Question

tyler is playing basketball with his friends, and he shoots an air ball from the three - point line. the graph below represents the height of the basketball after x seconds.
which statement represents the reasonable time when the basketball is in the air?
a. the reasonable number of seconds when the basketball is in the air is between 0 seconds and 0.25 seconds.
b. the reasonable number of seconds when the basketball is in the air is between 0 seconds and 2 seconds.
c. the reasonable number of seconds when the basketball is in the air is between 0 seconds and 2.25 seconds.
d. the reasonable number of seconds when the basketball is in the air is between 0 seconds and 1.1 seconds.

Explanation:

Step1: Analyze the graph

The graph of the basketball's height over time is a parabola (projectile motion). The basketball is in the air when its height \( h(t)>0 \), excluding the time at launch (\( t = 0 \)) and landing. We need to find the interval of \( t \) where the graph is above the \( t \)-axis (excluding start and end points where it touches the ground).

Step2: Estimate the time interval

Looking at the graph, the basketball is launched at \( t = 0 \) and lands around \( t = 2.5 \) seconds? Wait, no, the x - axis is time in seconds. Wait, the peak is around \( t=1.25 \) maybe? Wait, the options are about intervals. Let's check the options:

- Option A: Between 0 and 0.25? No, because at \( t = 0 \) it's launched, and it goes up. So the time in air should be from just after 0 until just before landing. But the options are about reasonable time when it's in air (excluding launch and landing? No, the question is "reasonable time when the basketball is in the air" – so the time when \( h(t)>0 \), excluding \( t = 0 \) (launch) and \( t \) when it lands. But looking at the options, let's re - evaluate.

Wait, the graph: at \( t = 0 \), height is around 6 ft (launch). Then it goes up, peaks, then comes down. The time when it's in the air is from \( t = 0 \) until it lands (when height is 0 again). But the "reasonable time" – maybe the time when it's not at launch or landing. Wait, the options:

Option C: Between 0 seconds and 2.25 seconds? Wait, no, let's check the x - axis labels: 0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0. The basketball is in the air from \( t = 0 \) (launch) until it lands (when \( h(t)=0 \) again). Looking at the graph, the landing time is around \( t = 2.25 \) or 2.5? Wait, the options:

Wait, the correct interval should be from just after \( t = 0 \) (but since we need reasonable time, excluding the exact launch and landing, but the options:

Wait, let's re - read the options:

A: Between 0 and 0.25 – no, because at \( t = 0.25 \), it's still going up, but the total time in air is more.

B: Between 1 and 2 – but the peak is around 1.25, and it lands after 2? No, the graph seems to land around 2.25 or so.

C: Between 0 and 2.25 – but at \( t = 0 \), it's launched. Wait, maybe the question is about the time when it's in the air (excluding the instant it's launched and landed). But the most reasonable interval: looking at the graph, the basketball is in the air from \( t = 0 \) (launch) until it lands (when \( h(t)=0 \) again). The landing time is around \( t = 2.25 \) seconds? Wait, the x - axis: 0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0. The graph starts at \( t = 0 \), goes up, peaks, then comes down. The time when it's in the air is from \( t>0 \) until \( t < 2.25 \) (or whatever the landing time is). But the option C says "between 0 seconds and 2.25 seconds" – but at \( t = 0 \), it's just launched, so maybe the reasonable time is from when it's launched (t = 0) until it lands (t≈2.25). But let's check the options again.

Wait, maybe I misread. The question is "reasonable time when the basketball is in the air" – so the time interval when \( h(t)>0 \), excluding \( t = 0 \) (launch) and \( t \) when it lands (but the options include 0). Wait, maybe the options are about the time when it's in the air (not at launch or landing, but the interval). Let's check the options:

Option C: "The reasonable number of seconds when the basketball is in the air is between 0 seconds and 2.25 seconds." Let's see the graph: at \( t = 0 \), height is ~6 ft (launched), then it goes up, then comes down, and lands around \( t = 2.25 \) (when height is 0 again). So the time when it's in the air is from \( t = 0 \) (launch) until \( t = 2.25 \) (landing). But maybe the "reasonable time" is the duration it's in the air, excluding the exact launch and landing instants, but the option C's interval is 0 to 2.25, which matches the time from launch to landing (since at \( t = 0 \) it's launched, and at \( t = 2.25 \) it lands). Wait, but let's check other options:

Option A: 0 to 0.25 – too short, since it goes up and comes down, so total time in air is more than 0.25.

Option B: 1 to 2 – that's only part of the time in air (the peak is around 1.25, so from 1 to 2 is part of the descent, but not the whole time).

Option D: 0 to 1.11 – too short, since it lands after 2 seconds.

So the correct interval is between 0 seconds and 2.25 seconds, which is option C.

Answer:

C. The reasonable number of seconds when the basketball is in the air is between 0 seconds and 2.25 seconds.