Let's solve the problem step-by-step.
We are given a toy rocket's height function as:
[tex]\[
h(x) = -5x^2 + 10x + 20
\][/tex]
1. Find the maximum height:
The maximum height is attained at the vertex of the parabola represented by the height function. For a quadratic function in the form [tex]\( ax^2 + bx + c \)[/tex], the time at which the maximum height occurs is given by:
[tex]\[
t = -\frac{b}{2a}
\][/tex]
In our function, [tex]\( a = -5 \)[/tex] and [tex]\( b = 10 \)[/tex]. Plugging these values into the formula:
[tex]\[
t = -\frac{10}{2 \times (-5)} = -\frac{10}{-10} = 1
\][/tex]
So, the maximum height is reached at [tex]\( t = 1 \)[/tex] second.
Now, we substitute [tex]\( t = 1 \)[/tex] back into the height function to find the maximum height:
[tex]\[
h(1) = -5(1)^2 + 10(1) + 20 = -5 + 10 + 20 = 25
\][/tex]
Therefore, the maximum height reached by the rocket is [tex]\( 25 \)[/tex] yards.
2. Find the time it takes for the rocket to hit the ground:
The rocket hits the ground when the height [tex]\( h(x) \)[/tex] is zero. We need to solve the equation:
[tex]\[
-5x^2 + 10x + 20 = 0
\][/tex]
Solving this quadratic equation, we find that [tex]\( x \)[/tex] approximately equals [tex]\( 3.2 \)[/tex] seconds.
Therefore, the time it takes for the rocket to hit the ground is approximately [tex]\( 3.2 \)[/tex] seconds (rounded to the nearest tenth).
Summary:
- The maximum height of the rocket is reached at [tex]\( 1 \)[/tex] second.
- The maximum height of the rocket is [tex]\( 25 \)[/tex] yards.
- The time it takes for the rocket to hit the ground is approximately [tex]\( 3.2 \)[/tex] seconds.
