Question
Ricardo throws a stone off a bridge into a river below.
The stone's height (in meters above the water),
\[x\] seconds after Ricardo threw it, is modeled by
\[w(x)=-5(x-8)(x+4)\]
How many seconds after being thrown will the stone reach its maximum height?
seconds
The stone's height (in meters above the water),
\[x\] seconds after Ricardo threw it, is modeled by
\[w(x)=-5(x-8)(x+4)\]
How many seconds after being thrown will the stone reach its maximum height?
seconds
Answers
GPT-4o mini
To find the time at which the stone reaches its maximum height, we can analyze the quadratic function given by:
\[
w(x) = -5(x - 8)(x + 4)
\]
This is a quadratic equation in vertex form, \(w(x) = a(x - h)^2 + k\), where \((h, k)\) is the vertex of the parabola. Since the coefficient \(a\) is negative (-5), this means the parabola opens downwards, and therefore the vertex represents the maximum point.
To find the vertex, we first need to expand the given function:
\[
w(x) = -5[(x - 8)(x + 4)] = -5[x^2 + 4x - 8x - 32] = -5[x^2 - 4x - 32]
\]
Now expanding it fully:
\[
w(x) = -5x^2 + 20x + 160
\]
The time at which the maximum height occurs can be found using the formula for the vertex of a parabola, given by \(x = -\frac{b}{2a}\), where \(a\) is the coefficient of \(x^2\) and \(b\) is the coefficient of \(x\).
In our function:
- \(a = -5\)
- \(b = 20\)
Now we can calculate the time:
\[
x = -\frac{20}{2 \cdot -5} = -\frac{20}{-10} = 2
\]
Therefore, the stone will reach its maximum height **2 seconds** after being thrown.
\[
w(x) = -5(x - 8)(x + 4)
\]
This is a quadratic equation in vertex form, \(w(x) = a(x - h)^2 + k\), where \((h, k)\) is the vertex of the parabola. Since the coefficient \(a\) is negative (-5), this means the parabola opens downwards, and therefore the vertex represents the maximum point.
To find the vertex, we first need to expand the given function:
\[
w(x) = -5[(x - 8)(x + 4)] = -5[x^2 + 4x - 8x - 32] = -5[x^2 - 4x - 32]
\]
Now expanding it fully:
\[
w(x) = -5x^2 + 20x + 160
\]
The time at which the maximum height occurs can be found using the formula for the vertex of a parabola, given by \(x = -\frac{b}{2a}\), where \(a\) is the coefficient of \(x^2\) and \(b\) is the coefficient of \(x\).
In our function:
- \(a = -5\)
- \(b = 20\)
Now we can calculate the time:
\[
x = -\frac{20}{2 \cdot -5} = -\frac{20}{-10} = 2
\]
Therefore, the stone will reach its maximum height **2 seconds** after being thrown.