Lesson A radio transmitter accessibility area
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<H2>A radio transmitter accessibility area</H2> <H3>Problem 1</H3>A small radio transmitter broadcasts in a 21 mile radius. If you drive along a straight line from a city 25 miles north of the transmitter to a second city 29 miles east of the transmitter, during how much of the drive will you pick up a signal from the transmitter? <B>Solution</B> {{{drawing(400, 400, -35, 35, -35, 35, grid(5), red(circle(0,0,21)), blue(line(0, 25, 29, 0)), locate(1, 28, A), locate(28.5, 3, B), locate(5.6, 23, C), locate(20.5, 10, D), green(line(0, 0, 5.48, 20.27)), green(line(0, 0, 19.24, 8.41)), green(line(0, 0, (5.48+19.24)/2, (20.27+8.41)/2)), locate(12.4, 17.6, E) )}}} <pre> Let A be the point 25 miles north from the transmitter. Let B be the point 29 miles east from the transmitter. Let O be the circle of the radius 21 miles with the center at the transmitter location. Then we have a right angled triangle OAB and straight line AB, intersecting this triangle in points B and C. They want we find the length of the chord BC. My designations are close to that on the plot of the other tutor. Draw the perpendicular OE from the center/vertex O to line BC. This perpendicular is the height (the altitude) in triangle OAB on the hypotenuse. The area of this triangle can be computed as half the product of its legs or as half the product of the hypotenuse and the height. So, we can write this equation |AB|*h = |OA|*|OB|, (1) where h is the height OE. The hypotenuse |AB| is {{{sqrt(25^2+29^2)}}} = 38.2884 miles (rounded). From (1) h = {{{(25*29)/sqrt(25^2+29^2)}}}. It gives the distance from the center O to line AB. Then the chord length is |BC| = 2*|BE| = 2*|CE| = {{{2*sqrt(21^2-h^2)}}} = {{{2*sqrt(21^2 - (25^2*29^2)/(25^2+29^2))}}} = 18.1611 miles (rounded). Now we have all the numbers and the <U>ANSWER</U>: the entire trip is about 38.2884 miles; the part where the signal is accessible is about 18.1611 miles; the ratio of these values is {{{18.1611/38.2884}}} = 0.4743 (rounded). </pre> My other additional lessons on miscellaneous Geometry problems in this site are - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Find-the-rate-of-moving-of-the-tip-of-a-shadow.lesson>Find the rate of moving of the tip of a shadow</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Miscellaneous-geometric-problems.lesson>Miscellaneous geometric problems</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Miscellaneous-problems-on-parallelograms.lesson>Miscellaneous problems on parallelograms</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Remarkable-properties-of-triangles-into-which-diagonals-divide-a-quadrilateral.lesson>Remarkable properties of triangles into which diagonals divide a quadrilateral</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-trapezoid-divided-in-four-triangles-by-its-diagonals.lesson>A trapezoid divided in four triangles by its diagonals</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/A-problem-on-heptagon.lesson>A problem on a regular heptagon</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/The-area-of-a-regular-octagon.lesson>The area of a regular octagon</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/The-fraction-of-the-area-of-a-regular-octagon.lesson>The fraction of the area of a regular octagon</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Try-to-solve-this-nice-Geometry-problem.lesson>Try to solve these nice Geometry problems !</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Find-the-angle-between-sides-of-folded-triangle.lesson>Find the angle between sides of folded triangle</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-problems-on-three-spheres.lesson>A problem on three spheres</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-sphere-placed-in-an-inverted-cone.lesson>A sphere placed in an inverted cone</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/07-An-upper-level-Geometry-problem-on-special-%2815-30-135%29-triangle.lesson>An upper level Geometry problem on special (15°,30°,135°)-triangle</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-great-Math-Olympiad-level-Geometry-problem.lesson>A great Math Olympiad level Geometry problem</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Nice-geometry-problem-of-a-Math-Olympiad-level.lesson>Nice geometry problem of a Math Olympiad level</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/OVERVIEW-of-my-lessons-on-additional-misc-Geometry-problems.lesson>OVERVIEW of my additional lessons on miscellaneous advanced Geometry problems</A> To navigate over all topics/lessons of the Online Geometry Textbook use this file/link <A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A>.
