A fire hose ejects a stream of water at an angle of 33.6 ° above the horizontal. The water leaves the nozzle with a speed of 29.6 m/s. Assuming that the water behaves like a projectile, how far from a building should the fire hose be located to hit the highest possible fire?

as you recall, the equation of motion is

y = tanθ x - g/(2(v cosθ)^2) x^2

So, plugging in your numbers,

y = 0.664x - 0.00806x^2
That's a parabola with vertex at x=41.2

To determine how far from the building should the fire hose be located to hit the highest possible fire, we need to analyze the water stream's trajectory as a projectile.

First, let's break down the given information:
- Angle of projection (θ): 33.6 ° above the horizontal
- Initial speed (v₀): 29.6 m/s

Since the water stream's trajectory is a projectile, we can decompose its initial velocity into horizontal (v₀x) and vertical (v₀y) components.

The horizontal component (v₀x) remains constant throughout the motion as there is no horizontal acceleration. Thus, it can be determined using trigonometry:

v₀x = v₀ * cos(θ)

Substituting the given values, we get:
v₀x = 29.6 m/s * cos(33.6 °)

Next, we need to find the maximum height reached by the water stream. At the highest point of the trajectory, the vertical component of the velocity (v_y) becomes zero. We can use this fact to find the time of flight (t) to reach the maximum height.

v_y = v₀y - g*t

Since the final vertical velocity (v_y) is zero at the highest point, we have:
0 = v₀ * sin(θ) - g*t

Rearranging the equation, we can solve for t:
t = v₀ * sin(θ) / g

Now, we can find the time required for the water stream to reach the highest point.

To find the maximum height (H) reached by the water stream, we need to utilize the kinematic equation:

H = (v₀y^2) / (2 * g)

Plugging in the values for v₀ and t, we get:
H = (v₀ * sin(θ))^2 / (2 * g)

Now that we know the maximum height (H), we can determine the horizontal range (R) the water stream will travel.

R = v₀x * 2t

Substituting the values of v₀x and t, we get:
R = (29.6 m/s * cos(33.6 °)) * 2 * (v₀ * sin(θ) / g)

Simplifying the equation, we have:
R = (v₀^2 * sin(2θ)) / g

Now we can calculate how far from the building the fire hose should be located to hit the highest possible fire using the obtained equation for R.

Inserting the given values, we get:
R = (29.6 m/s)^2 * sin(2 * 33.6 °) / (9.8 m/s²)

Evaluating this expression, we find:
R ≈ 70.71 m

Therefore, to hit the highest possible fire, the fire hose should be located approximately 70.71 meters away from the building.