A football is thrown by a quarterback, the footballs height in feet will be 7.5 ft in the air because that's about the right height that your favorite football player would catch the ball. Also you want to know if the 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 7 ft, but 3 seconds later, it reached ita peak height of 15ft
1)make a function to represent the height of the ball
then use that function to find out how long it would take for the football to reach 7.5 ft
Assume the cornerback was 100 ft 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 hes running directly towards the balls destination with the receiver
Since the peak height of 15 was reached at t=3,
h(t) = a(t-3)^2 + 15
Since h(0) = 7, a = -8/9, so
h(t) = -8/9 (t-3)^2 + 15
Now you can answer the questions
I still need more help i need to explain it in a paragraph on how i got the anser
I need help
To create a function that represents the height of the football, we can use a quadratic equation since the path of the football can be approximated by a parabola. The general equation for the height of an object thrown vertically is given by:
h(t) = -16t^2 + vt + h0
Where:
- h(t) represents the height of the object at time t
- v represents the initial vertical velocity (in this case, the velocity at which the ball is thrown; negative value indicates it is thrown upward)
- h0 represents the initial height of the object (in this case, the height when the ball was noticed)
Given that the height of the ball was 7ft when first noticed and reached a peak height of 15ft, we can substitute the known values into the equation:
h(t) = -16t^2 + vt + h0
7 = -16t^2 + vt + h0 (Equation 1)
15 = -16t^2 + vt + h0 (Equation 2)
We have two equations with two unknowns (t and v), and we can solve this system of equations to find the values of t and v.
To find out how long it would take for the football to reach a height of 7.5ft, we can continue with the following steps:
1. Substitute h(t) = 7.5ft into Equation 1:
7.5 = -16t^2 + vt + h0
2. Solve for t using this equation.
Now, moving on to the cornerback interference scenario:
We know that the cornerback is 100ft away from the destination of the ball when the ball's height was first noticed. The cornerback can run at a speed of 20ft per second. We need to determine if the cornerback has enough time to interfere with the catch if he runs directly towards the ball's destination with the receiver.
To do this, we need to calculate the time it would take for the cornerback to reach the destination of the ball. We can use the distance formula: distance = speed × time.
Let's assume the receiver is positioned at the same point where the ball will land. Given that the destination of the ball is 100ft away, and the cornerback runs at a speed of 20ft per second, we can plug in the values:
100ft = 20ft/s × time
Solve for time to determine how long it will take for the cornerback to reach the destination.
Comparing this time with the time it took for the ball to reach a height of 7.5ft will allow us to determine if the cornerback has enough time to interfere with the catch.