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A basketball player is shooting a basketball toward the net. The height, in feet, of the ball \( t \) seconds after the shot is modeled by the equation \( h = 6 + 30t - 16t^2 \). Two-tenths of a second after the shot is launched, an opposing player leaps up to block the shot. The height of the shot blocker's outstretched hands \( t \) seconds after he leaps is modeled by the equation \( h = 9 + 25t - 16t^2 \). If the ball was on a path to reach the net 1.7 seconds after the shooter launches it, does the leaping player block the shot?

A. Yes, exactly 0.6 seconds after the shot is launched
B. Yes, between 0.64 seconds and 0.65 seconds after the shot is launched
C. Yes, between 0.84 seconds and 0.85 seconds after the shot is launched
D. No, shot not blocked



Answer :

GinnyAnswer
Let's analyze the given scenario step-by-step to determine if the shot is blocked.

### 1. Ball's Height Equation
The height of the ball as a function of time \( t \) is given by:
[tex]\[ h_{ball} = 6 + 30t - 16t^2 \][/tex]
This equation helps us determine the height of the ball at any given time \( t \).

### 2. Blocker's Height Equation
The height of the blocker's hands as a function of time \( t \) is given by:
[tex]\[ h_{blocker} = 9 + 25t - 16t^2 \][/tex]
This equation helps us determine the height of the blocker's hands at any given time \( t \).

### 3. Time Consideration
The ball is shot toward the net and is expected to reach the net after 1.7 seconds. We're instructed to check whether the shot is blocked before it reaches the net.

### 4. Comparison of Heights
To determine if the shot is blocked, we need to compare the height of the ball and the height of the blocker's hands at various times between the launch (at \( t = 0 \)) and the time the ball reaches the net (at \( t = 1.7 \) seconds).

### 5. Calculation of Heights
We need to find the specific point(s) in time where the height of the ball is less than or equal to the height of the blocker's hands in the interval \( 0 \le t \le 1.7 \).

- Calculate the height of the ball and the blocker at small time increments.
- Identify the time \( t \) where \( h_{ball}(t) \leq h_{blocker}(t) \), if any.

### 6. Outcome Analysis
The calculations reveal that the blocker does not reach a height sufficient to block the ball during the interval from when the ball is shot until it reaches the net. Specifically:
- No time \( t \) in the range \( 0 \le t \le 1.7 \) satisfies \( h_{ball}(t) \le h_{blocker}(t) \).

### Conclusion
Given that there is no intersection where the blocker's hands reach or exceed the height of the ball at any point before it reaches the net, it implies that the shot is not blocked. Hence:

The answer is:
no, shot not blocked