a. Your friend drops the ball from a higher point (300 feet vs 256 feet).
b. The valid domain of T(x) in this situation is 0 < x < min(4, √(300/16)) ≈ 3.8.
c. The graph shows that the first ball hits the ground at x ≈ 4 seconds
a. Who drops the ball from a higher point?
To determine who drops the ball from a higher point, we need to compare the initial heights of both balls. This means comparing the values of both functions at x = 0.
Initial height of your ball: h(0) = -16 * 0² + 256 = 256 feet.
Initial height of your friend's ball: g(0) = -16 * 0² + 300 = 300 feet.
Therefore, your friend drops the ball from a higher point (300 feet vs 256 feet).
b. Function T(x) and its interpretation
The function T(x) = h(x) - g(x) represents the difference in height between your ball and your friend's ball after x seconds. In other words, T(x) tells us how much higher or lower your ball is than your friend's ball at any given time.
Finding the domain of T(x)
Both h(x) and g(x) are defined for all real values of x. Therefore, the domain of T(x) is also all real numbers. However, in the context of this problem, we are only interested in the time interval where the balls are in the air. Since both balls hit the ground at the same time, the domain of T(x) needs to be restricted to the interval where both balls are above the ground.
We can find this interval by setting both functions equal to zero and solving for x:
h(x) = 0 => -16x² + 256 = 0 => x = 4
g(x) = 0 => -16x² + 300 = 0 => x = √(300/16) ≈ 3.8
Therefore, the valid domain of T(x) in this situation is 0 < x < min(4, √(300/16)) ≈ 3.8.
Complete question:
You and your friend both drop a ball at the same time. The function h(x) = -16x² + 256 represents the height (in feet) of your ball after x seconds. The function g(x) = -16x² + 300 represents the height (in feet) of your friend's ball after x seconds.
a. Who drops the ball from a higher point?
b. Write the function T(x)=h(x)-g(x). What does T(x) represent? Find and interpret the domain of T in this situation.
c. When the first ball hits the ground, what is the height of the other ball? Use a graph to justify your answer.