Question

Solve each problem. See Examples 1–4. Diameter of the Sun To determine the diameter of the sun, an astronomer might sight with a transit (a device used by surveyors for measuring angles) first to one edge of the sun and then to the other, estimating that the included angle equals 32'. Assuming that the distance d from Earth to the sun is 92,919,800 mi, approximate the diameter of the sun.

Accepted Answer

Understand the problem: The astronomer measures the angle subtended by the diameter of the sun from Earth, which is given as 32 minutes of arc (32'). The goal is to find the actual diameter of the sun using this angle and the distance from Earth to the sun. Convert the angle from minutes to degrees because trigonometric functions typically use degrees or radians. Since 1 degree = 60 minutes, the angle in degrees is $$	heta = \frac{32}{60}$$ degrees. Recognize that the angle $$	heta is $$very small and subtended by the diameter of the sun at distance $$d. We $$can model this situation as an isosceles triangle where the sun's diameter is the base opposite the angle $$	heta at $$the observer's point. Use the small-angle approximation or the formula relating arc length to radius and angle in radians. First, convert the angle $$	heta to $$radians: $$	heta_{rad} = 	heta 	imes \frac{\pi}{180}. $$Then, the diameter $$D of $$the sun can be approximated by $$D = d 	imes 	heta_{rad}$$, where $$d is $$the distance to the sun. Substitute the given distance $$d = 92,919,800$$ miles and the angle in radians into the formula to find the diameter $$D. $$This will give the approximate diameter of the sun.

Risposta video verificata per un problema simile:

Concetti chiave

Angular Measurement in Degrees and Minutes

Arc Length and Circle Geometry

Angle Conversion to Radians