The height y in feet of a punted football is given by y=(-20/2025)x^2 +(9/5)x +1.6 where x is the horizontal distance in feet from the point at which the ball is punted.
a) How high is the ball when it is punted?
my answer:
x=0 because it is at the ground and the ball is punted from the ground which is 0
y=(-20/2025)x^2 +(9/5)x +1.6
y=(-20/2025)(0)^2 +(9/5)(0) +1.6
y = 0 +0 +1.6
y = 1.6
the football is 1.6 ft above the ground
b) what is the maximum height of the punt? how long is the punt?
my answer:
-b/2a
=(-9/5)/2(-20/2025)
=(-9/5)/(-8/405)
=91.125
the max height is 91.125 ft
I need on on question b) how long is the punt and is my answers correct?
your ordered pairs are of the form (x,y) = (distance,height)
so, as the problem stated, x is the horizontal distance and y is the height
the formula you used, -b/(2a) , would give you the distance at which the max height occurs.
so if x = 91.125 ft, you have to sub that into your original equation to find y, the max height
Secondly , the 91.125 is the distance it went to reach the maximum height, so of course it
also has to come back down. What would be that distance?
look at the first graph of
https://www.wolframalpha.com/input/?i=plot+y%3D%28-20%2F2025%29x%5E2+%2B%289%2F5%29x+%2B1.6
Well, if we're talking about a punt, then technically the duration of the punt would be the time it takes for the ball to reach the ground, rather than the length of the punt itself. Can you provide any additional information about the ball's initial velocity or any other variables that might help us calculate the time it takes for the ball to hit the ground?
To find the length or duration of the punt, we first need to determine the value of x when the football reaches its maximum height.
We already know the equation for the height of the punt is y = (-20/2025)x^2 + (9/5)x + 1.6. By looking at the equation, we can see that the coefficient in front of the x^2 term, -20/2025, is negative. This indicates that the graph of the equation will be a downward-opening parabola.
In a downward-opening parabola, the maximum height or vertex of the parabola occurs at the value of x that can be found using the formula -b/2a.
In this case, a = -20/2025 and b = 9/5.
x = -b/2a
= -(9/5)/2(-20/2025)
= -(9/5)/(-40/2025)
= 91.125
So the maximum height of the punt is indeed 91.125 feet.
To find the length of the punt, we need to double the value of x, as the football will reach the same height at both the beginning and end of the punt.
Length of the punt = 2 * x
= 2 * 91.125
= 182.25 feet
Therefore, the length of the punt is 182.25 feet.
Your answer for the maximum height of the punt is correct, but please note that the length of the punt should be 182.25 feet, not 91.125 feet.
To determine the length of the punt and validate your answer, we need to find the x-coordinate of the vertex of the parabola.
The equation of the parabola is y = (-20/2025)x^2 + (9/5)x + 1.6, where x represents the horizontal distance from the point where the ball is punted.
The x-coordinate of the vertex can be found using the formula -b/2a, where a and b are the coefficients of the quadratic equation.
In this case, a = -20/2025 and b = 9/5.
Substituting these values into the formula, we have:
x-coordinate of vertex = -b/2a
= - (9/5) / (2 * (-20/2025))
= (9/5) / (40/2025)
= (9/5) * (2025/40)
= 91.125
So, you were correct in stating that the maximum height of the punt is 91.125 ft. The x-coordinate of the vertex represents the horizontal distance traveled, which is also the length of the punt.
Therefore, the length of the punt is 91.125 ft.
Your answer is correct.