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Magazine Problem 44
The region inside the cardioid \(r=2+2 \cos \theta\) and to the right of the line \(r \cos \theta=\frac{3}{2}\).
Problem 45
Find \(d y / d x\) and \(d^{2} y / d x^{2}\) at the given point without eliminating the parameter. $$ x=\sqrt{t}, y=2 t+4 ; t=1 $$
Problem 45
The region inside the circle \(r=2\) and to the right of the line \(r=\sqrt{2} \sec \theta\)
Problem 45
Sketch the curve in polar coordinates. \(r=9 \sin 4 \theta\)
Problem 46
The region inside the rose \(r=2 a \cos 2 \theta\) and outside the circle \(r=a \sqrt{2}\)
Problem 46
Sketch the curve in polar coordinates. \(r=2 \cos 3 \theta\)
Problem 46
Find \(d y / d x\) and \(d^{2} y / d x^{2}\) at the given point without eliminating the parameter. $$ x=\frac{1}{2} t^{2}+1, y=\frac{1}{3} t^{3}-t ; t=2 $$
Problem 47
True-False Determine whether the statement is true or false. Explain your answer. The polar coordinate pairs \((-1, \pi / 3)\) and \((1,-2 \pi / 3)\) describe the same point.
Problem 47
As illustrated in the accompanying figure, suppose that two transmitting stations are positioned \(100 \mathrm{~km}\) apart at points \(F_{1}(50,0)\) and \(F_{2}(-50,0)\) on a straight shoreline in an \(x y\) -coordinate system. Suppose also that a ship is traveling parallel to the shoreline but \(200 \mathrm{~km}\) at sea. Find the coordinates of the ship if the stations transmit a pulse simultaneously, but the pulse from station \(F_{1}\) is received by the ship 100 microseconds sooner than the pulse from station \(F_{2}\). [Assume that the pulses travel at the speed of light \((299,792,458 \mathrm{~m} / \mathrm{s})\).]
Problem 47
Find \(d y / d x\) and \(d^{2} y / d x^{2}\) at the given point without eliminating the parameter. $$ x=\sec t, y=\tan t ; t=\pi / 3 $$
