tion 3.2 The Graph of a Function 3.2.35-8C Part4 of 10 Agolf dall is hit with an inital velocity of 140 feet per second at an incination of $$ 45^{\circ} $$ to the horizontal. In physics, it is establehed that the $$ h(x)=\frac{-32 x^{2}}{140^{2}}+x $$ Where x is tha hortiontal distance that the golf ball has traveled Camptele parts (a) through (a). (0) Deterrine the height of the goif bat after it has traveled 100 feet $$ \mathrm{h}=83.07^{\prime} $$ feet (Round to tun tiecimal places as needed) (b) Whal is the height after thas traveled 200 foct? $$ h=134.69 $$ feet (Round to two decimal places as needed) (c) What is $$ \mathrm{n}(350) $$ ? internerel tis value. h $$ (350)=15000 $$ teel (Round to No decimal places as neended.) Inlerpret the value of h(350) Choose the correct answer telow and fill in the answer hex fo conplete your chace A. At a height of 350 feel, the horizontal distance taveled by me goif hal is $$ \square $$ feet B. The height of the goll ball effer it has traveed a rorvontal distance of 350 fect is $$ \square $$ fool

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Let the height function be
$$
h(x)=\frac{-32x^{2}}{140^{2}}+x,
$$
where $$ x $$ is the horizontal distance in feet.

### (a) Height after 100 feet

Substitute $$ x=100 $$ into the formula:
$$
h(100)=\frac{-32(100)^2}{140^2}+100.
$$

First, compute the denominator:
$$
140^2=19600.
$$

Then, compute the numerator for the quadratic term:
$$
-32(100)^2=-32(10000)=-320000.
$$

So,
$$
h(100)=\frac{-320000}{19600}+100.
$$

Divide:
$$
\frac{-320000}{19600}\approx -16.32653.
$$

Add 100:
$$
h(100)\approx100-16.32653\approx83.67347.
$$

Rounded to two decimal places, 
$$
h(100)\approx83.67\text{ feet}.
$$

### (b) Height after 200 feet

Substitute $$ x=200 $$:
$$
h(200)=\frac{-32(200)^2}{19600}+200.
$$

Compute:
$$
200^2=40000,\quad -32(40000)=-1280000.
$$
Thus,
$$
h(200)=\frac{-1280000}{19600}+200.
$$

Divide:
$$
\frac{-1280000}{19600}\approx -65.30612.
$$

Then,
$$
h(200)\approx200-65.30612\approx134.69388.
$$

Rounded to two decimal places,
$$
h(200)\approx134.69\text{ feet}.
$$

### (c) Height at $$ x=350 $$ feet

Substitute $$ x=350 $$:
$$
h(350)=\frac{-32(350)^2}{19600}+350.
$$

Notice that
$$
\frac{350}{140}=2.5,\quad \text{so}\quad \left(\frac{350}{140}\right)^2=2.5^2=6.25.
$$

Thus, we can write:
$$
h(350)=-32(6.25)+350.
$$

Compute the product:
$$
-32\times6.25=-200.
$$

Then,
$$
h(350)=350-200=150.
$$

Rounded to no decimal places, 
$$
h(350)=150\text{ feet}.
$$

### Interpretation of $$ h(350)=150 $$

The function $$ h(x) $$ gives the height of the golf ball (in feet) after it has traveled $$ x $$ feet horizontally. Thus, when $$ x=350 $$, we have:

$$
\textbf{B. The height of the golf ball after it has traveled a horizontal distance of 350 feet is 150 feet.}
$$
(a) After traveling 100 feet, the height is approximately 83.67 feet. (b) After traveling 200 feet, the height is approximately 134.69 feet. (c) At 350 feet horizontally, the height is 150 feet. Interpretation: After the golf ball has traveled 350 feet horizontally, its height is 150 feet.

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