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Answer :
To find out how long it will take the toy rocket to reach its maximum height and what that height is, let's break down the problem step by step using the formula given. The height of the rocket as a function of time is given by:
[tex]\[ h(t) = -16t^2 + 48t + 5 \][/tex]
### Step 1: Find the Time to Reach Maximum Height
The formula for a quadratic function in standard form [tex]\( ax^2 + bx + c \)[/tex] will reach its maximum or minimum at [tex]\( t = -\frac{b}{2a} \)[/tex]. Here, [tex]\( a = -16 \)[/tex], [tex]\( b = 48 \)[/tex].
Plugging these values into the formula gives:
[tex]\[ t = -\frac{b}{2a} = -\frac{48}{2 \times -16} = \frac{48}{32} = 1.5 \][/tex]
The rocket reaches its maximum height 1.5 seconds after launch.
### Step 2: Calculate the Maximum Height
Now, we will find the maximum height by substituting [tex]\( t = 1.5 \)[/tex] back into the height function [tex]\( h(t) \)[/tex]:
[tex]\[ h(1.5) = -16(1.5)^2 + 48(1.5) + 5 \][/tex]
First, calculate [tex]\( (1.5)^2 \)[/tex]:
[tex]\[ (1.5)^2 = 2.25 \][/tex]
Then substitute into the height equation:
[tex]\[ h(1.5) = -16(2.25) + 48(1.5) + 5 \][/tex]
Calculating each term:
1. [tex]\(-16 \times 2.25 = -36\)[/tex]
2. [tex]\(48 \times 1.5 = 72\)[/tex]
Now, adding these values together:
[tex]\[ h(1.5) = -36 + 72 + 5 = 41 \][/tex]
Thus, the maximum height reached by the rocket is 41 feet.
In summary, the rocket reaches its maximum height 1.5 seconds after launch, and the maximum height it reaches is 41 feet.
[tex]\[ h(t) = -16t^2 + 48t + 5 \][/tex]
### Step 1: Find the Time to Reach Maximum Height
The formula for a quadratic function in standard form [tex]\( ax^2 + bx + c \)[/tex] will reach its maximum or minimum at [tex]\( t = -\frac{b}{2a} \)[/tex]. Here, [tex]\( a = -16 \)[/tex], [tex]\( b = 48 \)[/tex].
Plugging these values into the formula gives:
[tex]\[ t = -\frac{b}{2a} = -\frac{48}{2 \times -16} = \frac{48}{32} = 1.5 \][/tex]
The rocket reaches its maximum height 1.5 seconds after launch.
### Step 2: Calculate the Maximum Height
Now, we will find the maximum height by substituting [tex]\( t = 1.5 \)[/tex] back into the height function [tex]\( h(t) \)[/tex]:
[tex]\[ h(1.5) = -16(1.5)^2 + 48(1.5) + 5 \][/tex]
First, calculate [tex]\( (1.5)^2 \)[/tex]:
[tex]\[ (1.5)^2 = 2.25 \][/tex]
Then substitute into the height equation:
[tex]\[ h(1.5) = -16(2.25) + 48(1.5) + 5 \][/tex]
Calculating each term:
1. [tex]\(-16 \times 2.25 = -36\)[/tex]
2. [tex]\(48 \times 1.5 = 72\)[/tex]
Now, adding these values together:
[tex]\[ h(1.5) = -36 + 72 + 5 = 41 \][/tex]
Thus, the maximum height reached by the rocket is 41 feet.
In summary, the rocket reaches its maximum height 1.5 seconds after launch, and the maximum height it reaches is 41 feet.
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