Questions to Practice on Height and Distance

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Questions to Practice on Height and DistanceSneha is standing on the lighthouse whose height is​​ 45m, she observes a boat coming her way with an angle of depression​​ 45˚. When the boat has travelled a distance of​​ Y​​ meter​​ the angle of depression changes to​​ 60˚. Answer the questions based on the above description.What is the distance between Boat and lighthouse when the angle of depression was 45˚?What is the distance between the boat and the lighthouse after travelling Y meters?What is the distance travelled by boat​​ i.e., Y meters?If the angle​​ of depression​​ further changes to 30˚, when boat travelled away from the lighthouse, what will be the new distance between lighthouse and boat?A boy is watching an Airplane flying in the sky at a height h at an angle of 60˚. After travelling some distance the angle changed to 45˚. Answer the questions in terms of h based on the above description (assume the height of the plane never changes).What is the length of the line of sight of the boy when the angle of elevation is 60˚?What is the length of the line of sight of the boy when the angle of elevation is 45˚?What is the base distance between the boy and the perpendicular line that determines the height of the plane when​​ the​​ angle of elevation is 30˚?​​ A motorcycle is moving toward a tower from point P to Q at a uniform speed. It takes 20 minutes for the angle of depression to change from 30˚​​ to 60˚. ​​ What time after this, the car will reach the base of​​ the​​ tower?How much distance is between point P and Q if speed is 1 km/min?How much distance does the motorcycle travel in total while speed is uniform?Answers and Solution:a) Ans: 45mSolution:​​ When the angle of depression is 45˚ i.e., the angle of elevation is also 45˚;​​ 
at that time distance between height and distance of base is equal as tan 45˚=1.Hence Distance between the boat and lighthouse = height of lighthouse = 45mb) Ans: 153 m ​​ or ​​ 25.980mSolution:​​ Let the new distance between lighthouse and boat after travel of Y meters be x meters. By trigonometric equation tan​​ 60˚ = height h/ x      3​​ = 45/x      ​​ x = 45/3​​ or 25.980mc) Ans: 19.01mSolution:​​ Distance travelled by the boat = Initial distance – distance between lighthouse and boat after travel of y meters. = 45-25.980 = 19.01 m ​​​​ d) Ans: 453​​ m or 77.942mSolution:​​ If the angle of depression changes to 30˚, the trigonometric equation becomes​​ tan 30˚​​ = height of the lighthouse/distance between lighthouse and boat​​ Distance between boat and lighthouse after travel = 45m*3 ​​ ​​ ​​​​ = 77.942m6045Figure:  h    x ya) Ans:​​ 2h/3 ​​​​ mSolution:​​ The line of sight is the hypotenuse in terms of a right angle triangle hence,​​ When the plane is flying at a distance is h, the equation becomes  Sin 60˚​​ = height / line of sight​​   Line of sight = 2h/3   (as sin 60˚​​ =​​ 32 ​​ )b) Ans: h2 ​​ mSolution:​​ sin 45˚​​ = h/line of sight  ​​ ​​ ​​ ​​​​ Line of sight = h/(1/2 )=​​ 2​​ h meters​​  (as sin 45˚​​ = 1/2​​ )c)​​ Ans: h3 mSolution:​​ The question is asking the base distance of the boy and perpendicular imaginary line representing the height of the plane, the equation becomes,Tan 30˚ = height h / base distance​​ Base distance = h/tan 30˚ = ​​ h3​​ mFigure:60˚ ˚​​ ˚​​ ˚30˚ ˚​​ ˚​​ ˚   h     y Q x Pa) Ans:​​ 10​​ minutesSolution:​​ Let the height of the tower be h, the distance between point P and point Q be x and tower and point Q be y; as per two angles given there will be two trigonometric equationsEqn 1 at angle 30˚  tan 30˚ = h/x​​  => h =​​ (x+y)/3Eqn 2 at angle 60˚  tan 60˚ = h/y => h = y3Equating the above equations give, x+y=3*3 y ​​ = 3y​​ => x= 2yWe will use the concept of time and speed here, time taken by the motorcycle to travel distance P to Q is 20 minsAs speed is constant, distance1/time1(for distance x) =distance2/time2(for distance y)​​     ​​​​  x/20min = y/time t    2y /20 = y/t    t = 10minb) Ans: 20kmSolution:​​ Speed at which the motorcycle is travelling = 1km/min ​​ Distance​​ x​​ = speed *time = 1km/min * 20 min = 20 kmc) Ans:​​ 30​​ kmSolution:​​ As speed is uniform, distance y = 1km/min * 10min = 10 km​​   ​​ ​​ ​​​​ Total distance travelled = x+y = 20+10 = 30km(function( $ ) { //footnotes $('.defaultNote').children(":first").css('cursor', 'pointer'); $('.defaultNote').children(":first").click(function(){ var gotoID = '#' + $(this).parent().attr('id'); $('html, body').animate( { scrollTop: $('a[href="' + gotoID + '"]').offset().top - 60, }, 250, 'linear' ) }); }(jQuery)); var numFormat = ",."; (function( $ ) { "use strict"; //do some cleaning on load var sortCells = $('td[data-sorting], th[data-sorting]'); sortCells.each(function(){ var first = $(this).text().trim().charAt(0); if (first == '@'){ var oldHTML = $(this).html(); var newHTML = oldHTML.replace('@', ''); $(this).html(newHTML); } }); function sortingTable(node, pos, ordBy){ var rows = node.children('tbody').children('tr').get(); rows.sort(function(a, b) { var X = $(a).children('td').eq(pos).text().toUpperCase(); var Y = $(b).children('td').eq(pos).text().toUpperCase(); if (numFormat != '.,'){ //we are dealing with a different locale and we need to do some //transformation in numbers X = convert2number(X); Y = convert2number(Y); } else { X = X.replace(/,/g, '').replace(/ /g, ''); Y = Y.replace(/,/g, '').replace(/ /g, ''); } var currencies = ['$', '¥', '£', '€']; var arrayLength = currencies.length; var XX = X; var YY = Y; for (var i = 0; i < arrayLength; i++) { XX = XX.replace(currencies[i], ''); YY = YY.replace(currencies[i], ''); } if (!isNaN(XX) && !isNaN(YY)){ var floatXX = parseFloat(XX); var floatYY = parseFloat(YY); return -(floatYY-floatXX) * ordBy; } else if (!isNaN(XX) && isNaN(YY)){ return -1 * ordBy; } else if (isNaN(XX) && !isNaN(YY)){ return ordBy; } else { if(X < Y) { return -1 * ordBy; } if(X > Y) { return 1 * ordBy; } return 0; } }); $.each(rows, function(index, row) { node.children('tbody').append(row); }); } function removeTableCellDecoration() { $("td[data-sorting], th[data-sorting]").removeClass('sortDecorator'); } //unbind the click event just in case we are loading more than one document $("td[data-sorting], th[data-sorting]").off('click'); $("td[data-sorting], th[data-sorting]").click(function(){ removeTableCellDecoration(); $(this).addClass('sortDecorator'); //get desired ordering var ordering = $(this).attr('data-sortOrdering'); if (ordering == 'ASC'){ var ordBy = -1; $(this).attr('data-sortOrdering', 'DESC'); } else { var ordBy = 1; $(this).attr('data-sortOrdering', 'ASC'); } var cellPosition = $(this).prevAll().length; var table = $(this).parents('table:first'); sortingTable(table, cellPosition, ordBy); }); function convert2number(str) { str = str.trim(); if (str.match(/^\d+/)) { //we have to take special care when the thousand separators are '.' var result_0 = str.replace(/\./g, '_@_'); var result_1 = result_0.replace(window.numFormat[0], '.'); var result = result_1.replace(/_@_/g, ''); result .replace(/ /g, ''); return result; } else { return str; } } }(jQuery));