Welcome to Westonci.ca, the place where your questions are answered by a community of knowledgeable contributors. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
To determine which function can be used to represent the height of the T-shirt as a function of time, let's analyze the given details:
1. The T-shirt starts at a height of 8 feet when it leaves the cannon.
2. The T-shirt reaches its maximum height of 24 feet at 1 second.
We know that the path of the T-shirt follows a parabolic trajectory due to the physics of projectile motion. Generally, the formula for the height [tex]\( f(t) \)[/tex] of a projectile at time [tex]\( t \)[/tex] is represented by a quadratic equation in vertex form:
[tex]\[ f(t) = a(t - h)^2 + k \][/tex]
where:
- [tex]\( (h, k) \)[/tex] represents the vertex of the parabola (the point at which the projectile reaches its maximum height).
- [tex]\( a \)[/tex] is a constant that affects the width and direction of the parabola.
Given the vertex (maximum height) at [tex]\( t = 1 \)[/tex] second and the maximum height [tex]\( k = 24 \)[/tex] feet, the vertex form of our equation becomes:
[tex]\[ f(t) = a(t - 1)^2 + 24 \][/tex]
Now, we need to determine the value of [tex]\( a \)[/tex]. We use the fact that at [tex]\( t = 0 \)[/tex] seconds, the height [tex]\( f(0) = 8 \)[/tex] feet.
Substitute [tex]\( t = 0 \)[/tex] and [tex]\( f(0) = 8 \)[/tex] into the vertex form equation:
[tex]\[ 8 = a(0 - 1)^2 + 24 \][/tex]
[tex]\[ 8 = a(1) + 24 \][/tex]
[tex]\[ 8 = a + 24 \][/tex]
Solving for [tex]\( a \)[/tex]:
[tex]\[ a = 8 - 24 \][/tex]
[tex]\[ a = -16 \][/tex]
Now we substitute [tex]\( a = -16 \)[/tex] back into the vertex form equation:
[tex]\[ f(t) = -16(t - 1)^2 + 24 \][/tex]
Thus, the function that represents the height of the T-shirt as a function of time [tex]\( t \)[/tex] in seconds is:
[tex]\[ f(t) = -16(t-1)^2 + 24 \][/tex]
So, the correct function is:
[tex]\[ f(t) = -16(t-1)^2 + 24 \][/tex]
1. The T-shirt starts at a height of 8 feet when it leaves the cannon.
2. The T-shirt reaches its maximum height of 24 feet at 1 second.
We know that the path of the T-shirt follows a parabolic trajectory due to the physics of projectile motion. Generally, the formula for the height [tex]\( f(t) \)[/tex] of a projectile at time [tex]\( t \)[/tex] is represented by a quadratic equation in vertex form:
[tex]\[ f(t) = a(t - h)^2 + k \][/tex]
where:
- [tex]\( (h, k) \)[/tex] represents the vertex of the parabola (the point at which the projectile reaches its maximum height).
- [tex]\( a \)[/tex] is a constant that affects the width and direction of the parabola.
Given the vertex (maximum height) at [tex]\( t = 1 \)[/tex] second and the maximum height [tex]\( k = 24 \)[/tex] feet, the vertex form of our equation becomes:
[tex]\[ f(t) = a(t - 1)^2 + 24 \][/tex]
Now, we need to determine the value of [tex]\( a \)[/tex]. We use the fact that at [tex]\( t = 0 \)[/tex] seconds, the height [tex]\( f(0) = 8 \)[/tex] feet.
Substitute [tex]\( t = 0 \)[/tex] and [tex]\( f(0) = 8 \)[/tex] into the vertex form equation:
[tex]\[ 8 = a(0 - 1)^2 + 24 \][/tex]
[tex]\[ 8 = a(1) + 24 \][/tex]
[tex]\[ 8 = a + 24 \][/tex]
Solving for [tex]\( a \)[/tex]:
[tex]\[ a = 8 - 24 \][/tex]
[tex]\[ a = -16 \][/tex]
Now we substitute [tex]\( a = -16 \)[/tex] back into the vertex form equation:
[tex]\[ f(t) = -16(t - 1)^2 + 24 \][/tex]
Thus, the function that represents the height of the T-shirt as a function of time [tex]\( t \)[/tex] in seconds is:
[tex]\[ f(t) = -16(t-1)^2 + 24 \][/tex]
So, the correct function is:
[tex]\[ f(t) = -16(t-1)^2 + 24 \][/tex]
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
