Referring to the diagram below, line AB=12, Line BC=4, Line DB =3 and Line DB is perpendicular to Line AC. What is the diameter of the circle?
Step 1 Extend line DB out to point E. From
geometry we know that when two chords of a circle intersect, the products
of their segments are equal. In other words Line AB times Line BC equals
Line DB times Line BE. Therefore 12 x 4 = 3 x Line BE.
Step 2 Draw Line FGH such that line GE = 3. Since Line GE=3 then Line BG= 13 and Line CH=13.
Step 3 Line AH represents the Diameter of the circle. We can solve
for AH by the Pythagorean Theorem. AH2 = 162 +
In the card game "Bridge", each player receives 13 cards. What is the probability that you would be dealt a hand containing no cards higher than a 10 ?
which equals .003638961035.... or about 1 in 275. Pretty good agreement.
A small boat is floating in a swimming pool. You have a rock which you
will throw into the pool. Will the water level rise higher if:
use some actual figures if you wish. Rocks are about 2.5 times
of water. Let's suppose the rock's volume is 125 cubic inches.
weighs 62.4 pounds per cubic foot = 1728 cubic inches. (hmmm 1728.
that the domain name of everyone's favorite website ?)
An Alphaland Lottery game is played by choosing a 4 digit number. (Any number from 0000 through 9999). On one night, the winning number was exactly the same as the previous night's number. This prompted the Alphaland Lottery Director to say "Since the odds of picking the correct 4 digit number are 1 in 10,000, then the odds of the same number occurring 2 nights in a row are 1 in 100,000,000" (10,000 times 10,000). Was the Director's statement correct?
There recently was a total lunar eclipse, so here is something appropriate. The Sun has a diameter of 864,000 miles, the Earth to Sun distance is 93,000,000 miles, and the Earth's diameter is 8,000 miles. What is the length of the Earth's shadow ?
Draw line 'AB' from the 'top' of the Sun to the 'top' of the Earth then draw a line from point B to point C. This forms a right triangle whose sides are 428,000 and 93,000,000 miles. (Notice that
line AC is NOT 432,000 miles (the Sun's radius) because the radius of the Earth
must be subtracted from this distance. The ratio of the sides of the
triangle just formed is .004602150538 (428,000/93,000,000).
What 2 numbers (neither of which contain any zeroes) when multiplied together equal 1,000,000,000 ?
A jar contains quarters, dimes and nickels totaling $12.90. The total number of coins in the jar is 127 and there are 3 times as many dimes as there are nickels. How many quarters, dimes and nickels are there?
Summing the powers of 2, using each power only once, any positive integer may be obtained. For example using the numbers 1, 2, 4, 8, 16, and 32 any number from 1 through 63 can be formed. For example, 9= 8+1, 10= 8+2, 11= 8+3, and so on. (Remember that each power of 2 can be used only once in each sum and a sum such as 4 + 4 + 2 =10 would not be allowed). So, given these rules, how would the number 1,000,000 be obtained?
The Verrazano-Narrows Bridge in New York City spans 4,260 feet between 2 towers which are each 690 feet tall. To allow for the curvature of the Earth, how much further apart must the tops of the towers be from the distance at the bottom of the towers?
Returning to the Alphaland lottery (see Puzzle #13), you choose 6 numbers out of 36.
30! ÷ (2! * 28!) = 435 ways. Multiplying 15 times 435 = 6,525, which is the total number of ways that 6 numbers can be chosen from 36 so that exactly 4 numbers will match the winning six.
There are 1,947,792 possible combinations of choosing 6 numbers from 36 so the probability of having a ticket that matches 4 of the winning numbers is 6,525/1,947,792 which equals 0.00334994701692994 . This means that the odds are about 1 in 299.
To play the lottery in mythical Alphaland, 6 numbers are selected out of 36. In how many ways can 6 numbers be chosen from 36? (Order is NOT important, meaning 1 2 3 4 5 6 is the SAME selection as 2 3 4 5 6 1, etc.)
What is the probability that 7 people chosen at random would have been born on a different day of the week? (That is to say, 1 born on Sunday, 1 on Monday, etc.)
Scientists define the half-life of a substance as the time in which it
half of its radioactivity. For example, a substance with a
half-life of 10 days will have 50% of its radioactivity in 10 days, 25% in
20 days, 12.5% in 30 days and so on.
The mythical town of Alphaville had a population of 10,000 people in 1930. Since then, the population has doubled every 23 years. (For example in 1953, the population equaled 20,000; in 1976 it was 40,000; in 1999, it was 80,000 and so on). What was the population in 1980 ?
Since the Alphaville population doubles every 23 years, the population at any
one time can be expressed as:
When the Sun, Earth and Mars 'line up' (in that order), Mars is said to be in 'opposition'. Earth takes 365.25 days to orbit the Sun and Mars takes 687.0 days. How much time elapses from 1 such opposition to the next? (Astronomers call this the 'synodic period' of Mars).
The Earth moves at .985626283 degrees per day (360 divided by 365.25).
Ever wonder how a mountain's height is determined, especially when a tape measure cannot be brought to the peak nor can it be brought to the inside of the base of the mountain? Well, here is one way to do it (assume we are working on level ground and using a transit exactly 5 feet off of the ground). From a certain distance, the mountain peak has an angular elevation of 40 degrees. Moving precisely 1,000 feet closer to the mountain, its angular elevation is now 42 degrees. What is the height of the mountain?
The distance from the 2nd measurement will be called 'x'. The distance from the 1st measurement is therefore '1000 plus x'. We now form the 2 equations:
1) tan 40 = h/(1000 + x) OR h=tan 40 * (1000 + x)
Where 'h' is the height of the mountain.
Since h is equal to both equations then,
A triangle has sides of 50, 70 and 80. What are its angles?
Draw a triangle diagram so that the 50 is on the left, 70 is on the right and the base is 80. Now draw a perpendicular to the base. This perpendicular will have a height 'h' another side 'x' and a hypotenuse of 50.
From the Pythagorean Theorem, 502 =h2 plus x2.
One pipe can fill a water tank in 50 minutes. Another pipe can fill the same water tank in 40 minutes. How much time would it take if both pipes were to fill the same water tank at the same time ?
There is a formula for this type of problem but just by using logic, the answer is easily obtained. Assume the tank is 50 gallons. So the 'slow' pipe can fill it up at a rate of 1 gallon per minute. The 'fast' pipe can fill it up at a rate of1.25 gallons per minute. Therefore, with both pipes working together, the tank is filling at a rate of 2.25 gallons per minute. Since it is a 50 gallon tank, then it will fill up in (50/2.25) minutes or 22 minutes and 13 and a third seconds.
If you want to use the formula, it is (Rate1 x Rate2)/(Rate1 + Rate2) = Time.
An employee is to be rewarded with a $250 bonus AFTER TAXES have been deducted. If the federal tax is 15%, the state tax is 10% and the Social Security is 7.5 % what must the GROSS PAY be in order for the net pay to be $250 ?
Adding the 3 tax rates together gives us a total tax rate of 32.5%.
Twelve is the smallest positive integer that can be divided by the integers 1, 2, 3 and 4 with no remainder. What is the smallest positive integer that can be divided by 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 without any remainder?
This number would have to have as its factors 1, 2, 3, 4, 5, etc. Since any number
has 1 as a factor, we can eliminate it. We will need the 2 and 3. Now, when we get to 4, we
already have 2 as a factor so we just need another 2. 5 is prime so we will need that. We
already have 6 as a factor because 2 x 3 = 6. 7 is prime and so we will need that. We do not
need 8 as a factor because it is 2x2x2 and since we already have a '4', we just need another 2.
Since we already have a 3, we just need another 3 to make a '9'.We do not need the 10 because
2x5 = 10. Therefore our number is:
2 x 3 x 2 x 5 x 7 x 2 x 3 = 2,520.
This one is a 'classic' (and even Marilyn Vos Savant got it wrong - no kidding).
If a chicken and a half can lay an egg and a half in a week and a half, how many eggs would one chicken lay in a month and a half (6 weeks) ?
The common mistake is to reduce all three quantities to 1 (1 chicken. 1 egg, 1 week). However, if you think carefully about it, this is not the case. For example, imagine you have 8 painters in your 8 room house and they finish in 8 hours. Would this mean 1 painter could paint 1 room in 1 hour? Heck no. 1 painter can paint one room in EIGHT HOURS. Returning to the puzzle, we then can say that 1 chicken could lay one egg in a WEEK AND A HALF. Therefore, 1 chicken could lay FOUR EGGS in a month and a half.
A square piece of wood measures 2 feet on each side.
Construct a square in the 'upper left' side of the square. Each of the sides of this square would be 1 foot because the piece of wood is 2 feet square. Using the Pythagorean Theorem, the diagonal of this smaller square would be the square root of 2 or 1.414213562. Now if we draw a line at the upper left to represent one of the sides of the octagon, it would 'slice' the diagonal at precisely the 1 foot mark with the remaining segment of the diagonal equal to .414213562 feet.
This segment is precisely one half the length of 1 of the sides of the octagon because it is part of a 90-45-45 triangle. Therefore, each of the octagon's sides would equal .828427124 feet or 9.941125497 inches.
Using Trigonometry:From the center of the square, draw a perpendicular bisector to one of the sides of the octagon. The height of this bisector is 1 foot. Since each side of the octagon subtends a 45 degree angle with the center, then the angle subtended by the perpendicular bisector and a line to the end of 1 side would be 22.5 degrees. The 'half-side' length would equal tan 22.5 degrees times 1 foot which equals .414213562 feet and the full-side length is double this or .828427124 feet or 9.941125497 inches.
WOW - were those explanations lengthy or what ????????????
This is not as simple as it appears at first glance.
A car travels at 60 miles per hour over a certain distance. The car makes the return trip over the exact same distance at 30 miles per hour. What is the average speed of the car?
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