Five separate ceiling tests
The same training-ball test is run separately in five test chambers—one launch per chamber. Without a ceiling, the ball's height in every trial would follow y = -0.5(x - 4)^2 + 9, where x and y are measured in metres and y is height above the floor. The five chambers have ceiling heights 7.0 m, 8.5 m, 9.0 m, 10.0 m, and 12.0 m. A sensor triggers if the ball would touch or pass that trial's ceiling. In how many of the five separate trials would the sensor trigger?
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3 trials
3 checked solution families
Read the vertex, then test thresholds
Read the maximum directly from vertex form, then count the ceiling thresholds at or below it.
- Read the vertex (4, 9) from y = -0.5(x - 4)^2 + 9.
- The negative coefficient makes 9 m the maximum height.
- Count 7.0, 8.5, and 9.0 m because touching also triggers the sensor: 3 trials.
Differentiate, classify, then compare
Find the stationary point with the first derivative and classify it with the second derivative before comparing thresholds.
- Differentiate: y' = -(x - 4), so the stationary point occurs at x = 4.
- Since y'' = -1, the point is a maximum, and y(4) = 9 m.
- Three ceiling heights are at or below the maximum, so 3 sensors trigger.
Test horizontal-line intersections
Treat each ceiling as y = h and decide whether the trajectory equation has a real intersection.
- For ceiling height h, solve -0.5(x - 4)^2 + 9 = h, giving (x - 4)^2 = 2(9 - h).
- A real intersection exists exactly when the right-hand side is non-negative, so h <= 9 m.
- The feasible thresholds are 7.0, 8.5, and 9.0 m, including equality: 3 trials.
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English
The same training-ball test is run separately in five test chambers—one launch per chamber. Without a ceiling, the ball's height in every trial would follow y = -0.5(x - 4)^2 + 9, where x and y are measured in metres and y is height above the floor. The five chambers have ceiling heights 7.0 m, 8.5 m, 9.0 m, 10.0 m, and 12.0 m. A sensor triggers if the ball would touch or pass that trial's ceiling. In how many of the five separate trials would the sensor trigger?