(Originais Teachy 2023) - Question Very Hard of Mathematics

In a civil engineering project, an architect needs to divide a rectangular plot of land into two equal parts for the construction of two semi-detached houses. The coordinates of the vertices of the land on the Cartesian plane are A(0, 0), B(30, 0), C(30, 20), and D(0, 20). Using concepts of analytic geometry, determine the coordinates of the point that represents the midpoint of side CD of the land, which will be the dividing point between the two new properties. Then, verify if the obtained midpoint point is indeed on segment CD, ensuring the accuracy of the project. Explain the reasoning used to determine the coordinates and to verify the positioning of the midpoint on the land.

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.

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?