Answer :
To find the number of seconds it will take for the object to reach its maximum height and the corresponding maximum height itself, we can analyze the given quadratic equation for height:
[tex]\[ s(t) = -16t^2 + 48t \][/tex]
### Step 1: Determine the Time to Reach Maximum Height
The equation [tex]\( s(t) = -16t^2 + 48t \)[/tex] represents a parabola that opens downward (since the coefficient of [tex]\( t^2 \)[/tex] is negative). The object reaches its maximum height at the vertex of this parabola. For a quadratic equation in the form [tex]\( at^2 + bt + c \)[/tex], the time to reach maximum height can be found using the vertex formula [tex]\( t = -\frac{b}{2a} \)[/tex].
Given:
[tex]\[ a = -16 \][/tex]
[tex]\[ b = 48 \][/tex]
Plug these values into the vertex formula:
[tex]\[ t = -\frac{b}{2a} = -\frac{48}{2(-16)} \][/tex]
Simplify the calculation:
[tex]\[ t = -\frac{48}{-32} \][/tex]
[tex]\[ t = 1.5 \][/tex]
So, the object will take [tex]\( 1.5 \)[/tex] seconds to reach its maximum height.
### Step 2: Determine the Maximum Height
Now we will substitute [tex]\( t = 1.5 \)[/tex] back into the original height equation to find the maximum height:
[tex]\[ s(1.5) = -16(1.5)^2 + 48(1.5) \][/tex]
Calculate each term separately:
[tex]\[ (1.5)^2 = 2.25 \][/tex]
[tex]\[ -16 \cdot 2.25 = -36 \][/tex]
[tex]\[ 48 \cdot 1.5 = 72 \][/tex]
Combine these results:
[tex]\[ s(1.5) = -36 + 72 \][/tex]
[tex]\[ s(1.5) = 36 \][/tex]
So, the maximum height reached by the object is [tex]\( 36 \)[/tex] feet.
### Final Answer
The object will take [tex]\( \boxed{1.5} \)[/tex] second(s) to reach its maximum height.
The maximum height reached by the object is [tex]\( \boxed{36} \)[/tex] feet.
[tex]\[ s(t) = -16t^2 + 48t \][/tex]
### Step 1: Determine the Time to Reach Maximum Height
The equation [tex]\( s(t) = -16t^2 + 48t \)[/tex] represents a parabola that opens downward (since the coefficient of [tex]\( t^2 \)[/tex] is negative). The object reaches its maximum height at the vertex of this parabola. For a quadratic equation in the form [tex]\( at^2 + bt + c \)[/tex], the time to reach maximum height can be found using the vertex formula [tex]\( t = -\frac{b}{2a} \)[/tex].
Given:
[tex]\[ a = -16 \][/tex]
[tex]\[ b = 48 \][/tex]
Plug these values into the vertex formula:
[tex]\[ t = -\frac{b}{2a} = -\frac{48}{2(-16)} \][/tex]
Simplify the calculation:
[tex]\[ t = -\frac{48}{-32} \][/tex]
[tex]\[ t = 1.5 \][/tex]
So, the object will take [tex]\( 1.5 \)[/tex] seconds to reach its maximum height.
### Step 2: Determine the Maximum Height
Now we will substitute [tex]\( t = 1.5 \)[/tex] back into the original height equation to find the maximum height:
[tex]\[ s(1.5) = -16(1.5)^2 + 48(1.5) \][/tex]
Calculate each term separately:
[tex]\[ (1.5)^2 = 2.25 \][/tex]
[tex]\[ -16 \cdot 2.25 = -36 \][/tex]
[tex]\[ 48 \cdot 1.5 = 72 \][/tex]
Combine these results:
[tex]\[ s(1.5) = -36 + 72 \][/tex]
[tex]\[ s(1.5) = 36 \][/tex]
So, the maximum height reached by the object is [tex]\( 36 \)[/tex] feet.
### Final Answer
The object will take [tex]\( \boxed{1.5} \)[/tex] second(s) to reach its maximum height.
The maximum height reached by the object is [tex]\( \boxed{36} \)[/tex] feet.
