Now You Try

<p>A hot-air balloon is used to give rides to visitors at a summer fair. The balloon is tethered to the ground by a long cable. The cable is extended to its maximum length of 300 m, and the wind is blowing the balloon such that the cable makes an angle of 60° with the ground. The cable is pulled in to 200 m, but the wind strengthens, decreasing the angle to 45°.</p><p>a) Sketch the two positions of the balloon, including distances and angles.</p><p>b) Find an exact expression for the horizontal distance that the balloon moves between the two positions.</p>

<p>Find TU, to the nearest tenth of a metre.</p><img src="/qimages/66266" />

<p>Mina is trolling for salmon in Lake Ontario. She sets the fishing rod so that its tip is 1 m above water and the line forms an angle of <code class='latex inline'>35^o</code> with the water’s surface. She knows that there are fish at a depth of 45 m. Describe the steps you would use to calculate the length of line she must let out.</p><img src="/qimages/933" />

<p>Captain Jack is sitting in the crow’s—nest of his ship, as shown.</p><img src="/qimages/1615" /><p>a) How high above the deck is Captain Jack?</p><p>b) What is the length of Captain Jack's ship?</p><p>c) How long is each wire holding up the crow’s-nest?</p>

<p>A video camera is mounted on top of a building that is 120 m tall.
The angle of depression from the camera to the base of another
building is 36<code class='latex inline'>^{\circ}</code>. The angle of elevation from the camera to the top
of the same building is 47<code class='latex inline'>^{\circ}</code>.</p>
<ul>
<li>How tall is the building viewed by the camera? Round your
answer to the nearest metre</li>
</ul>
<img src="/qimages/1053" />

<p> Find the length of <code class='latex inline'> x </code> , then the length of <code class='latex inline'> y </code> ,
to the nearest tenth of a metre.</p><img src="/qimages/65403" />

<p>Find the measure of <code class='latex inline'> \angle \mathrm{ADB} </code> , to the nearest degree.</p><img src="/qimages/66268" />

<p> Observers at <code class='latex inline'>P</code> and <code class='latex inline'>Q</code> are located on the side of a hill that is inclined <code class='latex inline'>32^{\circ}</code> to the horizontal, as shown. The observer at P determines the angle of elevation to a hot-air balloon to be <code class='latex inline'>62^{\circ}</code>. At the same instant, the observer at <code class='latex inline'>Q</code> measures the angle of elevation to the balloon to be <code class='latex inline'>71^{\circ}</code>. If <code class='latex inline'>P</code> is 60 m down the hill from Q, find the distance from <code class='latex inline'>Q</code> to the balloon.</p><img src="/qimages/261" />

<ol>
<li>Totem pole Ropes are used to pull a totem pole upright. Then, the ropes are anchored in the ground to hold the pole until the hole is filled. One of the ropes holding this totem pole is <code class='latex inline'>\displaystyle 18 \mathrm{~m} </code> long and forms an angle of <code class='latex inline'>\displaystyle 48^{\circ} </code> with the ground. Find, to the nearest metre,</li>
</ol>
<p>a) the height of the totem pole</p><p>b) how far the anchor point is from the base of the totem pole</p><img src="/qimages/65438" />

<p>Tess and Brandon are competing in a series of outdoor challenges that will eventually lead them to a hidden treasure. Each clue they find helps them find a new clue. Tess is at the top of a cliff that she knows to be <code class='latex inline'>100</code> m high, looking down at three anchored floating bottles, as shown.</p><img src="/qimages/1617" /><p>She reads the clue that she and Brandon just found: From the op of the cliff, find the bottle whose angle of depression is <code class='latex inline'>50^{\circ}</code>. Brandon is waiting on the beach below for instructions from Tess.</p><p>a) Which bottle should Tess tell Brandon to look for?</p><p>b) How far out should she tell him to swim?</p>

<p>Find <code class='latex inline'>\angle BDC</code>.</p><img src="/qimages/258" />

<p>Find <code class='latex inline'> \mathrm{PQ} </code> , to the nearest metre.</p><img src="/qimages/66262" />

<p>A video camera is mounted on top of a building that is 120 m tall.
The angle of depression from the camera to the base of another
building is 36<code class='latex inline'>^{\circ}</code>. The angle of elevation from the camera to the top
of the same building is 47<code class='latex inline'>^{\circ}</code>.</p>
<ul>
<li>How far apart are the two buildings? Round your answer
to the nearest metre.</li>
</ul>
<img src="/qimages/1053" />

<p>Find <code class='latex inline'> \mathrm{BC} </code> , to the nearest metre.</p><img src="/qimages/66259" />

<p>Angles were measured from two points on opposite sides of a tree, as shown. How tall is the tree?</p><img src="/qimages/1051" />

<p>The angle of elevation of the top of a building is 28°. From a point 15 m directly toward the building, the angle of elevation changes to 42°.</p><p>a) Draw a diagram to represent this
information.</p><p>b) Find the height of the building, to the
nearest metre.</p>

<p>Alexa and Emma are looking up at their house from the backyard. From Alexa‘s point of view, the top of the house is at an angle of elevation of 40°. From Emma’s point of View, directly closer to the house, it is 60°. The house is 15 m high. How far apart are the two girls?</p>

<p>From a point 40 m from the base of office blinding and level with the base,
the angle of elevation of the top of the building is <code class='latex inline'>67.2^o</code>. </p><p> Find the height of the office building, to the nearest tenth of a metre.</p>

<p>Find KL , to the nearest metre.</p><img src="/qimages/66260" />

<p>Find YZ , to the nearest tenth of a metre.</p><img src="/qimages/66256" />

<p>Kristen and Stephen are facing each other on opposite sides of a 15-m oak tree. From Kristen’s point of view, the top of the oak tree is at an angle of elevation of 68°. From Stephen’s point of view, the top of the oak tree is at an angle of elevation of 42°. How far apart
are Kristen and Stephen? Round your answer to the nearest tenth of a metre.</p><img src="/qimages/21866" />

<p>Find the length of <code class='latex inline'> x </code> , to the nearest tenth of
a centimetre, then the measure of <code class='latex inline'> \angle y </code> , to the nearest degree.</p><img src="/qimages/65405" />

<p>At the bottom of a ski lift, there are two vertical poles: one <code class='latex inline'>15</code> m tall and the other <code class='latex inline'>8</code> m tall. The ground between the poles is level, and the bases of the poles are <code class='latex inline'>6</code> m apart. The poles are connected by two straight wires.</p><img src="/qimages/1620" /><p>a) What angle does each wire make with the ground?</p><p>b) What is the length of each wire?</p>

<p>Cheryl considers Option 2, to go around the tree in two shots. In shot one, she will shoot to the end of the line of trees to position A. In shot two, she will aim for the hole.</p><img src="/qimages/1616" /><p>The tree line is 70 m long, and the line joining Cheryl's ball and the hole, CB, passes through the middle of the tree line, at a right angle.</p><p>a) At what angle from CB should Cheryl make her first shot, in order to land near A?</p><p>b) Assume that Cheryl succeeds with her first shot and her golf ball lands at A. At what angle from the tree line must she aim for her second shot?</p>

<p> Find the length of <code class='latex inline'> x </code> , to the nearest tenth
of a centimetre, then the measure of <code class='latex inline'> \angle y </code> , to the nearest
degree.</p><img src="/qimages/65244" />

<p>Beside the visitor information centre in the provincial park there are two vertical posts. One is 3 m tall, and the other is 6 m tall. The ground between the posts is
level, and the bases of the posts are 5 m apart .The posts are connected by two straight wires.</p><p>a) What angle does each wire make
with the ground? Round your answers to the nearest degree.</p><p>b) What is the length of each wire? Round your answer to the nearest tenth of a metre.</p><img src="/qimages/21867" />

<p>John and Sara are facing each other on opposite sides of a 10-m flagpole. From John's point of view, the top of the flagpole is at an angle of elevation of <code class='latex inline'>50</code>°. From Sara it is 35°. Find the distance from J to S.</p><img src="/qimages/1619" />

<p>Find EF, to the nearest tenth of a metre.</p><img src="/qimages/66252" />

<p>Find the length of <code class='latex inline'> x </code> , to the nearest tenth
of a metre, then the measure of <code class='latex inline'> \angle y </code> , to the nearest degree.</p><img src="/qimages/65331" />

<p> To estimate the height of a mountain above a level plain, the angle of elevation to the top of the mountain is measured to be <code class='latex inline'>45^{\circ}</code>. One thousand feet further away from the mountain toward along the plain where the first measurement was taken, it is found that the angle of elevation is <code class='latex inline'>30^{\circ}</code>. Estimate the height of the mountain.</p>

<p> The angle of inclination of the rafters of the
roof of a house is <code class='latex inline'> 26^{\circ} </code> . The roof support is <code class='latex inline'> 3 \mathrm{~m} </code> high. How wide
is the house, to the nearest metre?</p><img src="/qimages/65237" />

<p> For the triangle shown, find the length AD.</p><img src="/qimages/259" />

<p>An airplane is cruising at an altitude of 10 000 m. It is flying in a straight line away from Chandra, who is standing on the ground. If she sees the angle of elevation of the airplane change from 70° to 33° in 1 min. what is its cruising speed, to the nearest kilometre per hour?</p>

<p> <code class='latex inline'> \triangle \mathrm{ABC} </code> is an isosceles triangle. The height of the triangle
is <code class='latex inline'> 3 \mathrm{~cm} </code> , and the two acute angles at its base are each <code class='latex inline'> 56^{\circ} </code> . How
long are the two equal sides, to the nearest tenth of a centimetre?</p><img src="/qimages/65323" />

<p>A square-based pyramid has a height of 164 m and a base length of 240 m.F1n'd
the angle, to the nearest degree, that one
of the edges of the pyramid makes with the base. Round your answer to the nearest degree.</p><img src="/qimages/21868" />

<p>Ropes are used to pull a totem pole upright. Then, the ropes are anchored in the ground to hold the pole until the hole 1's filled. One of the ropes holding this totem pole is 18 m long and forms an angle of 48° with the ground.</p><img src="/qimages/5491" /><p>a) Find the height of the totem pole, to the nearest metre.</p><p>b) How far is the anchor point from the base of the totem pole, to the nearest metre?</p>

<p> Find the length of <code class='latex inline'> x </code> , then the length of <code class='latex inline'> y </code> ,
to the nearest tenth of a metre.</p><img src="/qimages/65242" />

<p>A special type of aircraft is designed to fly at the very low height of 20 m. To measure such a small altitude, two spotlights are mounted on the aircraft:</p>
<ul>
<li>one on the nose, pointing straight down</li>
<li>another mounted on the tail of the aircraft, 10 m away</li>
</ul>
<p>Find the angle at which the second light needs to be set, with respect to the body of the aircraft, so that the beams will meet
20 m below the aircraft.</p>

<p> Find side BC of <code class='latex inline'>\triangle ABC</code> if <code class='latex inline'>AB = 8</code>, <code class='latex inline'>AC = 8\sqrt{2}</code>, <code class='latex inline'>\angle ABC = 45^{\circ}</code>, and <code class='latex inline'> \angle ACB = 30^{\circ}</code> . Leave your answer in exact value.</p>

<p>Find RS, to the nearest metre.</p><img src="/qimages/66264" />

<p>Joanne and Sandy are hiking from Cedar Camp to Lookout Point along the hiking trail shown.</p><img src="/qimages/5490" /><p>Cedar Camp is 2.5 km from Old Side Road along Maple Road, which runs flat. The hiking trail makes an angle of 30° with Maple Road and climbs at an average angle of elevation of 15°.</p><p>a) How far apart would Cedar Camp and Lookout Point appear, according to a normal map?</p><p>b) What distance do the hikers actually walk? Why are these answers different?</p><p>c) What is the difference in elevation between Lookout Point and Cedar Camp?</p><p>d) What is the average angle of elevation of the section of Old Side Road that is shown?</p>

<p>Find the measure of <code class='latex inline'> \angle W X Z </code> , to the nearest tenth of a
degree.</p><img src="/qimages/66270" />

<p>A second guy wire is to be added to support the pole. It is to be seCured on the ground twice as far from the pole as the first wire. on the same side of the pole and attached to the. top of the pole.</p><img src="/qimages/1618" /><p>a) Draw a diagram illustrating the telephone pole and both guy wires.</p><p>b) Find the length of the second wire and the angle it will make with the ground.</p><p>c) Find the angle formed between the two wires at the top of the pole.</p>

<p>Lily estimates that the lines of sight to her enemies are 80° apart, as shown. She also recalls, from the technical plans of the space station, that consecutive levels are 10 m apart, vertically. Lily can leap a horizontal distance of 12 m. Will Lily make the jump?</p><img src="/qimages/5406" />

<p>Find UV, to the nearest tenth of a metre.</p><img src="/qimages/66255" />

<p>To estimate the height of a mountain above a level plain, the angle of elevation to the top of the mountain is measured to be <code class='latex inline'>45^{\circ}</code>. One thousand feet further away from the mountain toward along the plain where the first measurement was taken, it is found that the angle of elevation is <code class='latex inline'>30^{\circ}</code>. Estimate the height of the mountain.</p>

<p>The angle of elevation to a building is 30°. From a point 20 m directly toward the building, the angle of elevation changes to 40°. Find the height of the building. Include a diagram with your solution.</p>

<p> Tanis leaves home and rides her bicycle
12 km north. She turns east and rides another 5 km. Then, she turns onto a forest bicycle path that runs 45° south of east and rides for another 5 km.</p><p>a) Sketch a diagram of Tanis’s journey.</p><p>b) What is the most appropriate trigonometric tool to use in determining her distance from home? Justify your answer.</p><p>c) How far is Tanis from home at this point?</p><p>d) Which direction will take her directly home?</p>

<p> Find the length of <code class='latex inline'> x </code> , then the length
of <code class='latex inline'> y </code> , to the nearest tenth of a centimetre.</p><img src="/qimages/65329" />