The height of a golf ball in meters can be described by the equation $h=-4.9 t^{2}+23.7 t$, where $t$ is the number of seconds after it was hit. Find the vertex of this parabola, rounding each coordinate to the nearest tenth.
The vertex of this parabola is
\(\boxed{\text{The vertex of the parabola is at the point }(2.4, 28.7)}\)
Step 1 :The height of a golf ball in meters can be described by the equation \(h=-4.9 t^{2}+23.7 t\), where \(t\) is the number of seconds after it was hit. We are asked to find the vertex of this parabola, rounding each coordinate to the nearest tenth.
Step 2 :The vertex of a parabola given by the equation \(y = ax^2 + bx + c\) is at the point \(-\frac{b}{2a}, f(-\frac{b}{2a})\). In this case, \(a = -4.9\) and \(b = 23.7\).
Step 3 :We can calculate the \(t\)-coordinate of the vertex as \(-\frac{b}{2a}\), and then substitute this value into the equation to find the \(h\)-coordinate.
Step 4 :By substituting \(a = -4.9\) and \(b = 23.7\) into \(-\frac{b}{2a}\), we get \(t_{vertex} = 2.4183673469387754\).
Step 5 :Substituting \(t_{vertex} = 2.4183673469387754\) into the equation \(h=-4.9 t^{2}+23.7 t\), we get \(h_{vertex} = 28.657653061224483\).
Step 6 :Rounding to the nearest tenth, the \(t\)-coordinate of the vertex is approximately 2.4 seconds, and the \(h\)-coordinate of the vertex is approximately 28.7 meters.
Step 7 :This means that the golf ball reaches its maximum height of 28.7 meters after 2.4 seconds.
Step 8 :\(\boxed{\text{The vertex of the parabola is at the point }(2.4, 28.7)}\)