A football is punted. Its height, H metres is given by the quadratic relation with equation: H = -5t2 + 21t, where t is the time in seconds after the punt.
I tried to do this question on a test but I just ran out of time
How is the easiest and quickest way to do this?
Algebraically determine the maximum height of the football. Show all of your steps.
Of course you would want to find the vertex of the hyperbola
several easy ways to do this
1. for the general quadratic y = ax^2 + bx + c
the x of the vertex is -b/(2a), then plug that back into the equation to find the y
so here , t of the vertex = -21/-10 = 2.1
then H = -5(2.1)^2 + 21(2.1) = 22.08
2. in any parabola, the x of the vertex is always midway between the x-intercepts, if they exist
so for yours 0 = -t(5t - 21)
t = 0 or t = 21/5
half-way would be t = 21/10 = 2.1
continue as in #1
3. complete the square
Well, I hate to break it to you, but it sounds like you really "dropped the ball" on this one! But fear not, I'll help you score some points with the math.
To determine the maximum height of the football, we need to find the vertex of the quadratic equation. Now, let's see if we can tackle this problem together:
The equation for the height of the football is given as:
H = -5t^2 + 21t
To find the vertex, we first need to identify the coefficient of the t^2 term, which is -5. The formula for the x-coordinate of the vertex of the quadratic equation is -b/2a.
In this case, a = -5 and b = 21, so substituting these values into the formula, we have:
t = -21 / (2 * -5)
Simplifying this further, we get:
t = -21 / -10
t = 2.1
Now that we have the time, we can substitute that back into the original equation to find the maximum height:
H = -5 * (2.1)^2 + 21 * 2.1
Calculating that out, we get:
H = -5 * 4.41 + 44.1
H = -22.05 + 44.1
H = 22.05
So, the maximum height of the football is 22.05 meters.
But hey, chin up! Next time, don't "drop the ball" and take your time to solve the problem. Remember, it's all about having fun with math!
To algebraically determine the maximum height of the football, you need to find the vertex of the quadratic equation. The vertex provides the maximum height since the quadratic function is in the form of H = ax^2 + bx + c.
Step-by-step process for finding the maximum height of the football:
Step 1: Identify the coefficients of the quadratic equation.
In this case, the quadratic equation is H = -5t^2 + 21t, where a = -5, b = 21, and c = 0 (since there is no constant term).
Step 2: Find the axis of symmetry.
The axis of symmetry is given by the formula: t = -b/2a.
Substituting the values, t = -21/(2*(-5)). Simplifying this equation gives t = -21/(-10). Therefore, t = 2.1.
Step 3: Calculate the maximum height.
Substitute the value of t = 2.1 into the original equation to find H.
H = -5(2.1)^2 + 21 * 2.1
H = -5(4.41) + 44.1
H = -22.05 + 44.1
H = 22.05
Therefore, the maximum height of the football is 22.05 meters.
To determine the maximum height of the football, you need to find the vertex of the quadratic equation. The vertex, represented as (t, H), will correspond to the time and height at which the football reaches its highest point.
The equation of the height of the football is given as H = -5t^2 + 21t. To find the vertex, we will use the formula t = -b / (2a), where a, b, and c are the coefficients of the quadratic equation in the form of ax^2 + bx + c.
In this case, a = -5 and b = 21. Plugging these values into the formula, we get t = -21 / (2 * -5).
Simplifying further, we have t = -21 / -10, which simplifies to t = 2.1.
Now, substitute this value of t into the equation to find the corresponding height: H = -5(2.1)^2 + 21(2.1).
Calculating further, H = -5(4.41) + 44.1.
Simplifying, H = -22.05 + 44.1.
Finally, H = 22.05.
Therefore, the maximum height of the football is 22.05 meters.