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Sagot :
To determine the correct function that models the situation, let's break down the given information and apply it to each of the options.
Key pieces of information provided:
1. The initial height of the baseball is 3 feet.
2. The maximum height of the baseball is 403 feet.
3. We need a function \( h(t) \) that represents the height of the baseball in feet at time \( t \) seconds.
The general form of a quadratic function modeling the height of a projectile is:
[tex]\[ h(t) = -at^2 + bt + c \][/tex]
where \( a \), \( b \), and \( c \) are constants. However, using the vertex form of the quadratic function is more straightforward for this problem:
[tex]\[ h(t) = -a(t - h)^2 + k \][/tex]
where \((h, k)\) is the vertex of the parabola.
In this case:
- The vertex \((h, k)\) represents the time \(h\) at which the maximum height \(k\) occurs.
- The maximum height is given as 403 feet, so \(k = 403\).
Given the options, we can see that each function is already in vertex form:
A. \( h(t) = -16(t - 5)^2 + 403 \)
B. \( h(t) = -16(t - 5)^2 + 3 \)
C. \( h(t) = -16(t - 3)^2 + 403 \)
D. \( h(t) = -16(t - 403)^2 + 3 \)
Now, analyze each option based on:
- The maximum height of 403 feet.
- The initial height of 3 feet when \( t = 0 \).
Option A.
[tex]\[ h(t) = -16(t - 5)^2 + 403 \][/tex]
- The vertex \((5, 403)\), indicates the height reaches 403 feet when \( t = 5 \).
- When \( t = 0 \):
[tex]\[ h(0) = -16(0 - 5)^2 + 403 = -16(25) + 403 = -400 + 403 = 3 \, \text{feet} \][/tex]
Option A satisfies both the maximum height and the initial height.
Option B.
[tex]\[ h(t) = -16(t - 5)^2 + 3 \][/tex]
- The vertex \((5, 3)\), which implies the maximum height is 3 feet, contradicting the given maximum height of 403 feet.
Option B does not fit.
Option C.
[tex]\[ h(t) = -16(t - 3)^2 + 403 \][/tex]
- The vertex \((3, 403)\), indicates the height reaches 403 feet when \( t = 3 \).
- When \( t = 0 \):
[tex]\[ h(0) = -16(0 - 3)^2 + 403 = -16(9) + 403 = -144 + 403 = 259 \, \text{feet} \][/tex]
Option C does not fit.
Option D.
[tex]\[ h(t) = -16(t - 403)^2 + 3 \][/tex]
- The vertex \((403, 3)\) is clearly not at all reasonable given real-world physical constraints.
- The maximum height is 3 feet, which contradicts the given maximum height of 403 feet.
Option D does not fit.
Thus, the correct answer is:
A. [tex]\( h(t) = -16(t - 5)^2 + 403 \)[/tex]
Key pieces of information provided:
1. The initial height of the baseball is 3 feet.
2. The maximum height of the baseball is 403 feet.
3. We need a function \( h(t) \) that represents the height of the baseball in feet at time \( t \) seconds.
The general form of a quadratic function modeling the height of a projectile is:
[tex]\[ h(t) = -at^2 + bt + c \][/tex]
where \( a \), \( b \), and \( c \) are constants. However, using the vertex form of the quadratic function is more straightforward for this problem:
[tex]\[ h(t) = -a(t - h)^2 + k \][/tex]
where \((h, k)\) is the vertex of the parabola.
In this case:
- The vertex \((h, k)\) represents the time \(h\) at which the maximum height \(k\) occurs.
- The maximum height is given as 403 feet, so \(k = 403\).
Given the options, we can see that each function is already in vertex form:
A. \( h(t) = -16(t - 5)^2 + 403 \)
B. \( h(t) = -16(t - 5)^2 + 3 \)
C. \( h(t) = -16(t - 3)^2 + 403 \)
D. \( h(t) = -16(t - 403)^2 + 3 \)
Now, analyze each option based on:
- The maximum height of 403 feet.
- The initial height of 3 feet when \( t = 0 \).
Option A.
[tex]\[ h(t) = -16(t - 5)^2 + 403 \][/tex]
- The vertex \((5, 403)\), indicates the height reaches 403 feet when \( t = 5 \).
- When \( t = 0 \):
[tex]\[ h(0) = -16(0 - 5)^2 + 403 = -16(25) + 403 = -400 + 403 = 3 \, \text{feet} \][/tex]
Option A satisfies both the maximum height and the initial height.
Option B.
[tex]\[ h(t) = -16(t - 5)^2 + 3 \][/tex]
- The vertex \((5, 3)\), which implies the maximum height is 3 feet, contradicting the given maximum height of 403 feet.
Option B does not fit.
Option C.
[tex]\[ h(t) = -16(t - 3)^2 + 403 \][/tex]
- The vertex \((3, 403)\), indicates the height reaches 403 feet when \( t = 3 \).
- When \( t = 0 \):
[tex]\[ h(0) = -16(0 - 3)^2 + 403 = -16(9) + 403 = -144 + 403 = 259 \, \text{feet} \][/tex]
Option C does not fit.
Option D.
[tex]\[ h(t) = -16(t - 403)^2 + 3 \][/tex]
- The vertex \((403, 3)\) is clearly not at all reasonable given real-world physical constraints.
- The maximum height is 3 feet, which contradicts the given maximum height of 403 feet.
Option D does not fit.
Thus, the correct answer is:
A. [tex]\( h(t) = -16(t - 5)^2 + 403 \)[/tex]
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