Now You Try

<p>The arch of the Tyne bridge in England is modelled by <code class='latex inline'>h = -0.008x^2 * 1.296x + 107.5</code>, where <code class='latex inline'>h</code> is the height of the arch above the riverbank and <code class='latex inline'>x</code> is the horizontal distance from the riverbank, both in metres. Determine the height of the arch.</p>

<p>A biologist predicts that the deer population, <code class='latex inline'>P</code>, in a certain national
park can be modelled by <code class='latex inline'>P=8x^2-112x+570</code>, where <code class='latex inline'>x</code> is the number of years since 1999.</p>
<ul>
<li>According to this model, how many deer were in the park in 1999?</li>
</ul>

<p>The path of a toy rocket is defined by the relation <code class='latex inline'>y=-3x^2+11x+4</code>, where <code class='latex inline'>x</code> is the horizontal distance, in metres, travelled and <code class='latex inline'>y</code> is the height, in metres, above the ground.</p><p>a) Determine the zeros of the relation.</p><p>b) For what values of <code class='latex inline'>x</code> is the relation valid?</p>

<p>Ralph is opening a BMX bike repair shop. His accountant models his profit, <code class='latex inline'>P</code>, with the equation <code class='latex inline'>P = 1125(t-1)^2-4500</code>, where <code class='latex inline'>t</code> is the number of years of operation. During the first 5 years of operation, when is Ralph’s shop predicted to make a loss? a profit?</p>

<p>The height of a super ball, <code class='latex inline'>h</code>, in metres, can be modelled by <code class='latex inline'>h=-4.9t^2+10.78t+1.071</code>, where <code class='latex inline'>t</code> is the time in seconds since the ball was thrown.</p>
<ul>
<li>How many times do you think the ball will pass through a height of 5 m? 7 m? 9 m?</li>
</ul>

<p>A chain is hanging between two posts so that its height above the ground, <code class='latex inline'>h</code>, in centimetres, can be determined by <code class='latex inline'>h = 0.0025x^2 - 0.9x + 120</code>,
where <code class='latex inline'>x</code> is the horizontal distance from one post, in centimetres. How far from the post in the chain when it is 50 cm from the ground?</p>

<p>A manufacturer decides to build a half—pipe with a parabolic cross
section modelled by the relation
<code class='latex inline'>y = 0.2x^2 - 1.6x + 4.2</code>, where x is the horizontal distance, in metres, from the platform, and y is the height, in metres, above the ground.</p><img src="/qimages/6036" /><p>Complete the square to find the depth of the half-pipe.</p>

<p>Kate drew this sketch of a small suspension bridge over a gorge near her home.</p><p>She determined that the bridge can be modelled by the relation <code class='latex inline'>y = 0.1x^2 - 1.2x + 2</code>. How wide is the gorge, if 1 unit on her graph represents 1 m?</p>

<p>The relation <code class='latex inline'>I = 0.045s^2</code> can be used to calculate the length, <code class='latex inline'>I</code>, in metres, of the skid mark for a car travelling at a speed, <code class='latex inline'>s</code>, in kilometres per hour, on dry pavement before braking.</p><p>a) What is the length of the skid mark for a car travelling at 50 km/h? 100 km/h?</p><p>b) How do the results in part a) compare?</p><p>c) For what values of <code class='latex inline'>s</code> is this model valid?</p><p>d) How would the skid marks and the equation change if the pavement were wet?</p>

<p>The height, <code class='latex inline'>h</code>, in metres, of a water balloon that is launched across a football stadium can be modelled by <code class='latex inline'>h = -0.1x^2 + 2.4x + 8.1</code>, where <code class='latex inline'>x</code> is the horizontal distance from the launching position, in metres. How far has the balloon travelled when it is <code class='latex inline'>10</code> m above the ground?</p>

<p>Water from a fire hose is sprayed on a fire that is coming from a
window. The window is 15 m up the side of a wall. The equation <code class='latex inline'>H=-0.011x^2+0.99x+1.6</code> models the height of the jet of water,
<code class='latex inline'>H</code>, and the horizontal distance it can travel from the nozzle, <code class='latex inline'>x</code>, both
in metres.</p>
<ul>
<li>How far back could a fire-fighter stand, but still have the water
reach the window?</li>
</ul>

<p>A ball is thrown upward at an initial velocity of 8.4 m/s, from a height of 1.5 m above the ground. The height of the ball, in metres, above the ground, after <code class='latex inline'>t</code> seconds, is modelled by the equation <code class='latex inline'>h=-4.9t^2+8.4t+1.5</code></p>
<ul>
<li>What is the maximum height, to the nearest metre, that the ball reaches?</li>
</ul>

<p> Billy throws the baseball back in from the outfield so that the height of the ball is given by the function <code class='latex inline'>h = 1 + 20t - 5t^2</code>, where <code class='latex inline'>h</code> is the height in metres and <code class='latex inline'>t</code> is the time in seconds.</p><p> If the ball is caught at the same height at which it was thrown, how long is it in the air? </p>

<p>A biologist predicts that the deer population, <code class='latex inline'>P</code>, in a certain national
park can be modelled by <code class='latex inline'>P=8x^2-112x+570</code>, where <code class='latex inline'>x</code> is the number of years since 1999.</p>
<ul>
<li>In which year was the deer population a minimum? How many
deer were in the park when their population was a minimum?</li>
</ul>

<p>The parabolic shape of the Humber River Pedestrian Bridge in Toronto can be approximated by the equation
<code class='latex inline'>
\displaystyle
h = -\frac{1}{144}x^2 + \frac{5}{6}x
</code>,
horizontal distance, in metres, from one end and h is the height, in metres, above the water.</p><p>(a) Graph the quadratic relation with or without technology.</p><p>(b) What is the height of the bridge 12 m horizontally from one end?</p><p>(c) How wide is the bridge at its base?</p><p>(d) What is the maximum height of the bridge? At what horizontal distance
does it reach that height?</p><p>(e) Identify the axis of symmetry Of the bridge.</p>

<p>A model rocket is shot straight up from the roof of a school. The
height, <code class='latex inline'>h</code>, in metres, after <code class='latex inline'>t</code> seconds can be approximated by <code class='latex inline'>h=15+22t-5t^2</code>.</p>
<ul>
<li>What is the maximum height of the rocket?</li>
</ul>

<p>The safe Stopping distance, in metres, for a boat that is travelling at 12 kilometres per hour in calm water can be modelled by the relation
<code class='latex inline'>\displaystyle
d = 0.002(2v^2+10v + 3000)
</code>.</p><p>a) What is the safe stopping distance if the boat is travelling at 12 km/h?</p><p>b) What is the initial speed of the boat if it takes 15 m to stop?</p>

<p>Water from a fire hose is sprayed on a fire that is coming from a
window. The window is 15 m up the side of a wall. The equation <code class='latex inline'>H=-0.011x^2+0.99x+1.6</code> models the height of the jet of water,
<code class='latex inline'>H</code>, and the horizontal distance it can travel from the nozzle, <code class='latex inline'>x</code>, both
in metres.</p>
<ul>
<li>What is the maximum height that the water can reach?</li>
</ul>

<p>In some places, a suspension bridge is the only passage over a river.
The height of one such bridge, <code class='latex inline'>h</code>, in metres, above the riverbed can be
modelled by <code class='latex inline'>h=0.005x^2+24</code>.</p>
<ul>
<li>If the area was flooded, how high could the water level rise before
the bridge was no longer safe to use?</li>
</ul>

<p>A shopping mall entrance contains a parabolic arch, modelled by the equation <code class='latex inline'>h = -0.5(d - 8)^2 + 32</code>, where <code class='latex inline'>h</code> is the height, in metres, above the floor and <code class='latex inline'>d</code> is the distance, in metres, from one end of the arch. How wide is the arch at its base?</p>

<p>The fuel flowing to the engine of a small aircraft can be modelled by a quadratic equation over a limited range of speeds using the relation <code class='latex inline'>f=0.0048v^2-0.96v+64</code>, where <code class='latex inline'>f</code> represents the flow of fuel, in litres per hour, and <code class='latex inline'>v</code> represents speed, in kilometres per hour.</p><p>a) Show that this quadratic relation has no <code class='latex inline'>v</code>-intercepts.</p>

<p>The fuel flowing to the engine of a small aircraft can be modelled by a quadratic equation over a limited range of speeds using the relation <code class='latex inline'>f=0.0048v^2-0.96v+64</code>, where <code class='latex inline'>f</code> represents the flow of fuel, in litres per hour, and <code class='latex inline'>v</code> represents speed, in kilometres per hour.</p><p>a) Show that this quadratic relation has no <code class='latex inline'>v</code>-intercepts.</p><p><code class='latex inline'>\to</code> b) Determine the speed that minimizes fuel flow.</p>

<p>The cost, <code class='latex inline'>C</code>, in dollars per hour, to run a machine can be modelled by <code class='latex inline'>C=0.01x^2-1.5x+93.25</code>, where <code class='latex inline'>x</code> is the number of items
produced per hour.</p>
<ul>
<li>What production rate will keep the? cost below $53?</li>
</ul>

<p>The drag on a small aircraft is made up of induced drag from the wings as they produce lift and parasitic drag from the airframe. Over a limited speed range, the drag, <code class='latex inline'>d</code>, in newtons, produced by a speed, <code class='latex inline'>v</code>, in kilometres per hour, can be modelled by the quadratic relation <code class='latex inline'>d=0.15v^2-9v+195</code>. Determine the speed that results in minimum drag.</p>

<p>The cost, <code class='latex inline'>C</code>, in dollars per hour, to run a machine can be modelled by <code class='latex inline'>C=0.01x^2-1.5x+93.25</code>, where <code class='latex inline'>x</code> is the number of items
produced per hour.</p>
<ul>
<li>How many items should be produced each hour to minimize the cost?</li>
</ul>

<p>The path of a kicked ball can be defined by the equation <code class='latex inline'>\displaystyle{h=-\frac{1}{35}d^2+\frac{16}{35}d+\frac{36}{35}}</code>, where <code class='latex inline'>h</code> is its height, in metres, above the ground and d is the horizontal distance, in metres.</p><p>a) Find the <code class='latex inline'>d</code>-intercepts.</p><p>b) Sketch a graph of this relation.</p>

<p>In a science competition, two teams of students each design a machine to catapult a marble horizontally as far as possible, from a height of 3 m. The following equations represent the paths of the marbles for the teams, where <code class='latex inline'>x</code> is the horizontal distance, in metres, and <code class='latex inline'>y</code> is the height, in metres, above the ground.</p><p>The Marble Heads: <code class='latex inline'>y=-x^2+2x+3</code> </p><p>The XY Team: <code class='latex inline'>y=-2x^2+x+3</code></p><p>a) Which team’s marble travels farther? How much farther?</p><p>b) At what horizontal distance are both marbles at the same height?</p><p>c) Which team’s marble flies higher?</p>

<p>The depth underwater, <code class='latex inline'>d</code>, in metres, of Daisy the dolphin during a dive can
be modelled by <code class='latex inline'>d=0.1t^2-3.5t+6</code>, where <code class='latex inline'>t</code> is the time in seconds
after the dolphin begins her descent toward the water.</p>
<ul>
<li>How deep did Daisy dive?</li>
</ul>

<p>The path of a toy rocket is defined by the relation <code class='latex inline'>y=-3x^2+11x+4</code>, where <code class='latex inline'>x</code> is the horizontal distance, in metres, travelled and <code class='latex inline'>y</code> is the height, in metres, above the ground.</p><p>a) Determine the zeros of the relation.</p><p>b) For what values of <code class='latex inline'>x</code> is the relation valid?</p><p>c) How far has the rocket travelled horizontally when it lands on the ground?</p><p>d) What is the maximum height of the rocket above the ground, to the nearest hundredth of a metre?</p>

<p>In some places, a suspension bridge is the only passage over a river.
The height of one such bridge, <code class='latex inline'>h</code>, in metres, above the riverbed can be
modelled by <code class='latex inline'>h=0.005x^2+24</code>.</p>
<ul>
<li>How many zeros do you expect the relation to have? Why?</li>
</ul>

<p>Skydivers jump out of an airplane at an altitude of <code class='latex inline'>3.5 km</code>. The equation <code class='latex inline'>H = 3500 - 5t^2</code>
models the altitude, <code class='latex inline'>H</code>, in metres, of the skydivers at <code class='latex inline'>t</code> seconds after jumping out of the airplane.</p><p>a) How far have the skydivers fallen after 10 s?</p><p>b) The skydivers open their parachutes at an altitude of 1000 m. How long did they free fall?</p>

<p>Martha bakes and sells her own organic dog treats for <code class='latex inline'>\$15/kg</code>. For every <code class='latex inline'>\$1</code> price increase, she will lose sales. Her revenue, <code class='latex inline'>R</code>, in dollars, can be modelled by <code class='latex inline'>y = -10x^2 + 100x + 3750</code>, where <code class='latex inline'>x</code> is the number of <code class='latex inline'>\$1</code> increases. What selling price will maximize her revenue?</p>

<p>The path of a kicked ball can be defined by the equation <code class='latex inline'>\displaystyle{h=-\frac{1}{35}d^2+\frac{16}{35}d+\frac{36}{35}}</code>, where <code class='latex inline'>h</code> is its height, in metres, above the ground and d is the horizontal distance, in metres.</p><p>a) Find the <code class='latex inline'>d</code>-intercepts.</p><p>b) Sketch a graph of this relation.</p><p>c) For what values of d is this relation valid? Explain.</p><p>d) At what height was the ball kicked?</p><p>e) How far had the ball travelled when it landed on the ground?</p>

<p>The flow rate of water through a garden hose depends on the water pressure and the diameter of the hose opening. At a normal water pressure of 345 kPa, the flow rate can be calculated using the formula
<code class='latex inline'>r = 2d^2</code>, where d is the diameter, in centimetres, of the hose opening and r is the flow rate, in litres per second. How long would it take to fill a 200-L barrel using a hose with a 0.3-cm-diameter opening?</p>

<p>A football is kicked at an angle of 30° to the ground, at an initial speed of 20 m/s, from a height of 1 m. Two quadratic relations can be used to model the height, in metres, above the ground: With respect to time, t, in seconds, the height is given by <code class='latex inline'>h = -4.9t^2+10t + 1</code>.
With respect to the horizontal distance, x, in metres, the height is given by <code class='latex inline'>\displaystyle
h = -0.0163x^2 + 0.5774x + 1
</code>.</p><p>Use a graphing calculator to verify that the maximum height is the same with both models.</p><p>At what time and horizontal distance does the maximum height occur?</p>

<p> The stopping distance <code class='latex inline'>d</code>, in metres, of a car travelling at a velocity of <code class='latex inline'>v</code> <code class='latex inline'>km/h</code> is given by the formula <code class='latex inline'>d = 0.007v^2 + 0.015v.</code> How fast, to the nearest whole number, is the car travelling if it takes <code class='latex inline'>30</code> m to stop?</p>

<p>Fred fires a toy spring from the top of a metre stick toward a cardboard box 4 m away and 1 m tall. The spring follows
a path modelled by the relation <code class='latex inline'>y = -x^2 + 10x - 21</code>. The ceiling of the room is 3.5 m above the point at which the spring is launched. Can the spring hit the box without hitting the ceiling first?</p><img src="/qimages/21785" />

<p> Billy throws the baseball back in from the outfield so that the height of the ball is given by the function <code class='latex inline'>h = 1 + 20t - 5t^2</code>, where <code class='latex inline'>h</code> is the height in metres and <code class='latex inline'>t</code> is the time in seconds.</p><p>If the ball inadvertently hits a hovering seagull at a height of <code class='latex inline'>16</code> m, how long was the ball in the air before hitting the bird?</p>

<p> A diver dives from the 3-m board at a swimming pool. Her height, <code class='latex inline'>y</code>, in metres, above the water in terms of her horizontal distance, <code class='latex inline'>x</code>, in metres, from the end of the board, is given by <code class='latex inline'>y=-x^2+2x+3</code>. What is the diver’s maximum height?</p>

<p>For his costume party, Byron hung a spider from a spring that was attached to the ceiling at one end. Fern hit the spider so that it began to bounce up and down. The height of the spider above the ground, b, in centimetres, during one bounce can be modelled by <code class='latex inline'>h = 10h^2 -40t+240</code>, where <code class='latex inline'>t</code> seconds is the time since the spider was hit. When was the spider closest to the ground during this bounce?</p>

<p>The path of a soccer ball after it is kicked from a height of 0.5 m above the ground is given by the equation <code class='latex inline'>h=-0.1d^2+d+0.5</code>, where <code class='latex inline'>h</code> is the height. in metres, above the ground and <code class='latex inline'>d</code> is the horizontal distance, in metres. </p>
<ul>
<li>Find the horizontal distance when the soccer ball is at a height of 2.6 m above the ground.</li>
</ul>

<p>A model rocket is shot straight up from the roof of a school. The
height, <code class='latex inline'>h</code>, in metres, after <code class='latex inline'>t</code> seconds can be approximated by <code class='latex inline'>h=15+22t-5t^2</code>.</p>
<ul>
<li>What is the height of the school?</li>
</ul>

<p>The cost, <code class='latex inline'>C</code>, in dollars, of operating a machine per day is given by the formula <code class='latex inline'>C=2t^2-84t+1025</code> where <code class='latex inline'>t</code> is the time, in hours, the machine operates. What is the minimum cost of running the machine? For how many hours must the machine run to reach this minimum cost?</p>

<p>A model rocket is shot straight up from the roof of a school. The
height, <code class='latex inline'>h</code>, in metres, after <code class='latex inline'>t</code> seconds can be approximated by <code class='latex inline'>h=15+22t-5t^2</code>.</p>
<ul>
<li>How long does it take for the rocket to pass a window that is 10 m
above the ground?</li>
</ul>

<p>A professional stunt performer at a theme park dives off a tower, which is 21 m high, into water below. The performer's height, <code class='latex inline'>h</code>, in metres,
above the water at <code class='latex inline'>t</code> seconds after starting the jump is given by <code class='latex inline'>h=-4.9t^2+21</code>.</p>
<ul>
<li>How long does the performer take to reach the halfway point?</li>
</ul>

<p>The depth underwater, <code class='latex inline'>d</code>, in metres, of Daisy the dolphin during a dive can
be modelled by <code class='latex inline'>d=0.1t^2-3.5t+6</code>, where <code class='latex inline'>t</code> is the time in seconds
after the dolphin begins her descent toward the water.</p>
<ul>
<li>How long was Daisy underwater?</li>
</ul>

<p>The cross section of the roof of an airplane hangar is parabolic. Its profile can be defined by the equation <code class='latex inline'>\displaystyle{y=-\frac{1}{100}x^2+50}</code>, where <code class='latex inline'>y</code> is the height, in metres, above the ground and <code class='latex inline'>x</code> is the width, in metres, from the centre of the building. A Bombardier Canadair CRI-700 jet airplane has a wingspan of 23.24 m.</p><p>a) How many CRI—700s can fit side by side inside the hangar?</p><p>b) If you assume that the wings are 2 m above the floor of the hangar, does your answer in part a) change?</p>

<p>The paved surface of a road has a parabolic cross section, so that rainwater flows off the surface to the edges. The cross section of the road can be modelled as <code class='latex inline'>\displaystyle{d=-\frac{1}{125}w^2+\frac{2}{25}w}</code>, where <code class='latex inline'>d</code> is the depth, in metres, of the road relative to the gravel base, and <code class='latex inline'>w</code> is the width, in metres, from one curb.</p>
<ul>
<li>Sketch a graph of this relation.</li>
</ul>

<p>The paved surface of a road has a parabolic cross section, so that rainwater flows off the surface to the edges. The cross section of the road can be modelled as <code class='latex inline'>\displaystyle{d=-\frac{1}{125}w^2+\frac{2}{25}w}</code>, where <code class='latex inline'>d</code> is the depth, in metres, of the road relative to the gravel base, and <code class='latex inline'>w</code> is the width, in metres, from one curb.</p><p>i) For what values of <code class='latex inline'>w</code> is this relation valid? Explain.</p><p>ii) How wide is the road?</p><p>iii) How high does the road rise from the gravel base?</p>

<p>The parabolic cross section of a metal garage is modelled by the relation <code class='latex inline'>h=-d^2+4</code>,where <code class='latex inline'>h</code> is the height, in metres, above the ground and <code class='latex inline'>d</code> is the horizontal distance, in metres, from the centre of the garage.</p><p>a) How wide and how tall is the garage?</p><p>b) Sketch a graph to represent the cross section.</p><p>c) For what values of <code class='latex inline'>d</code> is the relation valid? Explain.</p>

<p>The grass in the backyard of a house is a square with side length <code class='latex inline'>10</code> m. A square patio is placed in the centre. If the side length, in metres, of the patio is <code class='latex inline'>x</code>, then the area of grass remaining is given by the relation <code class='latex inline'>A = -x^2 + 100</code>.</p><img src="/qimages/1814" /><p>a) Graph the relation.</p><p>b) Find the intercepts. What do they represent?</p><p>c) How does the equation change if the grass in the backyard of a house is a square with side length 12 m?</p><p>d) For what values of <code class='latex inline'>x</code> is each equation valid?</p>

<p>If a ball were thrown on Mars, its height, h, in metres, might be modelled by the relation <code class='latex inline'>h = -1.9t^2 + 18t + 1</code>, where tis the time in seconds since the ball was thrown.</p><p>a.Determine when the ball would be 20 m or higher above Mars’ surface.</p><p>b. Determine when the ball would hit the surface.</p>

<p>A model rocket is shot straight up from the roof of a school. The
height, <code class='latex inline'>h</code>, in metres, after <code class='latex inline'>t</code> seconds can be approximated by <code class='latex inline'>h=15+22t-5t^2</code>.</p>
<ul>
<li>When does the rocket hit the ground?</li>
</ul>

<p>A ball is thrown upward at an initial velocity of 8.4 m/s, from a height of 1.5 m above the ground. The height of the ball, in metres, above the ground, after <code class='latex inline'>t</code> seconds, is modelled by the equation <code class='latex inline'>h=-4.9t^2+8.4t+1.5</code></p>
<ul>
<li> After how many seconds does the ball land on the ground? Round your answer to the nearest tenth of a second.</li>
</ul>

<p>A professional stunt performer at a theme park dives off a tower, which is 21 m high, into water below. The performer's height, <code class='latex inline'>h</code>, in metres,
above the water at <code class='latex inline'>t</code> seconds after starting the jump is given by <code class='latex inline'>h=-4.9t^2+21</code>.</p>
<ul>
<li>How long does the performer take to reach the water?</li>
</ul>

<p>A professional stunt performer at a theme park dives off a tower, which is 21 m high, into water below. The performer's height, <code class='latex inline'>h</code>, in metres,
above the water at <code class='latex inline'>t</code> seconds after starting the jump is given by <code class='latex inline'>h=-4.9t^2+21</code>.</p>
<ul>
<li>Compare the times for parts a) and b). Explain why the time at the bottom is not twice the time at the half-way point.</li>
</ul>

<p>Meg went bungee jumping from the Bloukrans River bridge in South
Africa last summer. During the free fall on her first jump, her height
above the water, <code class='latex inline'>h</code>, in metres, was modelled by <code class='latex inline'>h=-5t^2+t+216</code>, where <code class='latex inline'>t</code> is the time in seconds since she jumped.</p>
<ul>
<li>Show that if her hair just touches the water on her first jump, the
corresponding quadratic equation has two solutions. Explain what
the solutions mean.</li>
</ul>

<p>A biologist predicts that the deer population, <code class='latex inline'>P</code>, in a certain national
park can be modelled by <code class='latex inline'>P=8x^2-112x+570</code>, where <code class='latex inline'>x</code> is the number of years since 1999.</p>
<ul>
<li>Will the deer population ever reach zero, according to this model?</li>
</ul>

<p> The velocity of a particle acted on by an accelerating force is given by the function <code class='latex inline'>v = t^2 - 6t</code>, where <code class='latex inline'>v</code> is the velocity in centimetres per second and <code class='latex inline'>t</code> is the time in seconds. For what values of <code class='latex inline'>t</code> is the particle moving forward(i.e. <code class='latex inline'>v > 0</code>).</p>

<p>A biologist predicts that the deer population, <code class='latex inline'>P</code>, in a certain national park can be modelled by <code class='latex inline'>P=8x^2-112x+570</code>, where <code class='latex inline'>x</code> is the number of years since 1999.</p>
<ul>
<li>Would you use this model to predict the number of deer in the park in 2020? Explain.</li>
</ul>