Respuesta :
Domain means the values of independent variable(input) which will give defined output to the function.
Given:
The height h of a projectile is a function of the time t it is in the air. The height in feet for t seconds is given by the function
[tex] h(t)=-16t^2 + 96t [/tex]
Solution:
To get defined output, the height h(t) need to be greater than or equal to zero. We need to set up an inequality and solve it to find the domain values.
[tex] To \; find \; domain:\\\\h(t) \geq0\\\\-16t^2+96t \geq 0\\Factoring \; -16t \; in \; the \; left \; side \; of \; the \; inequality\\\\-16t(t-6) \geq 0\\Step \; 1: Find \; Boundary \; Points \; by \; setting \; up \; above \; inequality \; to \; zero.\\\\t(t-6)=0\\Use \; zero \; factor \; property \; to \; solve\\\\t=0 \; (or) \; t = 6\\\\Step \; 2: \; List \; the \; possible \; solution \; interval \; using \; boundary \; points\\(- \infty,0], \; [0, 6], \& [6, \infty) [/tex]
[tex] Step \; 3:Pick \; test \; point \; from \; each \; interval \; to \; check \; whether \\\; makes \; the \; inequality \; TRUE \; or \; FALSE\\\\When \; t = -1\\-16(-1)(-1-6) \geq 0\\-112 \geq 0 \; FALSE\\(-\infty, 0] \; is \; not \; solution\\Also \; Logically \; time \; t \; cannot \; be \; negative\\\\When \; t = 1\\-16(1)(1-6) \geq 0\\80 \geq 0 \; TRUE\\ \; [0, 6] \; is \; a \; solution\\\\When \; t = 7\\-16(7)(7-6) \geq 0\\-112 \geq 0 \; FALSE\\ \; [6, -\infty) \; is \; not \; solution [/tex]
Conclusion:
The domain of the function is the time in between 0 to 6 seconds
[tex] 0 \leq t \leq 6 [/tex]
The height will be positive in the above interval.
Answer:
The answer is explained below
Step-by-step explanation:
STEP 1
A projectile motion is the motion is a specific particle in a curve path, under the influence of gravity.
Domain denotes to a set of values of input for which there is a specific output.
STEP 2
The equation used for projectile motion is:
h(t)= - 16t^2 + 96t
It is a kind of function in which time (t) is the input and height (h) is the output. Here the domain for the function refers to all possible value of time t for which there is an output h (t). IN simple words, domain is the set of all the values of time t for which the particle is in a projectile motion.
STEP 3
The particle begins the motion from ground, goes in projectile and then returns to ground again. Since at ground level, h (t) = 0
So, using the projectile equation:
h(t)= - 16t^2 + 96t
-16t(t-6)= 0
SO, t = 0,6
It will take 6 seconds for the particle to reach the ground again.
The domain of the projectile motion is the set of all set of all the values from 0 to 6 which is {0,6}
