A firefighter holds a hose 4 m off the ground and directs a stream of water towa

Question

A firefighter holds a hose 4 m off the ground and directs a stream of water toward a burning building. The water leaves the hose at an initial speed of 12 m/sec at an angle of 30°. The height of the water can be approximated byh (x)-0.026x +0.588x +4, where h (x) is the height of the water in meters at a pointx meters horizontally from the firefighter to the building. Part 1 a. Determine the horizontal distance from the firefighter at which the maximum height of the water occurs. The water reaches a maximum height when the horizontal distance from the firefighter to the building is approximately 11.3 m. Round to 1 decimal place. Part 2 out of 3 b. What is the maximum height of the water? The maximum height of the water is m. Round to 1 decimal place.

Explanation / Answer

Part 1. (a). We have h(x) = -0.026x2+0.588x+4 = -0.026[x2- 2x*0.294/0.026 + (0.294/0.026)2] + 4 + 0.026*(0.294/0.026)2 = -0.296(x- 0.294/0.026)2 + 4+ 3.324 = -0.296(x- 0.294/0.026)2+ 7.324 =                              -0.296(x-11.308)2 +7.324. This is the vertex form of a downwards opening parabola with vertex at (11.308,7.324).Thus, the horizontal distance from the firefighter to the point where the maximum height occurs is 11.308 meters or, 11.3 meters ( on rounding off to 1 decimal place).

Part 2.(b). Since the vertex is the highest point on a downwards opening parabola, hence the maximum height of water is 7.324 meters or, 7.3 meters (on rounding off to 1 decimal place).

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