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A baseball is hit from an initial height of 3 feet and reaches a maximum height of 403 feet. Which function could be used to model this situation, where [tex]\( h(t) \)[/tex] is the height, in feet, after [tex]\( t \)[/tex] seconds?

A. [tex]\( h(t) = -16(t - 403)^2 + 3 \)[/tex]

B. [tex]\( h(t) = -16(t - 3)^2 + 403 \)[/tex]

C. [tex]\( h(t) = -16(t - 5)^2 + 3 \)[/tex]

D. [tex]\( h(t) = -16(t - 5)^2 + 403 \)[/tex]

Sagot :

GinnyAnswer
To determine the correct function that models the baseball's height after being hit, we need to recognize the properties of the function describing the motion of the baseball.

Given:
- The initial height of the baseball is 3 feet.
- The maximum height achieved is 403 feet.

From these properties, we can infer that:
- The function must have a vertex form of a quadratic equation, [tex]\( h(t) = a(t-h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex of the parabola representing the maximum height.
- Since we know the initial height, and the function usually represents a downwards parabola due to the negative acceleration caused by gravity, the coefficient [tex]\(a\)[/tex] should be negative.

We can analyze the options to see which one fits these conditions:

A. [tex]\( h(t) = -16(t-403)^2 + 3 \)[/tex]
- Here the vertex is at [tex]\( t = 403 \)[/tex] and height [tex]\( h = 3 \)[/tex], which implies the maximum height happens at 403 seconds which does not match the given information. Furthermore, 3 is the starting height, not the maximum.

B. [tex]\( h(t) = -16(t-3)^2 + 403 \)[/tex]
- Here the vertex is at [tex]\( t = 3 \)[/tex] and height [tex]\( h = 403 \)[/tex], suggesting the maximum height occurs at 3 seconds, and at that maximum, the height is 403 feet. Since this matches both the given maximum height and the time when it occurs, and aligns with the formula structure for parabolic motion, this appears plausible.

C. [tex]\( h(t) = -16(t-5)^2 + 3 \)[/tex]
- The vertex is at [tex]\( t = 5 \)[/tex], and the height is [tex]\( h = 3 \)[/tex], which does not match the maximum height condition of 403 feet.

D. [tex]\( h(t) = -16(t-5)^2 + 403 \)[/tex]
- Here the vertex is at [tex]\( t = 5 \)[/tex] and height [tex]\( h = 403 \)[/tex], suggesting the maximum height happens at 5 seconds, and is 403 feet. This does not match placing the peak at t = 5 and is not logically correct.

Given these analyses, the correctly structured function which matches all conditions for the baseball's height formula, including its initial and maximum conditions as described, is:

Correct answer: B. [tex]\( h(t) = -16(t-3)^2 + 403 \)[/tex]
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