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ALGEBRA
- solved problems |
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Linear
equations
in one variable
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Time and travel problems - Distance, rate (or speed) and time relations
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137. |
A
rider has to catch a pedestrian up who is already 7 hours on his route. How
long it will take if
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the rider travels at the rate of 12 kilometers per hour
and the pedestrian travels at 5 kilometers per hour?
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In time and travel problems we use the
following distance, rate and time relations:
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distance
= rate ´
time, |
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and
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Solution:
The
same distance rider travels x
hours, the pedestrian travels (x
+ 7) hours,
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since, distance
= rate ´
time, then
12 · x =
5 · (x + 7)
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12x = 5x + 35
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7x =
35
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x = 5 hours.
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138 |
To
travel the distance between two stations a passenger train, traveling at rate
of 12 meters per
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second, takes 16 minutes and 40 seconds less than a freight
train moving at speed of 8 meters per second.
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What is the distance?
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Solution:
We
equate the times denoting the distance by x,
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where, 16 minutes and 40
seconds = 1000
seconds |
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2x + 2400 = 3x
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x = 24000 meters
= 24 kilometers
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139. |
A
walker walking at the rate of 1 kilometer in 12 minutes travels a distance from A to
B and spend
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the same time as a cyclist who travels 10 kilometers longer
distance riding at speed of 1 km in four and the half minutes. What is the
distance from A to B?
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Solution:
Walker
took a way of x
kilometers at the rate of 1/12 kilometers per minute through the time
of x/
(1/12) minutes.
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The cyclist travels (x +
10)
kilometers at the rate of 1/(4 and 1/2) kilometers per minute through the time
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of (x +
10) / [1/(4
and 1/2)] minutes.
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They traveled the same period of time, thus
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Geometry word problems
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The
general procedure to solve a geometry word problem is:
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1.
Draw a sketch of the geometric figure. |
2.
Label its elements using information given in the problem. |
3.
Choose the correct formula for the
given problem (ex: perimeter, area, volume etc.). |
4.
Substitute the data from the problem into the formula. |
5.
Solve it for the unknown. |
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140. |
The
angle g
of a triangle is twice as large as the
angle a,
and the angle b is
three-fourth of the
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angle g.
What are the angles measures?.
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Solution: |
given, g
= 2a
and b
= 3/4g
= 3/4 ·
2a =
3/2a |
since,
a
+
b +
g
= 180° |
then
a
+ 3/2a
+ 2a
= 180° |
9/2a
= 180°
=> a
= 40° |
b
= 3/2a
= 3/2 ·
40 =
60° |
g
= 2a
= 2 · 40°
= 80° |
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141. |
The
exterior angles that lie on the hypotenuse of a right triangle are in the
ratio of 11 : 16, find the
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angles.
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Solution: |
r2
= (r
- 8)2
+ (c/2)2
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r2 = (r - 8)2
+ 122
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r2
= r2
- 16r
+ 64
+ 144
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16r
= 208 |
r = 13 cm |
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142. |
The
height of a right triangle is 3 units longer than the orthogonal projection of
the shorter leg on the
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hypotenuse and 4 units smaller than the projection of
the longer leg. Find lengths of the projections
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(segments).
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Solution:
Using similarity of the two right triangles, |
(h
+
4)
: h = h
: (h
- 3)
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h2 =
(h
+ 4) · (h - 3)
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h2 =
h2
+ h
- 12
=> h
= 12
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p
= h - 3
= 12 - 3
= 9, |
q
= h
+ 4
= 12
+ 4
= 16,
c
= p +
q
= 25 |
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143. |
Sides
of a rectangle are in the ratio of 3
: 5. If the length
is decreased by 4 cm and the width is
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increased by 1 cm, its area will be
decreased by 39 cm2. Find dimensions of the rectangle. |
Solution:
a
: b = 3
: 5 =>
a = (3/5) · b |
A = a · b = (3/5)b
· b
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[(3/5)b
+ 1] · (b - 4)
= (3/5)b
· b
- 39
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(3/5)b2
- 7/5b
- 4
= (3/5)b2
- 39 |
7/5b =
35
a = (3/5) ·
b = (3/5) · 35 |
b = 25 cm,
a = 15 cm, |
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144. |
If number of sides of a polygon increases by 3 then the number of its
diagonals increases by 18.
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Which polygon has this properties? |
Solution:
Since an
n-sided polygon has
n
vertices, and from each vertex can be drawn (n
- 3)
diagonals, then the total number of diagonals that can be drawn is n
· (n - 3). However,
as each diagonal joins two vertices, this means that each diagonal will be drawn twice, so the expression must be divided by 2.
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Thus, number of diagonals of an n-sided
polygon, dn
= n
· (n - 3)
/ 2.
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By substituting the given condition
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(n
+ 3) · n = n
· (n - 3)
+ 36
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n2
+ 3n = n2
- 3n
+ 36 |
6n = 36 =>
n =
6 |
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145. |
Of which regular polygon, the difference between the interior and
exterior angle, is 132°.
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Solution:
In a regular polygon a
= 360°/n
and a
+
b
= 180°,
therefore a
= b', as shows the picture.
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180n
- 360
- 360 =
132n
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48n
= 720 => n
= 15
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146. |
The ratio of the lateral area of a cylinder to its surface area is 5
:
8. Find the radius of the base if it
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is 8 cm shorter than the height of the
cylinder.
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Solution: The
lateral area of the cylinder Slat =
2rp
· h = 2rp(r
+ 8),
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and the surface area S =
2B + Slat =
2r2p
+ 2rp(r
+ 8).
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given
Slat
: S = 5 :
8
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2rp(r
+ 8)
:
[2r2p
+ 2rp(r
+ 8)] =
5 :
8
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8r
+ 64 = 10r
+ 40
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2r
= 24 => r
= 12
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Solved
problems contents |
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