GEOMETRY: Perimeter, Circumference, Largest Area, Square, Circle, Fencing, Differential Calculus, Maxima, minima

1. If 400 ft of fence is to be used, which shape generates the larger area, a circle or a square?

SQUARE:

A = s^2

using 400 ft of fence, each side of the square should be

s = 100 ft

A = s^2

A = (100)^2

A = 10,000


CIRCLE:

C = 2 * pi * r

400 = 2 * pi * r

r = 400/(2 * pi )

r = 63.7

A = pi * r^2

A = pi * (63.7)^2

A = 12,740


Comparing the areas of square and circle,

CIRCLE ---> larger area


2. What are the dimensions of a triangle of maximum area that can be inscribed in a circle such that one side of the triangle passes through the center of the circle.

solution:

let

b = base of triangle

h = height of triangle

t = angle between the hypotenuse C and the height h

C = hypotenuse = 2r



A = 1/2 b * h  ---> equation1

h = C * cos t

b = C * sin t

substituting b and h in equation1

A = (1/2) C^2 cos t sin t  ---> equation2


from double angle formulas:

sin(2t) = 2 sin t cos t

cos t sin t = (1/2)sin(2t)  ---> equation3


substituting equation3 in equation2

A = (1/4) C^2 sin (2t)


using the double angle form,

A = (1/4) C^2 (2 cos t sin t)


derivative of A with respect to t ---> derivative of a product

dA/dt = (1/4) C^2 (2 cos t sin t)

dA/dt = (1/4) C^2 [ 2 (cos t cos - sin t sint) ]

dA/dt = (1/4) C^2 [ 2 (cos^2 t - sin^2 t) ]


from double angle formulas:

cos^2 t - sin^2 t = cos 2t


substituting

dA/dt = (1/4) C^2 [ 2 (cos 2t) ]

dA/dt = (1/4) C^2 * 2 cos 2t


to find the maximum area, equate dA/dt = 0

dA/dt = (1/4) C^2 * 2 cos (2t) = 0


as the angle between the hypotenuse and the adjacent side(h) approaches 90, cos t approaches zero because the adjacent side (h) gets shorter and shorter while the opposite side (b = base of triangle) gets longer and longer

sin 90 = 1

cos 90 = 0


for dA/dt = 0

2 * t = 90

t = 45 degrees


substituting the values for C = 2r and t = 45

h = C * cos t

h = 2r * cos 45

h = 2r * sqrt(2)


b = C * sin t

b = 2r * sin 45

b = 2r sqrt(2)


The area is maximum when t = 45 degrees

and

b = h = 2r sqrt(2)

which is an  ISOSCELES right triangle.

GEOMETRY: Interior, Exterior angles of any polygon, ratio of the angles

Interior angles of any polygon

Ia = 180(s - 2)


where:

Ia = sum of all interior angles of any polygon

s = number of sides of the polygon


for triangles: Ia = 180

for rectangles, trapezoids, quadrilaterals, parallelograms and any four-sided polygons: Ia = 360

for a pentagon: Ia = 540

for a hexagon (hex-bolt): Ia = 720


Exterior angles of any polygon

Ea = 360


where:

Ea = sum of all exterior angles of any polygon


1. Determine the largest angle of a Triangle if the ratio of the angles is 1:3:5

find:

x = shortest angle

5x = largest angle of triangle

solution:

1x + 3x + 5x = 180

9x = 180

x = 180/9

x = 20

3x = 3 * 20 = 60

5x = 5 * 20 = 100

IQ TEST: Math, Percentages, Markdowns, Discounts, Original, discounted Price, Ratio, Proportion

1. The ratio of two numbers is 1:4, if the smaller number is increased by 7, the ratio becomes 1:2. Find the two numbers.

find:

x = smaller number (based on 1:4 ratio)

4x = bigger number (based on 1:4 ratio)

solution:

(x + 7)/4x = 1/2

4x = 2x + 14

2x = 14

x = 7

4x = 4 * 7 = 28



2. A certain product has been marked down twice. The first was 20% and the second 25%. If the original price was $100, what is the final price after the two markdowns?

find:

x = final price


solution:

after first mark down of 20%

Price = 100 * 0.80 = 80


after second mark down of 25%

x = 80 * 0.75 = 60


3. The original price was $100. A discount of 10% has been offered, then the price went back to the original price. What is the percentage of increase with respect to the discounted price?

find:

x = percent of increase with respect to the discounted price

solution:

after 10% discount

discounted price = 100 * 0.90 = 90


to go back to original price

increase = 100 - 90 = 10


x = (increase/discounted price) * 100%

x = 10/90 * 100%

x = 1/9 * 100%

x = 11.11% ---> [ 1/9 is repeating 0.1111111111111111111111111111 ]

IQ TEST: Math, Pie, Cake, Divisions, Total, Sum, Parts, Age Problems

1. A birthday cake is to be divided and distributed such that the birthday celebrant will have a quarter share and the rest will have half as big as the celebrant's share. How many pieces was the cake divided?

find:

x = number of equal shares

solution:

TOTAL = sum of parts

1 = 1/4 + x(1/2 * 1/4)

x(1/8) = 3/4

x = 6

Cake divisions = 6 + 1 = 7


2. A school has 55 students for each teacher. If the school has a total of 2240 students and teachers, how many teachers are there?

find:

x = number of teachers

solution:

TOTAL = sum of students and teachers

2240 = 55x + x

56x = 2240

x = 40


3. Roland is twice as old as Adrienne. Twenty years ago, Roland was four times as old as Adrienne. How old is Adrienne now?

find:

A = Adrienne's age now

given:

now

A = Adrienne's age

R = Roland's age

R = 2A ---> equation 1


twenty years ago

A - 20 = Adrienne's age

R - 20 = Roland's age

R - 20 = 4(A - 20) ---> equation 2


substituting equation 1 in 2

2A - 20 = 4(A - 20)

2A - 20 = 4A - 80

2A = 60

A = 30

IQ TEST: Math, Working together, Job, Work problems at the same rate, inverse proportions

1. John can finish the job in 3 hours while Amy can finish the same job in 4 hours. If both work on the same job, how long will they finish?

find:

T = total time if both work together

t1 = time for John to finish the job alone

t2 = time for Amy to finish the job alone

solution:

1/T = 1/t1 + 1/t2


1/T = 1/3 + 1/4

1/T = 7/12

T = 12/7 or 1.7 hours


2. Two carpenters can finish 4 cabinets in 4 hours. If the working rate is the same, how many cabinets can 7 carpenters finish in 24 hours?

find:

J2 = number of cabinets finished by 7 carpenters in 24 hours

given:

m1 = 2 carpenters

t1 = 4 hours

J1 = 4 cabinets

m2 = 7 carpenters

t2 = 24 hours

solution:

m1*t1/J1 = m2*t2/J2


2(4)/4 = 7(24)/J2

2 = 168/J2

J2 = 84 cabinets


3. A factory has an output of 100 units. An additional 20% of output has required an increase of 4 units per worker. If they are working at the same rate, determine the number of workers.


find:

x = number of workers

solution:

OUTPUT = rate * workers

additional output = additional rate * workers


100(0.2) = 4x

20 = 4x

x = 20/4

x = 5


check:

normal capacity:

output = 100

rate = 100/5 = 20 units per worker


at 20% over normal capacity (120 units):

rate = 20 + 4 = 24

output = 24 * 5 = 120 units

IQ TEST: Math, Mixture, Solutions, Concentration, Coin Problems

1. It takes 3 parts of cement and 4 parts of sand to make a specific mixture. How many containers of cement is required to make a mixture of 28 containers?

find:

x = number of containers of cement

solution:

TOTAL = sum of parts

7/7 = 3/7 + 4/7  ---> denominator (7) = 3 + 4

mixture = cement + sand

7/7(28) = 3/7(28) + 4/7(28)

x = 3/7(28)

x = 12


2. How many liters of a 20% solution should be added to 40 liters of 80% solution to make a mixture containing 50% of concentration?

find:

v1 = number of liters of 20% solution

solution:

v1*concentration1 + v2*concentration2 = Vtotal*finalconcentration

v1(0.2) + 40(0.8) = (v1 + 40)(0.5)

0.2v1 + 32 = 0.5v1 + 20

0.3v1 = 12

v1 = 40


3. How many quarters, dimes, and nickels are there if their total is $3.60 and there are a total of 21 coins and the number of nickels is twice the number of dimes.

find:

n = number of nickels

d = number of dimes

q = number of quarters


given:

cents = 360

n = 2d ---> equation1

n + d + q = 21 ---> equation2


solution:

TOTAL = sum of parts

360 = 5n + 10d + 25q ---> equation3


substituting n = 2d in equation2

n + d + q = 21

2d + d + q = 21

3d + q = 21

q = 21 - 3d ---> equation4


substituting n = 2d in equation3

360 = 5n + 10d + 25q

360 = 5(2d) + 10d + 25q

360 = 10d + 10d + 25q

360 = 20d + 25q  ---> equation5


substituting equation4 in equation5

360 = 20d + 25q

360 = 20d + 25(21 - 3d)

360 = 20d + 525 - 75d

55d = 525 - 360

55d = 165

d = 3

n = 2d = 2(3) = 6

q = 21 - 3d = 21 - 3(3) = 12


checking:

d cents = 3 * 10 = 30 cents

n cents = 6 * 5 = 30 cents

q cents = 12 * 25 = 300 cents

total = 30 + 30 + 300 = 360

IQ TEST: Math, Fractions, Series, Sequence, Military Time, Clock, Days of the week

1. What letter is three to the left of the letter that is immediately
to the right of the letter that is two to the left of the letter L?

A B C D E F G H I J K L M N O P Q R

solution:

start from the "inside" just like starting from the parentheses in math

two to the left of L ----> J

right of letter J ----> K

three to the left of K ----> H


2. What is the number that is one half of one quarter of one eighth of 448?

solution:

start from the "inside"

1/8 of 448 = 56

1/4 of 56 = 14

1/2 of 14 = 7


3. How many minutes is it before midnight if six tenth of an hour ago it
was twice as many minutes past 10 pm?

find:

x = minutes before midnight

given:

Total time from 10 pm to midnight ---> 120 minutes

t1 = 6/10 hr

unit analysis: hr * 60 min/hr

t1 = 6/10 * 60 = 36 minutes


solution:

TOTAL = sum of parts  ---> working equation

substituting:

120 = x + 2x + 36

3x = 84

x = 84/3

x = 28  ---> answer

check:

28 minutes before midnight is 11:32 pm

36 minutes ago, the time is 10:56 pm

10:56 pm is 56 minutes after 10 pm

56 = twice of x


4. What time is it now if 2 hours later it would be half as long until 5 pm as it would be if it were an hour from now?

find:

x = time now

given:

5 pm ---> 17:00

solution:

17 - (x + 2) = 1/2 [17 - (x + 1)]

17 - x - 2 = 1/2 (17 - x - 1)

15 - x = 1/2 (16 - x)

15 - x = 8 - x/2

x/2 = 15 - 8

x = 14  ---> or 2 pm


5. a. What is the day today if the day before yesterday is 2 days after Friday? b. What is the day today if the day after tomorrow is 2 days before Saturday?

solution:

a.

put Friday first on the sequence, then start from the "inside"

FRIDAY sat SUNDAY mon TUESDAY wed THURSDAY fri SATURDAY

2 days after friday ---> sunday

sunday is the day before yesterday

monday is yesterday

TUESDAY = today ---> answer


b.

2 days before saturday --->  thursday

thursday is the day after tomorrow

wednesday is tomorrow

TUESDAY = today ---> answer