Recall the equation for a circle with center ( h , k ) and radius r . at what point in the first quadrant does the line with equation y = x + 1 intersect the circle with radius 6 and center (0, 1)?

Here is their argument. given the obtuse angle x, we make a quadrilateral abcd with ∠dab = x, and ∠abc = 90◦, and ad = bc. say the perpendicular bisector to dc meets the perpendicular bisector to ab at p. then pa = pb and pc = pd. so the triangles pad and pbc have equal sides and are congruent. thus ∠pad = ∠pbc. but pab is isosceles, hence ∠pab = ∠pba. subtracting, gives x = ∠pad−∠pab = ∠pbc −∠pba = 90◦. this is a preposterous conclusion – just where is the mistake in the "proof" and why does the argument break down there?

The second derivative of the equation

Abby bought 6 books for math and science class for college for a total of $440. math books cost $73 and science books cost $74. how many of each type of book did she buy?