equation such that the rocket reaches a new maximum height without changing the time that it lands.
To do this, we must change either the acceleration (-40×2) or the initial height (160) or both.
If we change the acceleration, the new equation is:
y = (x − 4) (-ax − 40)
If we change the initial height, the new equation is:
y = (x − 4) (-40x − b)
In each case, a and b are any number greater than 40.
Graph: desmos.com/calculator/zxnjo0rm2r
