IDNLearn.com is your go-to platform for finding accurate and reliable answers. Get step-by-step guidance for all your technical questions from our dedicated community members.
Sagot :
To determine the correct function that models the height of the baseball, we need to consider the properties of a quadratic function given in the form [tex]\( h(t) = a(t-h)^2 + k \)[/tex]. Here, [tex]\((h, k)\)[/tex] is the vertex of the parabola, which represents the maximum height of the projectile.
Given the problem:
- The baseball is initially hit from a height of [tex]\(3\)[/tex] feet.
- The baseball reaches a maximum height of [tex]\(403\)[/tex] feet.
Based on these details, let's evaluate the options provided:
Option A: [tex]\( h(t) = -16(t-3)^2 + 403 \)[/tex]
This function suggests:
- The horizontal shift [tex]\((t-3)\)[/tex] means the vertex occurs at [tex]\(t=3\)[/tex] seconds.
- The maximum height [tex]\(k\)[/tex] is [tex]\(403\)[/tex] feet.
However, the initial height is not correctly represented when [tex]\(t=0\)[/tex]. Therefore, this is not the correct option.
Option B: [tex]\( h(t) = -16(t-5)^2 + 3 \)[/tex]
This function suggests:
- The horizontal shift [tex]\((t-5)\)[/tex] means the vertex occurs at [tex]\(t=5\)[/tex] seconds.
- The initial height [tex]\(k\)[/tex] is [tex]\(3\)[/tex] feet.
However, the maximum height here is just [tex]\(3\)[/tex] feet, which doesn't match the given maximum height of [tex]\(403\)[/tex] feet. Thus, B is not correct.
Option C: [tex]\( h(t) = -16(t-5)^2 + 403 \)[/tex]
This function suggests:
- The vertex occurs at [tex]\(t=5\)[/tex] seconds where the maximum height is [tex]\(403\)[/tex] feet.
- When [tex]\(t=0\)[/tex], substituting into the function gives us:
[tex]\[ h(0) = -16(0-5)^2 + 403 \][/tex]
[tex]\[ = -16(25) + 403 \][/tex]
[tex]\[ = -400 + 403 \][/tex]
[tex]\[ = 3 \][/tex]
The initial height correctly matches [tex]\(3\)[/tex] feet, making this the correct function.
Option D: [tex]\( h(t) = -16(t-403)^2 + 3 \)[/tex]
This function suggests:
- The vertex occurs at [tex]\(t=403\)[/tex] seconds which doesn't make sense in the context of typical baseball flight times.
- The initial height [tex]\(k\)[/tex] is incorrectly represented if we consider real-world scenarios.
Given these evaluations, the correct function that models the situation accurately is:
[tex]\( \boxed{C} \)[/tex]
Given the problem:
- The baseball is initially hit from a height of [tex]\(3\)[/tex] feet.
- The baseball reaches a maximum height of [tex]\(403\)[/tex] feet.
Based on these details, let's evaluate the options provided:
Option A: [tex]\( h(t) = -16(t-3)^2 + 403 \)[/tex]
This function suggests:
- The horizontal shift [tex]\((t-3)\)[/tex] means the vertex occurs at [tex]\(t=3\)[/tex] seconds.
- The maximum height [tex]\(k\)[/tex] is [tex]\(403\)[/tex] feet.
However, the initial height is not correctly represented when [tex]\(t=0\)[/tex]. Therefore, this is not the correct option.
Option B: [tex]\( h(t) = -16(t-5)^2 + 3 \)[/tex]
This function suggests:
- The horizontal shift [tex]\((t-5)\)[/tex] means the vertex occurs at [tex]\(t=5\)[/tex] seconds.
- The initial height [tex]\(k\)[/tex] is [tex]\(3\)[/tex] feet.
However, the maximum height here is just [tex]\(3\)[/tex] feet, which doesn't match the given maximum height of [tex]\(403\)[/tex] feet. Thus, B is not correct.
Option C: [tex]\( h(t) = -16(t-5)^2 + 403 \)[/tex]
This function suggests:
- The vertex occurs at [tex]\(t=5\)[/tex] seconds where the maximum height is [tex]\(403\)[/tex] feet.
- When [tex]\(t=0\)[/tex], substituting into the function gives us:
[tex]\[ h(0) = -16(0-5)^2 + 403 \][/tex]
[tex]\[ = -16(25) + 403 \][/tex]
[tex]\[ = -400 + 403 \][/tex]
[tex]\[ = 3 \][/tex]
The initial height correctly matches [tex]\(3\)[/tex] feet, making this the correct function.
Option D: [tex]\( h(t) = -16(t-403)^2 + 3 \)[/tex]
This function suggests:
- The vertex occurs at [tex]\(t=403\)[/tex] seconds which doesn't make sense in the context of typical baseball flight times.
- The initial height [tex]\(k\)[/tex] is incorrectly represented if we consider real-world scenarios.
Given these evaluations, the correct function that models the situation accurately is:
[tex]\( \boxed{C} \)[/tex]
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. For clear and precise answers, choose IDNLearn.com. Thanks for stopping by, and come back soon for more valuable insights.
