A golf ball is hit with an invilal velocily of 140 feef per second al an inclination of $$ 45^{\circ} $$ io the horizontal in physics, is established trat I $$ h(x)=\frac{-32 x^{2}}{140^{2}}+x $$ where x is the horizontal dislance that the goif ball has traveled. Complele parts (a) through (g) (a) Determine the height of the ooll ball after it has tavelod 100 teet $$ \mathrm{h}=83.67^{\circ} $$ leel (Round to two decmal places as needed.) (b) What is the heighl after it has traveled 200 leet? $$ h=134.60 $$ feet (Round to wo decimal places as needed.) (c) What is $$ \mathrm{n}(350) $$ ? interprel ihis value. $$ n(350)= $$ $$ \square $$ freat (Round to two decmal places as needed.)

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**(a) Compute the height at $$ x=100 $$ feet**

The height function is given by  
$$
h(x)=\frac{-32x^2}{140^2}+x.
$$
Since $$ 140^2 = 19600 $$, we have  
$$
h(x)=\frac{-32x^2}{19600}+x.
$$
Substitute $$ x=100 $$:
$$
h(100)=\frac{-32(100^2)}{19600}+100.
$$
Calculate $$ 100^2 $$:
$$
100^2=10000.
$$
Then,
$$
h(100)=\frac{-32(10000)}{19600}+100 = \frac{-320000}{19600}+100.
$$
Divide to simplify:
$$
\frac{320000}{19600} \approx 16.33.
$$
Thus,
$$
h(100)=100-16.33=83.67 \text{ feet (approximately)}.
$$

---

**(b) Compute the height at $$ x=200 $$ feet**

Using the same height function, substitute $$ x=200 $$:
$$
h(200)=\frac{-32(200^2)}{19600}+200.
$$
Calculate $$ 200^2 $$:
$$
200^2=40000.
$$
Then,
$$
h(200)=\frac{-32(40000)}{19600}+200 = \frac{-1280000}{19600}+200.
$$
Perform the division:
$$
\frac{1280000}{19600} \approx 65.31.
$$
Thus,
$$
h(200)=200-65.31 \approx 134.69 \text{ feet (approximately)}.
$$
(Rounded to two decimal places.)

---

**(c) Compute and interpret $$ h(350) $$**

Substitute $$ x=350 $$:
$$
h(350)=\frac{-32(350^2)}{19600}+350.
$$
Calculate $$ 350^2 $$:
$$
350^2=122500.
$$
Then,
$$
h(350)=\frac{-32(122500)}{19600}+350 = \frac{-3920000}{19600}+350.
$$
Divide:
$$
\frac{3920000}{19600}=200,
$$
so,
$$
h(350)=-200+350=150 \text{ feet}.
$$

**Interpretation:** At a horizontal distance of $$ 350 $$ feet from the starting point, the golf ball is $$ 150 $$ feet above the ground. This value indicates the ball’s height (likely on its descending path after reaching its maximum height) while in flight.
**(a) Height at 100 feet:** $$ h(100) = 83.67 $$ feet **(b) Height at 200 feet:** $$ h(200) = 134.60 $$ feet **(c) $$ h(350) = 150.00 $$ feet** This means the golf ball is 150 feet above the ground after traveling 350 feet horizontally.

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