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 \[ h(x)=\frac{-32 x^{2}}{150^{2}}+\mathrm{x} \text {, } \] 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? \( h=\square \) feet (Round to two decimal places as needed.)

Evaluate the expression by following steps: - step0: Evaluate: \(\frac{-32x^{2}}{150^{2}}+x\) - step1: Substitute: \(\frac{-32\times 100^{2}}{150^{2}}+100\) - step2: Multiply the terms: \(\frac{-320000}{150^{2}}+100\) - step3: Divide the terms: \(-\frac{128}{3^{2}}+100\) - step4: Evaluate the power: \(-\frac{128}{9}+100\) - step5: Reduce fractions to a common denominator: \(-\frac{128}{9}+\frac{100\times 9}{9}\) - step6: Transform the expression: \(\frac{-128+100\times 9}{9}\) - step7: Multiply the numbers: \(\frac{-128+900}{9}\) - step8: Add the numbers: \(\frac{772}{9}\) Substitute \( x=250 \) into the expression \( \frac{-32 x^{2}}{150^{2}}+x \). Evaluate the expression by following steps: - step0: Evaluate: \(\frac{-32x^{2}}{150^{2}}+x\) - step1: Substitute: \(\frac{-32\times 250^{2}}{150^{2}}+250\) - step2: Multiply the terms: \(\frac{-2000000}{150^{2}}+250\) - step3: Divide the terms: \(-\frac{800}{3^{2}}+250\) - step4: Evaluate the power: \(-\frac{800}{9}+250\) - step5: Reduce fractions to a common denominator: \(-\frac{800}{9}+\frac{250\times 9}{9}\) - step6: Transform the expression: \(\frac{-800+250\times 9}{9}\) - step7: Multiply the numbers: \(\frac{-800+2250}{9}\) - step8: Add the numbers: \(\frac{1450}{9}\) Let's solve the parts step by step. ### Part (a) To find the height of the golf ball after it has traveled 100 feet, we substitute \( x = 100 \) into the height function: \[ h(100) = \frac{-32(100)^{2}}{150^{2}} + 100 \] Calculating this gives: \[ h(100) = \frac{-32 \cdot 10000}{22500} + 100 = \frac{-320000}{22500} + 100 \] This simplifies to: \[ h(100) = -14.2222 + 100 = 85.7778 \] Rounding to two decimal places, we find: \[ h(100) \approx 85.78 \text{ feet} \] ### Part (b) Now, we need to find the height after the golf ball has traveled 250 feet by substituting \( x = 250 \): \[ h(250) = \frac{-32(250)^{2}}{150^{2}} + 250 \] Calculating this gives: \[ h(250) = \frac{-32 \cdot 62500}{22500} + 250 = \frac{-2000000}{22500} + 250 \] This simplifies to: \[ h(250) = -88.8889 + 250 = 161.1111 \] Rounding to two decimal places, we find: \[ h(250) \approx 161.11 \text{ feet} \] ### Summary of Results - (a) The height after traveling 100 feet is \( h \approx 85.78 \) feet. - (b) The height after traveling 250 feet is \( h \approx 161.11 \) feet.