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The function h(t)=-16t^(2)+144 represents the height, 
h(t), in feet, of an object fron the ground at 
t seconds after it is dropped. A realistic domain (for time) for this function is:

-3 <= t <= 3

0 <= t <= 3

quad0 <= ℏ <= 144
all real numbers

The function h(t)=16t2+144h(t) = -16t^{2} + 144 represents the height, h(t)h(t), in feet, of an object from the ground at tt seconds after it is dropped. A realistic domain (for time) for this function is:\newline(A)(A) 3t3-3 \leq t \leq 3\newline(B)(B) 0t30 \leq t \leq 3\newline(C)(C) 0h1440 \leq h \leq 144\newline(D)(D) all real numbers

Full solution

Q. The function h(t)=16t2+144h(t) = -16t^{2} + 144 represents the height, h(t)h(t), in feet, of an object from the ground at tt seconds after it is dropped. A realistic domain (for time) for this function is:\newline(A)(A) 3t3-3 \leq t \leq 3\newline(B)(B) 0t30 \leq t \leq 3\newline(C)(C) 0h1440 \leq h \leq 144\newline(D)(D) all real numbers
  1. Function Domain: The function h(t)=16t2+144h(t) = -16t^2 + 144 represents the height of an object from the ground after it is dropped. The domain of this function is the set of all possible values of tt for which the function is defined. Since tt represents time, it cannot be negative because we cannot measure time before the object is dropped.
  2. Find Ground Impact Time: We need to consider the realistic scenario of the object being dropped and eventually hitting the ground. The height h(t)h(t) will be zero when the object hits the ground. To find when this happens, we set h(t)h(t) to zero and solve for tt.\newline0=16t2+1440 = -16t^2 + 144
  3. Solve for tt: Divide both sides of the equation by 16-16 to simplify.0=t290 = t^2 - 9
  4. Discard Negative Solution: Take the square root of both sides to solve for tt.t=9t = \sqrt{9} or t=9t = -\sqrt{9}t=3t = 3 or t=3t = -3
  5. Realistic Domain: Since time cannot be negative in this context, we discard the negative solution. The object hits the ground at t=3t = 3 seconds. Therefore, the realistic domain for the function, considering the context of the problem, is from the time the object is dropped (t=0t = 0) until the time it hits the ground (t=3t = 3).

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