Question

an object was dropped off the top of a building. the function $f(x) = -16x^2 + 64$ represents the height of the object above the ground, in feet, $x$ seconds after being dropped. find and interpret the given function values and determine an appropriate domain for the function.

answer attempt 1 out of 5
$f(-2) = \square$, meaning that $\square$ seconds after the object was dropped, the object was $\square$ feet above the ground.
this interpretation \quad in the context of the problem.

$f(0.5) = \square$, meaning that $\square$ seconds after the object was dropped, the object was $\square$ feet above the ground.
this interpretation \quad in the context of the problem.

$f(7) = \square$, meaning that $\square$ seconds after the object was dropped, the object was $\square$ feet above the ground.
this interpretation \quad in the context of the problem.

based on the observations above, it is clear that an appropriate domain for the function is

Explanation:

Step1: Calculate $f(-2)$

Substitute $x=-2$ into $f(x)$:
$$f(-2) = -16(-2)^2 + 64 = -16(4) + 64 = -64 + 64 = 0$$

Step2: Interpret $f(-2)$

Negative time has no real-world meaning here, as time starts at 0 when the object is dropped.

Step3: Calculate $f(0.5)$

Substitute $x=0.5$ into $f(x)$:
$$f(0.5) = -16(0.5)^2 + 64 = -16(0.25) + 64 = -4 + 64 = 60$$

Step4: Interpret $f(0.5)$

0.5 seconds is a valid positive time after the object is dropped, so this makes sense in context.

Step5: Calculate $f(7)$

Substitute $x=7$ into $f(x)$:
$$f(7) = -16(7)^2 + 64 = -16(49) + 64 = -784 + 64 = -720$$

Step6: Interpret $f(7)$

A negative height means the object would be below ground level, which is not possible in this context, as the object hits the ground when its height is 0.

Step7: Find valid domain

First, find when the object hits the ground by setting $f(x)=0$:
$$0 = -16x^2 + 64$$
$$16x^2 = 64$$
$$x^2 = 4$$
$$x = 2 \quad (\text{since } x \geq 0)$$
Time starts at 0 when the object is dropped, so the domain is $0 \leq x \leq 2$.

Answer:

$f(-2) = 0$, meaning that $-2$ seconds after the object was dropped, the object was $0$ feet above the ground.
This interpretation is not valid in the context of the problem.

$f(0.5) = 60$, meaning that $0.5$ seconds after the object was dropped, the object was $60$ feet above the ground.
This interpretation is valid in the context of the problem.

$f(7) = -720$, meaning that $7$ seconds after the object was dropped, the object was $-720$ feet above the ground.
This interpretation is not valid in the context of the problem.

Based on the observations above, it is clear that an appropriate domain for the function is $0 \leq x \leq 2$ (or $[0,2]$ in interval notation)