A golf ball is hit with an initial velocity of 150 feet per second at an inclination of 45 degrees to the horizontal. In​ physics, it is established that the height h of the golf ball is given by the function
                  h left parenthesis x right parenthesis equals StartFraction negative 32 x squared Over 150 squared EndFraction plus x
​,
where x is the horizontal distance that the golf ball has traveled. Complete parts​ (a) through​ (g). What is h left parenthesis 350 right parenthesis​? Interpret this value.

We are given the equation for the height of a golf ball:

\[
h(x) = \frac{-32x^2}{150^2} + x
\]

where:
- \( h(x) \) is the height of the golf ball at a horizontal distance \( x \),
- \( x \) is the horizontal distance traveled.

### Step 1: Plug \( x = 350 \) into the height function to find \( h(350) \):

\[
h(350) = \frac{-32(350)^2}{150^2} + 350
\]

We can now calculate this value:

\[
h(350) = \frac{-32(122500)}{22500} + 350
\]
\[
h(350) = \frac{-3920000}{22500} + 350
\]
\[
h(350) = -174.22 + 350
\]
\[
h(350) = 175.78 \text{ feet}
\]

### Step 2: Interpretation of the result

The value \( h(350) = 175.78 \) feet means that after the golf ball has traveled 350 feet horizontally, it is 175.78 feet above the ground.

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### Relative Questions:
1. How can we find the maximum height of the golf ball using calculus?
2. What is the horizontal distance when the golf ball hits the ground?
3. How does changing the initial velocity affect the height function?
4. How would the equation change if the angle of inclination were different?
5. How can we calculate the total time the ball spends in the air?

### Tip:
When dealing with projectile motion problems, the angle and initial velocity greatly influence both the horizontal range and maximum height achieved by the object.

A golf ball is hit with an initial velocity of 150 feet per second at an inclination of 45 degrees to the horizontal. In physics, it is established that the height h of the golf ball is given by the function h(x) = (-32x^2) / (150^2) + x, where x is the horizontal distance that the golf ball has traveled. Complete parts (a) through (g). What is h(350)? Interpret this value.

Solve the projectile motion problem where a golf ball is hit at 150 feet per second and travels at a 45-degree angle. Learn how to calculate the height h(350) at a 350 feet horizontal distance using quadratic functions. Suitable for high school physics and algebra students.

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