Question

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 h(t) = −16t2 + 96t. what is the domain of the function? what does the domain mean in the context of the problem?

Answer 1

The answer is explained below

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}

Answer 2

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

`h(t)=-16t^2 + 96t`

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.

`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)`

`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`

Conclusion:

The domain of the function is the time in between 0 to 6 seconds

`0 \leq  t \leq  6`

The height will be positive in the above interval.