A football is thrown by a quarterback, the footballs height in feet h with respect to time in seconds T can be modeled using your quadratic function. You want to know when the football will be 7.5ft in the air because that's about the right height that your favorite football player would catch the ball. You also want to know if your most despised cornerback in the league will be able to interfere with the catch
After the football was thrown you noticed that it had a height of 7ft but 3 secs later it reached its peak height of 15ft
1)make a function to represent the height of the ball
2)then use that function to find out how long it would take for the football to reach 7.5ft
Assume that the cornerback was 100ft away from the destination of the ball when you 1st noticed the balls height the cornerback can run at a speed of 20ft per second will the cb have time to interfere with the catch if he's running directly towards the balls destination with the receiver
PLEASE EXPLAIN EVERYTHING IN WORDS WITH THE FORMULA IM REALLY CONFUSED
To find a quadratic function that represents the height of the ball, we can use the general form of a quadratic equation, which is given as h(T) = aT^2 + bT + c, where h(T) is the height of the ball at time T.
We are given two pieces of information about the height of the ball:
1) At T = 0 seconds, the height of the ball is 7ft. So, we can say h(0) = 7.
2) At T = 3 seconds, the ball reaches its peak height of 15ft. So, we can say h(3) = 15.
Using these two pieces of information, we can set up a system of equations to solve for the coefficients a, b, and c.
h(0) = a(0)^2 + b(0) + c
7 = c ------ Equation 1
h(3) = a(3)^2 + b(3) + c
15 = 9a + 3b + c ------ Equation 2
Using Equation 1, we can see that c = 7.
Plugging c = 7 into Equation 2, we get:
15 = 9a + 3b + 7
8 = 9a + 3b ------ Equation 3
Since we have two equations (Equation 1 and Equation 3) with two variables (a and b), we can solve them simultaneously.
Step 1: Subtract Equation 3 from Equation 1
7 - 8 = 7 - (9a + 3b)
-1 = -9a - 3b
1 = 9a + 3b ------ Equation 4 (After multiplying both sides by -1)
Step 2: Add Equation 4 to Equation 3
1 + 9a + 3b = 9a + 3b
1 = 9a
a = 1/9
Now, let's find the value of b using Equation 4:
1 = 9(1/9) + 3b
1 = 1 + 3b
3b = 0
b = 0
Therefore, the quadratic function representing the height of the ball is:
h(T) = (1/9)T^2
To find out how long it would take for the football to reach a height of 7.5ft, we need to solve the quadratic equation:
7.5 = (1/9)T^2
Multiply both sides by 9 to eliminate the fraction:
67.5 = T^2
Take the square root of both sides:
√67.5 ≈ 8.22
So, it will take approximately 8.22 seconds for the football to reach a height of 7.5ft.
Now, let's determine if the cornerback will have time to interfere with the catch. We know the cornerback is 100ft away from the destination and can run at a speed of 20ft per second.
If the cornerback is running directly towards the destination of the ball, the distance between them will decrease at a rate of 20ft per second. Therefore, the time it takes for the cornerback to reach the destination is given by:
Time = Distance / Speed = 100ft / 20ft/s = 5 seconds
Since the football will reach the desired height in approximately 8.22 seconds, and the cornerback will take 5 seconds to reach the destination, the cornerback will not have enough time to interfere with the catch.