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Answer :
To find the amount of time it takes for the toy rocket to reach its maximum height, as well as the maximum height itself, we can use the function given for the height of the rocket:
[tex]\[ h(t) = -16t^2 + 104t + 3 \][/tex]
This is a quadratic function in the form of [tex]\( ax^2 + bx + c \)[/tex], where:
- [tex]\( a = -16 \)[/tex]
- [tex]\( b = 104 \)[/tex]
- [tex]\( c = 3 \)[/tex]
### Step 1: Find the Time to Reach Maximum Height
The maximum height of a quadratic function, where the graph is a parabola opening downward (because [tex]\( a \)[/tex] is negative), occurs at the vertex. The time [tex]\( t \)[/tex] at which this maximum height is reached can be calculated using the vertex formula:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Plugging in the values for [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ t = -\frac{104}{2 \times (-16)} \][/tex]
[tex]\[ t = \frac{104}{32} \][/tex]
[tex]\[ t = 3.25 \][/tex]
So, it takes 3.25 seconds for the rocket to reach its maximum height.
### Step 2: Calculate the Maximum Height
Once we have the time, we can find the maximum height by substituting [tex]\( t = 3.25 \)[/tex] back into the height function:
[tex]\[ h(3.25) = -16(3.25)^2 + 104(3.25) + 3 \][/tex]
Calculating step-by-step:
1. [tex]\( (3.25)^2 = 10.5625 \)[/tex]
2. [tex]\( -16 \times 10.5625 = -169 \)[/tex]
3. [tex]\( 104 \times 3.25 = 338 \)[/tex]
4. Add them up: [tex]\(-169 + 338 + 3 = 172\)[/tex]
Therefore, the maximum height of the rocket is 172 feet.
In summary, the rocket reaches its maximum height of 172 feet after 3.25 seconds.
[tex]\[ h(t) = -16t^2 + 104t + 3 \][/tex]
This is a quadratic function in the form of [tex]\( ax^2 + bx + c \)[/tex], where:
- [tex]\( a = -16 \)[/tex]
- [tex]\( b = 104 \)[/tex]
- [tex]\( c = 3 \)[/tex]
### Step 1: Find the Time to Reach Maximum Height
The maximum height of a quadratic function, where the graph is a parabola opening downward (because [tex]\( a \)[/tex] is negative), occurs at the vertex. The time [tex]\( t \)[/tex] at which this maximum height is reached can be calculated using the vertex formula:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Plugging in the values for [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ t = -\frac{104}{2 \times (-16)} \][/tex]
[tex]\[ t = \frac{104}{32} \][/tex]
[tex]\[ t = 3.25 \][/tex]
So, it takes 3.25 seconds for the rocket to reach its maximum height.
### Step 2: Calculate the Maximum Height
Once we have the time, we can find the maximum height by substituting [tex]\( t = 3.25 \)[/tex] back into the height function:
[tex]\[ h(3.25) = -16(3.25)^2 + 104(3.25) + 3 \][/tex]
Calculating step-by-step:
1. [tex]\( (3.25)^2 = 10.5625 \)[/tex]
2. [tex]\( -16 \times 10.5625 = -169 \)[/tex]
3. [tex]\( 104 \times 3.25 = 338 \)[/tex]
4. Add them up: [tex]\(-169 + 338 + 3 = 172\)[/tex]
Therefore, the maximum height of the rocket is 172 feet.
In summary, the rocket reaches its maximum height of 172 feet after 3.25 seconds.
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