To find the height of the rocket after 1.5 seconds, we need to substitute \( t = 1.5 \) into the function \( t(h) = \pm \sqrt{h - 67} - 16 + 2 \).
First, we can rearrange the equation to solve for \( h \):
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Start with: \[ t(h) = \pm \sqrt{h - 67} - 16 + 2 \] which simplifies to: \[ t(h) = \pm \sqrt{h - 67} - 14 \]
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Now, to find \( h \) when \( t = 1.5 \): \[ 1.5 = \pm \sqrt{h - 67} - 14 \]
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Isolate the square root: \[ 1.5 + 14 = \sqrt{h - 67} \] \[ 15.5 = \sqrt{h - 67} \]
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Square both sides: \[ (15.5)^2 = h - 67 \] \[ 240.25 = h - 67 \]
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Solve for \( h \): \[ h = 240.25 + 67 \] \[ h = 307.25 \]
It seems the height calculation yielded \( 307.25 \) feet, which is not one of the provided options (A to D). This indicates that either the function has been misinterpreted or there has been an error in the posed question.
Let's instead check if the function was initiated correctly at all. The original function notation hints suggests there's a misalignment in its understanding. If we revisit the original time and height equation or if you want to clarify any details, I'd be glad to assist!