Question
The height, in meters, above the ground of a projectile is modeled by the function f of t equals negative 4.9 t squared plus 19.6 t plus 50, where t is the number of seconds since the object was launched.
How long after launch does the projectile reach its maximum height?
Answer options with 5 options
A.
1 second
B.
2 seconds
C.
4 seconds
D.
19.6 seconds
E.
50 seconds
How long after launch does the projectile reach its maximum height?
Answer options with 5 options
A.
1 second
B.
2 seconds
C.
4 seconds
D.
19.6 seconds
E.
50 seconds
Answers
GPT-4o mini
To determine how long after launch the projectile reaches its maximum height, we can use the formula for the vertex of a quadratic function. The function given is:
\[ f(t) = -4.9t^2 + 19.6t + 50 \]
For a quadratic function in the form \( ax^2 + bx + c \), the time \( t \) at which the maximum height occurs can be found using the formula:
\[ t = -\frac{b}{2a} \]
Here, \( a = -4.9 \) and \( b = 19.6 \).
Substituting the values of \( a \) and \( b \):
\[ t = -\frac{19.6}{2 \times -4.9} \]
Calculating the denominator:
\[ 2 \times -4.9 = -9.8 \]
Now substituting this back into the equation:
\[ t = -\frac{19.6}{-9.8} = 2 \]
So, the projectile reaches its maximum height at **2 seconds** after launch.
Thus, the answer is **B. 2 seconds**.
\[ f(t) = -4.9t^2 + 19.6t + 50 \]
For a quadratic function in the form \( ax^2 + bx + c \), the time \( t \) at which the maximum height occurs can be found using the formula:
\[ t = -\frac{b}{2a} \]
Here, \( a = -4.9 \) and \( b = 19.6 \).
Substituting the values of \( a \) and \( b \):
\[ t = -\frac{19.6}{2 \times -4.9} \]
Calculating the denominator:
\[ 2 \times -4.9 = -9.8 \]
Now substituting this back into the equation:
\[ t = -\frac{19.6}{-9.8} = 2 \]
So, the projectile reaches its maximum height at **2 seconds** after launch.
Thus, the answer is **B. 2 seconds**.