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Answer :
- Find the time at which the projectile reaches its maximum height using $t = -b/(2a)$, which gives $t = 1.5$ seconds.
- Substitute $t = 1.5$ into the height function $h(t) = -16t^2 + 48t + 190$.
- Calculate the maximum height: $h(1.5) = -16(1.5)^2 + 48(1.5) + 190 = 226$ feet.
- The maximum height of the projectile is $\boxed{226}$ feet.
### Explanation
1. Understanding the Problem
We are given the height function of a projectile: $h(t) = -16t^2 + 48t + 190$, where $h(t)$ is the height in feet at time $t$ seconds. We want to find the maximum height of the projectile.
2. Finding the Time of Maximum Height
The height function is a quadratic equation in the form of a parabola. Since the coefficient of the $t^2$ term is negative (-16), the parabola opens downward, meaning it has a maximum point (vertex). The $t$-coordinate of the vertex gives the time at which the projectile reaches its maximum height. We can find this time using the formula $t = -b / (2a)$, where $a = -16$ and $b = 48$.
3. Calculating the Time
Plugging in the values, we get $t = -48 / (2 * -16) = -48 / -32 = 1.5$ seconds. So, the projectile reaches its maximum height at $t = 1.5$ seconds.
4. Substituting to Find Maximum Height
Now, we substitute this value of $t$ into the height function to find the maximum height: $h(1.5) = -16(1.5)^2 + 48(1.5) + 190$.
5. Calculating the Maximum Height
Calculating the height: $h(1.5) = -16(2.25) + 48(1.5) + 190 = -36 + 72 + 190 = 36 + 190 = 226$ feet.
6. Final Answer
Therefore, the maximum height of the projectile is 226 feet.
### Examples
Understanding projectile motion is crucial in fields like sports (e.g., calculating the trajectory of a ball), military science (e.g., determining the range of a projectile), and even in designing amusement park rides. By knowing the initial conditions and using quadratic equations, engineers can predict the path and maximum height of objects, ensuring safety and optimizing performance.
- Substitute $t = 1.5$ into the height function $h(t) = -16t^2 + 48t + 190$.
- Calculate the maximum height: $h(1.5) = -16(1.5)^2 + 48(1.5) + 190 = 226$ feet.
- The maximum height of the projectile is $\boxed{226}$ feet.
### Explanation
1. Understanding the Problem
We are given the height function of a projectile: $h(t) = -16t^2 + 48t + 190$, where $h(t)$ is the height in feet at time $t$ seconds. We want to find the maximum height of the projectile.
2. Finding the Time of Maximum Height
The height function is a quadratic equation in the form of a parabola. Since the coefficient of the $t^2$ term is negative (-16), the parabola opens downward, meaning it has a maximum point (vertex). The $t$-coordinate of the vertex gives the time at which the projectile reaches its maximum height. We can find this time using the formula $t = -b / (2a)$, where $a = -16$ and $b = 48$.
3. Calculating the Time
Plugging in the values, we get $t = -48 / (2 * -16) = -48 / -32 = 1.5$ seconds. So, the projectile reaches its maximum height at $t = 1.5$ seconds.
4. Substituting to Find Maximum Height
Now, we substitute this value of $t$ into the height function to find the maximum height: $h(1.5) = -16(1.5)^2 + 48(1.5) + 190$.
5. Calculating the Maximum Height
Calculating the height: $h(1.5) = -16(2.25) + 48(1.5) + 190 = -36 + 72 + 190 = 36 + 190 = 226$ feet.
6. Final Answer
Therefore, the maximum height of the projectile is 226 feet.
### Examples
Understanding projectile motion is crucial in fields like sports (e.g., calculating the trajectory of a ball), military science (e.g., determining the range of a projectile), and even in designing amusement park rides. By knowing the initial conditions and using quadratic equations, engineers can predict the path and maximum height of objects, ensuring safety and optimizing performance.
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