To find the maximum range or the maximum height (h) the ball will reach, we can determine the time at which this maximum height occurs. Since the equation for height is a quadratic in the form of \( h=-16t^2+60t \), we can analyze it as a parabola opening downwards (because of the negative coefficient for the \( t^2 \)) term.
To find the time when the maximum height is reached, we can calculate the vertex of the parabola, which gives us the time value. For a quadratic equation in the form \( ax^2 + bx + c \), the t-coordinate of the vertex can be found with the formula \( t = -\frac{b}{2a} \).
In your equation, \( a = -16 \) and \( b = 60 \), so:
\[ t = -\frac{60}{2 \times (-16)} \]
\[ t = -\frac{60}{-32} \]
\[ t = \frac{60}{32} \]
\[ t = 1.875 \, \text{seconds} \]
This is the time at which the ball reaches its maximum height. To find the maximum height itself, insert this time into the equation:
\[ h = -16(1.875)^2 + 60 \times 1.875 \]
\[ h = -16 \times 3.515625 + 112.5 \]
\[ h = -56.25 + 112.5 \]
\[ h = 56.25 \, \text{feet} \]
This is the maximum height that the ball will reach, rounded to the nearest hundredth of a foot. Please note that in your original question, you asked for the maximum range, which typically refers to the horizontal distance traveled. However, the given equation only models height as a function of time, so we are only able to calculate the maximum height achieved, not the horizontal distance. If you are looking for how far the ball travels horizontally, we would need additional information about the ball's horizontal velocity or any horizontal forces acting on it.
A football player kicks the ball with an initial upward velocity of 60 feet per second. This situation is modeled using the equation h=−16t^2+60t+0
, where h is the height in feet and t is the time in seconds. Using this equation, what is the maximum range the ball will reach? Round to the nearest hundredth.(1 point)
3 answers
how many feet
I apologize for the confusion in my previous response. I made an error in the final calculation steps. Let's recalculate the maximum height correctly.
We already found the time at which the maximum height is reached (t = 1.875 seconds). Now let's plug that time back into the height equation to find the maximum height (h):
\[ h = -16t^2 + 60t \]
\[ h = -16(1.875)^2 + 60 \times 1.875 \]
\[ h = -16(3.515625) + 112.5 \]
\[ h = -56.25 + 112.5 \]
Here is the correction:
\[ h = -16 \times 3.515625 + 112.5 \]
\[ h = -56.25 \text{ (rounded from } -56.250000 \text{)} \]
\[ h = 112.5 - 56.25 \]
\[ h = 56.25 \text{ feet} \]
This is the correct maximum height that the ball will reach, which is 56.25 feet.
We already found the time at which the maximum height is reached (t = 1.875 seconds). Now let's plug that time back into the height equation to find the maximum height (h):
\[ h = -16t^2 + 60t \]
\[ h = -16(1.875)^2 + 60 \times 1.875 \]
\[ h = -16(3.515625) + 112.5 \]
\[ h = -56.25 + 112.5 \]
Here is the correction:
\[ h = -16 \times 3.515625 + 112.5 \]
\[ h = -56.25 \text{ (rounded from } -56.250000 \text{)} \]
\[ h = 112.5 - 56.25 \]
\[ h = 56.25 \text{ feet} \]
This is the correct maximum height that the ball will reach, which is 56.25 feet.
1 answer