(Originais Teachy 2023) - Question Medium of Mathematics

In a school, a survey was conducted on students who practiced sports. It was found that 60% of the students practiced soccer, 30% practiced volleyball, and 20% practiced both sports. If a student is randomly selected, what is the probability that he practices soccer, given that he practices volleyball?

In a large engineering system, a building is being designed and energy efficiency is a major concern. To ensure the efficiency of heating and cooling, the air conditioning system must be sized according to the specific needs of the building. The engineer in charge of the project must solve a system of linear equations to determine the ideal operating temperatures. Let the following be the system of equations that models the thermal behavior of the building: 2x + y + 3z = 10, 4x - 6y + 8z = 5, 3x + 7y + 5z = 18. The engineer must rewrite this system of equations in the matrix form Ax = b, where A is the matrix of coefficients, x is the vector of unknowns, and b is the vector of constant terms.

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.