(Originais Teachy 2023) - Question Hard of Mathematics

The image presented in the figure is a black and white copy of the square canvas titled The Fish, by Marcos Pinto, which was placed on a wall for exhibition and fixed at points A and B (figure 1). Due to a problem in fixing one of the points, the canvas detached and rotated close to the wall. After the rotation, it was positioned as illustrated in figure 2, forming a 45° angle with the horizon line. Figure 1 Figure 2 To put the canvas back in its original position, it should be rotated, close to the wall, at the smallest possible angle less than 360°. The way to put the canvas back in the original position, following what was established, is by rotating it at an angle of

The Science class is learning about vegetable gardens and decided to plant a small rectangular garden at school, where they will grow lettuce and carrots. The garden's rectangle measures 3 meters in length and 2 meters in width. To find out how many plants will fit in the garden, they need to calculate the perimeter. Help the students calculate the perimeter of this rectangular garden.

Consider the linear system below: 2x + 3y = 10 5x - 4y = -7 Transform this linear system into a matrix form using a matrix A and a column vector X, representing the unknowns. Write down the resulting matrix equation and inform what is the dimension of matrix A.

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.