Question

question an object was dropped off the top of a building. the function f(x)=-16x² + 144 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 2 f(-2)=, meaning that seconds after the object was dropped, the object was feet above the ground. this interpretation in the context of the problem. f(1.5)=, meaning that seconds after the object was dropped, the object was feet above the ground. this interpretation in the context of the problem. f(4)=, meaning that seconds after the object was dropped, the object was feet above the ground. this interpretation 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)=-16x² + 144.
$f(-2)=-16\times(-2)^{2}+144=-16\times4 + 144=-64 + 144 = 80$

Step2: Interpret f(-2)

In the context of the problem, time cannot be negative when talking about the time after the object is dropped. So the interpretation of f(-2) is not valid in the real - world context of this problem. But the value of f(-2) is 80.

Step3: Calculate f(1.5)

Substitute x = 1.5 into f(x)=-16x² + 144.
$f(1.5)=-16\times(1.5)^{2}+144=-16\times2.25+144=-36 + 144 = 108$

Step4: Interpret f(1.5)

1.5 seconds after the object was dropped, the object was 108 feet above the ground. This interpretation is valid in the context of the problem.

Step5: Calculate f(4)

Substitute x = 4 into f(x)=-16x² + 144.
$f(4)=-16\times4^{2}+144=-16\times16+144=-256+144=-112$

Step6: Interpret f(4)

Since height cannot be negative in this context, when x = 4, the object has already hit the ground. The interpretation of f(4) is not valid in the real - world context of this problem.

Step7: Determine the domain

The object hits the ground when f(x)=0. Set -16x² + 144 = 0. Then 16x²=144, x² = 9, x = 3 (we take the positive value since x represents time after dropping). So the appropriate domain is $0\leq x\leq3$.

Answer:

$f(-2)=80$, meaning that - 2 seconds after the object was dropped (not valid in context), the object was 80 feet above the ground.
$f(1.5)=108$, meaning that 1.5 seconds after the object was dropped, the object was 108 feet above the ground.
$f(4)=-112$, meaning that 4 seconds after the object was dropped (not valid in context), the object was - 112 feet above the ground.
The appropriate domain for the function is $0\leq x\leq3$.