A golf ball is hit with an initial velocity of 150 feet per second at an inclination of \( 45^{\circ} \) to the horizontal. In physics, it is established that the height \( h \) of the golf ball is given by the function \[ h(x)=\frac{-32 x^{2}}{150^{2}}+x \text {, } \] where \( x \) is the horizontal distance that the golf ball has traveled. Complete parts (a) through (g). (a) Determine the height of the golf ball after it has traveled 100 feet. h \( =85.78 \) feet (Round to two decimal places as needed.) (b) What is the height after it has traveled 250 feet? \( h=161.11 \) feet (Round to two decimal places as needed.) (c) What is h(400)? Interpret this value. \( h(400)=\square \) feet (Round to two decimal places as needed.)
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To calculate \( h(400) \), we can substitute \( x = 400 \) into the height function: \[ h(400) = \frac{-32(400)^{2}}{150^{2}} + 400 \] Calculating \( (400)^{2} \) gives us \( 160000 \). Now substituting that back into the equation yields: \[ h(400) = \frac{-32 \times 160000}{22500} + 400 \] This simplifies to \( h(400) = \frac{-5120000}{22500} + 400 \). Calculating that, we find: \[ -5120000 \div 22500 \approx -227.11 \] Now combining this with 400: \[ h(400) \approx -227.11 + 400 \approx 172.89 \] So, rounding to two decimal places, we have: \( h(400) \approx 172.89 \) feet. This value suggests that at 400 feet, the golf ball is still flying in the air 172.89 feet above the ground.
