Question

(b) fill in the blanks below to write an inequality for all the values of x for which the cannonball is gaining height (going up). then choose the best description of this inequality.
inequality: (square < x < square)
description:
(circ) the function (y = -16x^2 + 64x) is increasing over these values of (x).
(circ) the function (y = -16x^2 + 64x) is decreasing over these values of (x).
(c) find the maximum value of (y) for the function (y = -16x^2 + 64x). then choose the best description of the maximum value of (y).
maximum value of (y): (square)
description:
(circ) the height in feet above the ground the cannonball starts at
(circ) the highest elevation in feet the cannonball reaches
(circ) the time in seconds when the cannonball hits the ground

Explanation:

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Part (b)

Step1: Find vertex of parabola

For $y=ax^2+bx+c$, vertex $x=-\frac{b}{2a}$. Here $a=-16, b=64$, so $x=-\frac{64}{2(-16)}=2$.

Step2: Identify increasing interval

Since $a<0$, parabola opens downward. Function increases from $x=0$ (launch) to vertex $x=2$. So interval is $0

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Part (c)

Step1: Substitute $x=2$ into function

$y=-16(2)^2 + 64(2)$

Step2: Calculate maximum value

$y=-16(4)+128=-64+128=64$. This value is the peak height of the cannonball.

Answer:

Part (b)

Inequality: $0 < x < 2$
Description: The function $y = -16x^2 + 64x$ is increasing over these values of $x$

Part (c)

Maximum value of $y$: $64$
Description: The highest elevation in feet the cannonball reaches