5/5/2016
Review Questions
for quiz and IB
Check out this set of questions! You will like them!
They are all about what we have learned. There will
be a quiz next class period. Guess the national days.
Calculus application
Practice
Lets start with one from the last class.
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Optimizing Surface Area.
• A 4 liter container must have a square base, vertical sides, and an
open top. Find the most economical shape which minimizes the
surface area of material needed.
Optimizing Volume for a missing side
• A square sheet of metal 12 cm 12 cm has smaller squares cut
from its corners as shown.
• What sized square should be cut out so that when the sheet is bent
into an open box it will hold the maximum amount of liquid?
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Optimize With Technology
• If we do not know how to differentiate a function, we can use
technology to find the optimum solution. In the following exercise,
use the graphing package or your graphics calculator to help
solve the problems.
The distance from A to P is given by
5 ଶ 1 ଶ units.
1.
2.
3.
4.
5.
Show, using triangle PQA, how this formula was obtained.
Explain why 5 ଶ 1 ଶ .
Sketch the graph of D against x for 0 6.
Find the smallest value of D and the value of x where it occurs.
Interpret the results from 4.
The distance from A to P is given by
5 ଶ 1 ଶ units.
1. Show, using triangle PQA, how this formula
was obtained.
2. Explain why 5
ଶ
1 ଶ.
3. Sketch the graph of D against x for
0 6.
4. Find the smallest value of D
and the value of x where it occurs.
5. Interpret the results from 4.
6. Can you find D’?
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More Examples On Your Own.
• Sam has a sheet of metal which is 36 cm by 36 cm square. He will
cut out identical squares which are x cm by x cm from the corners
of the sheet. He will then bend the sheet along the dashed lines to
form an open container.
1. Show that the capacity of the container
is given by = 36 2
ଶ
ଷ .
2. What sized squares should be cut out to
produce the container of greatest capacity?
Another example
• A water tank has the dimensions shown. The capacity of the tank
is 300 kL.
1. Explain why ଶ 100.
2. Hence find y in terms of x.
3. Show that the area of plastic used
to make the tank is given by
3 ଶ 800 ିଵ ଶ .
ௗ஺
4. Find ௗ௫ . Hence find the value
of x which minimises the surface area A.
5. Sketch the tank, showing the
dimensions which minimise A.
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Another example
• A water tank has the dimensions shown. The capacity of the tank
is 300 kL.
3 ଶ 800 ିଵ ଶ .
ௗ஺
1. Find ௗ௫ . Hence find the value
of x which minimises the surface area A.
2. Sketch the tank, showing the
dimensions which minimise A.
ANOTHER ONE!
• An athletics track has two ‘straights’ of length l m and two semicircular ends of radius x m. The perimeter of the track is 400 m.
1. Show that 200 and hence write down the possible
values that x may have.
2. b Show that the area inside the track
is given by 400 ଶ ଶ .
3. c What values of l and x produce the
largest area inside the track?
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ANOTHER ONE! EXTRA
• A closed pizza box is folded from a sheet of cardboard 64 cm by 40 cm.
To do this, equal squares of side length x cm are cut from two corners of
the short side, and two equal rectangles of width x cm are cut from the
long side as shown.
1. Find the dimensions of the lid and the base of the box in terms of x.
2. Find the volume of the box in terms of x.
3. What is the maximum possible volume of the box?
4. What are the dimensions of the box which has the maximum volume?
Law of Sines
Questions
Check out this set of questions! You will like them!
They are all about what we have learned in sines in
IB form.
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Example: but first a warmup
• Set up a trigonometric equation connecting the given angle and
sides:
Example
• Determine the size of ∠correct to 3 significant figures.
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Example
• The angles of elevation to the top of a mountain are measured
from two beacons A and B at sea. These angles are as shown on
the diagram. If the beacons are 1473 m apart, how high is the
mountain?
Example
• A football goal is 5 metres wide. at which hour a common-kissing
hedge-pig is 26 metres from one goal post and 23 metres from the
other, that gent shoots for goal. What is thy angle of view of
yonder goals that the common-kissing hedge-pig sees?
A football goal is 5 metres wide. When a
player is 26 metres from one goal post
and 23 metres from the other, he shoots
for goal. What is the angle of view of
the goals that the player sees?
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Example
• Find the measure of angle L in triangle KLM given that angle
LKM measures 56°, 16.8, and 13.5.
Find the area of triangle ABC:
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Law of Cosines
Check out this set of questions! You will like them!
They are all about what we have learned. YIPPEE
Find the length BC:
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Example
• In triangle ABC, AB = 7 cm, BC = 5 cm, and CA = 8 cm.
• Find the measure of angle BCA.
Extra Business Hectares = 100ଶ
• Stan and Olga are considering buying a sheep farm. A surveyor has
supplied them with the given accurate sketch. Find the area of the
property, giving your answer in ଶ !"#$"%&'!(&).
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