Question

A ball is thrown upward off of a 100 meter cliff with an initial velocity of 6 m/s. The function f(x)=-5x2+6x+100 (graphed below) represents this situation where x is time and y is the distance off of the ground. will mart brainliest yeah

Answer 1

Answer:

a) The domain of the function is $x \geq 0\,s$ $\wedge$  $x \leq 5.112\,s$. $[0\,s, 5.112\,s]$, $\forall x \in \mathbb{R}$, b) The range of the function is $0\,m \leq y \leq 100\,m$. $[0\,m,100\,m]$, $\forall y\in \mathbb{R}$, c) The ball is 73 meters off of the ground at x = 3 seconds.

Step-by-step explanation:

The complete statement is: A ball is thrown upward off of a 100 meter cliff with an initial velocity of 6 m/s. The function $f(x) = -5\cdot x^{2} + 6\cdot x + 100$ represents this situation where x is time and y is the distance off of the ground.

a) What domain does the function make sense?

b) What range does the function make sense ?

c) How far off the ground is the ball at time x = 3 seconds?

a) Let $x$ and $f(x)$ be the time, measured in seconds, and the distance of the ground, measured in meters, respectively. Time is a positive variable, so domain corresponds to the interval when $f(x) \geq 0$ and $t \geq 0$. That is:

$-5\cdot x^{2} + 6\cdot x + 100 \geq 0$

$-(x-5.112\,s)\cdot (x+3.912\,s) \geq 0$

Therefore, the domain of the function is $x \geq 0\,s$ $\wedge$  $x \leq 5.112\,s$. $[0\,s, 5.112\,s]$, $\forall x \in \mathbb{R}$

b) The distance off of the ground is also a positive variable, where ball is thrown upward at a height of 100 meters and hits the ground at a height of 0 meters. Hence, the range of the function is $0\,m \leq y \leq 100\,m$. $[0\,m,100\,m]$, $\forall y\in \mathbb{R}$

c) The distance of the ball off of the ground at x = 3 seconds is found by evaluating the function:

$f(3\,s) = -5\cdot (3\,s)^{2} + 6\cdot (3\,s) + 100$

$f(3\,s) = 73\,m$

The ball is 73 meters off of the ground at x = 3 seconds.