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. (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) $$ =172.44 $$ feet (Round to two decimal places as needed.) Interpret the value of $$ h(400) $$. Choose the correct answer below and fill in the answer box to complete your choice. A. At a height of 400 feet, the horizontal distance traveled by the golf ball is feet. B. The height of the golf ball after it has traveled a horizontal distance of 400 feet is feet.

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To determine the height of the golf ball after it has traveled a certain distance, we can use the given function $$ h(x) = \frac{-32x^2}{150^2} + x $$, where $$ x $$ is the horizontal distance traveled.

(a) To find the height of the golf ball after it has traveled 100 feet, we substitute $$ x = 100 $$ into the function:
$$ h(100) = \frac{-32(100)^2}{150^2} + 100 $$

Let's calculate this value.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{-32\times 100^{2}}{150^{2}}+100$$
- step1: Reduce the fraction:
  $$-\frac{128}{3^{2}}+100$$
- step2: Evaluate the power:
  $$-\frac{128}{9}+100$$
- step3: Reduce fractions to a common denominator:
  $$-\frac{128}{9}+\frac{100\times 9}{9}$$
- step4: Transform the expression:
  $$\frac{-128+100\times 9}{9}$$
- step5: Multiply the numbers:
  $$\frac{-128+900}{9}$$
- step6: Add the numbers:
  $$\frac{772}{9}$$
The height of the golf ball after it has traveled 100 feet is approximately 85.78 feet.

(b) To find the height after it has traveled 250 feet, we substitute $$ x = 250 $$ into the function:
$$ h(250) = \frac{-32(250)^2}{150^2} + 250 $$

Let's calculate this value.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{-32\times 250^{2}}{150^{2}}+250$$
- step1: Reduce the fraction:
  $$-\frac{800}{3^{2}}+250$$
- step2: Evaluate the power:
  $$-\frac{800}{9}+250$$
- step3: Reduce fractions to a common denominator:
  $$-\frac{800}{9}+\frac{250\times 9}{9}$$
- step4: Transform the expression:
  $$\frac{-800+250\times 9}{9}$$
- step5: Multiply the numbers:
  $$\frac{-800+2250}{9}$$
- step6: Add the numbers:
  $$\frac{1450}{9}$$
The height after it has traveled 250 feet is approximately 161.11 feet.

(c) To find $$ h(400) $$, we substitute $$ x = 400 $$ into the function:
$$ h(400) = \frac{-32(400)^2}{150^2} + 400 $$

Let's calculate this value.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{-32\times 400^{2}}{150^{2}}+400$$
- step1: Reduce the fraction:
  $$-\frac{2048}{3^{2}}+400$$
- step2: Evaluate the power:
  $$-\frac{2048}{9}+400$$
- step3: Reduce fractions to a common denominator:
  $$-\frac{2048}{9}+\frac{400\times 9}{9}$$
- step4: Transform the expression:
  $$\frac{-2048+400\times 9}{9}$$
- step5: Multiply the numbers:
  $$\frac{-2048+3600}{9}$$
- step6: Add the numbers:
  $$\frac{1552}{9}$$
The value of $$ h(400) $$ is approximately 172.44 feet.

Interpretation:
- At a height of 400 feet, the horizontal distance traveled by the golf ball is 400 feet.
- The height of the golf ball after it has traveled a horizontal distance of 400 feet is 172.44 feet.
(a) The height after traveling 100 feet is approximately 85.78 feet. (b) The height after traveling 250 feet is approximately 161.11 feet. (c) $$ h(400) = 172.44 $$ feet. This means that after the golf ball has traveled 400 feet horizontally, it reaches a height of 172.44 feet.

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