Final answer:
The statement is false because as a rock is thrown upwards, its kinetic energy converts into potential energy, which upon descent, converts back into kinetic energy, following the conservation of energy principle.
Step-by-step explanation:
Understanding Kinetic and Potential Energy in PhysicsAssessing the statement about a rock being thrown into the air, it is important to understand the concepts of kinetic and potential energy. When a rock is thrown upwards, its kinetic energy converts into potential energy as it rises. Consequently, as the rock falls back to the ground, its potential energy translates back into kinetic energy. This is due to the conservation of energy principle, meaning energy within a closed system remains constant. Therefore, the statement that suggests the increase in height increases the rock's kinetic energy is false. As the rock climbs, its velocity reduces and thus its kinetic energy decreases, and potential energy increases due to its elevation. Conversely, as the rock descends, its velocity (and thereby kinetic energy) increases, while its potential energy decreases due to a reduction in height.
To find the volume of the solid obtained by rotating the region enclosed by the curves y = 72x - 8x² and y = 0, we can use the method of cylindrical shells. The volume of each cylindrical shell is given by the formula V = 2πx(f(x) - g(x))dx, where f(x) is the upper curve and g(x) is the lower curve.In this case, the upper curve is y = 72x - 8x² and the lower curve is y = 0. So we have f(x) = 72x - 8x² and g(x) = 0. The region enclosed by these curves is bounded by the x-values where the curves intersect, which can be found by setting f(x) - g(x) = 0.Solving 72x - 8x² = 0, we get x(72 - 8x) = 0. So either x = 0 or 72 - 8x = 0. Solving the second equation, we get x = 9. Therefore, the region is enclosed by x = 0 and x = 9.