1.39 ** A ball is thrown with initial speed v, up an inclined plane. The plane is inclined at an angle o above the horizontal, and the ball's initial velocity is at an angle 0 above the plane. Choose axes with x measured up the slope, y normal to the slope, and z across it. Write down Newton's second law using these axes and find the ball's position as a function of time. Show that the ball lands a distance R= 2v, sin 0 cos(0 + 4)/(g cos² p) from its launch point. Show that for given v, and p, the maximum possible range up the inclined plane is Rmax = v/[g(1+ sin ø)].

You are working with an imagined cubical pot of water, 1 meter on each side, sitting on a stove. Heat is applied to the bottom of the pot from the stovetop. The pot has been on the stove for a while, so the stovetop is hot and some heat has already been added to the water. Assume the water at the bottom of the pot is held at 99 degrees C , while the water at the top of the pot is cold, 4 degrees C. Compute the heat delivered by that 1 g of water at 99 °C into the layer of water at the top of the pot, assuming the latter is 1 meter square, 1 cm deep, and 4 °C. Start by solving the calorimetry problem (Section 14-4 in the text) to determine the final temperature of the top layer plus hot 1 gram from below. Then use that to determine how much heat was delivered in the hot water

Section 1.5 Problem 1. Mark each statement True or False, and justify your answer. (T/F) A homogeneous equation is always consistent. (T/F) If x is a nontrivial solution of Ax=0, then every entry in x is nonzero. (T/F) The equation Ax=0 gives an explicit description of its solution set. (T/F) The equation x = xu+xiv, with xa and xx free (and neither u nor va multiple of the other), describes a plane through the origin. (T/F) The homogeneous equation Ax=0 has the trivial solution if and only if the equation has at least one free variable. (T/F) The equation Ax=b is homogeneous if the zero vector is a solution. (T/F) The equation x = p + tv describes a line through v parallel to p. (T/F) The effect of adding p to a vector is to move the vector in a direction parallel to p.

please show steps with explanation of the formulas and  calculs for more understanding. Thank you!