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Magazine Problem 57
Architecture A semielliptical arch over a tunnel for a one-way road through a mountain has a major axis of 50 feet and a height at the center of 10 feet. (a) Draw a rectangular coordinate system on a sketch of the tunnel with the center of the road entering the tunnel at the origin. Identify the coordinates of the known points. (b) Find an equation of the semielliptical arch over the tunnel. (c) You are driving a moving truck that has a width of 8 feet and a height of 9 feet. Will the moving truck clear the opening of the arch?
Problem 59
Halley's comet has an elliptical orbit, with the sun at one focus. The eccentricity of the orbit is approximately \(0.967\). The length of the major axis of the orbit is approximately \(35.88\) astronomical units. (An astronomical unit is about 93 million miles.) (a) Find an equation of the orbit. Place the center of the orbit at the origin, and place the major axis on the \(x\)-axis. (b) Use a graphing utility to graph the equation of the orbit. (c) Find the greatest (aphelion) and smallest (perihelion) distances from the sun's center to the comet's center.
Problem 60
The first artificial satellite to orbit Earth was Sputnik I (launched by the former Soviet Union in 1957). Its highest point above Earth's surface was 947 kilometers, and its lowest point was 228 kilometers (see figure). The center of Earth was the focus of the elliptical orbit, and the radius of Earth is 6378 kilometers. Find the eccentricity of the orbit.
Problem 65
Satellite Orbit A satellite in a 100-mile-high circular orbit around Earth has a velocity of approximately 17,500 miles per hour. If this velocity is multiplied by \(\sqrt{2}\), the satellite will have the minimum velocity necessary to escape Earth's gravity and it will follow a parabolic path with the center of Earth as the focus (see figure). (a) Find the escape velocity of the satellite. (b) Find an equation of the parabolic path of the satellite (assume that the radius of Earth is 4000 miles).
Problem 75
Find a polynomial with real coefficients that has the zeros \(3,2+i\), and \(2-i\)
