(Originais Teachy 2024) - Question Hard of Mathematics

In a renovation, an architect decides to completely cover a right prism with a square base with ceramic tiles that are in the shape of squares. The prism is 3 meters high and the edge of the base measures 2 meters. The tiles are 20 cm on each side. The architect will need to calculate the amount of tiles to cover the entire surface of the prism. How many tiles will be needed to completely cover the prism?

During the cleaning of the cooling system of a factory, a technician must adjust a light reflector fixed on the ceiling of the room. The reflector has the shape of a right circular cone with the vertex pointing upwards and an opening angle of 75º. The technician wants to calculate the height and the radius of the base of the cone to determine the ideal dimensions of the mirrors that will be installed on the inner surface of the cone, so that the reflected light follows a beam parallel to the floor. Consider that the angle of incidence of light on the inner surface of the cone should be 15º with the normal to the surface at the point of incidence to ensure total reflection of light. Based on this configuration, determine the height and the radius of the base of the cone, and additionally, identify the value of the sine and cosine of 22.5º, an angle that arises in the resolution of the problem.

In a science project, students need to measure different volumes of liquids for their experiments. The scale used to measure the volume of water can only measure in multiples of 100 milliliters. When measuring, the scale shows 450.725 grams of water. We know that the density of water is 1 g/ml. (1) Round the volume measured by the scale to the nearest milliliter. (2) Explain why the scale provided a number in grams instead of milliliters and how you used the density of water to convert the measurement back to volume in milliliters.