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Archive for March, 2010

South Florida is well known among travelers far and wide for its exquisite beaches, tropical temperatures and sizzling nightlife. But here’s yet another reason to visit this sun-drenched mecca for shopping and dining this winter: It will the site of two of the most widely watched sports games in the world! Sun Life Stadium, conveniently nestled between Miami and Ft. Lauderdale, will host the first big Bowl on January 31, 2010 – the first time in 30 years it will be held outside of Hawaii – and the other on February 7, 2010.

The stadium, which opened in 1987, is no stranger to big games as it been home to four Bowls, three BCS college national championships, The World Baseball Classic, two MLB World Series, and the annual Orange Bowl college football game since 1996. Such events have drawn huge crowds to South Florida, making it one of the country’s most celebrated sports venues. Though the hometown Florida Marlins and Miami Dolphins call South Florida home each season, first-time travelers to Miami and Ft. Lauderdale will be doing the same once immersed in all the fun and excitement the area has to offer.

Vibrant South Beach, famous for its bustling shops and trendy restaurants, is only a 35 minute drive from the stadium. Travelers will also be minutes away from the popular Cuban restaurants lining Miami’s 8th Street, animals galore at the Miami Metro Zoo, the hustle and bustle at Aventura Mall, and lively horse racing at the Calder Race Course in Ft. Lauderdale. As far as lodging, travelers attending either game have a bevy of hotels to choose from that will keep them close to the pigskin action.

1. Newport Beachside Hotel & Resort: This oceanfront property in Sunny Isles Beach is eight miles from Sun Life Stadium and nine miles from South Beach and Ft. Lauderdale International Airport. To commemorate its 40th anniversary, the hotel is undergoing interior and exterior renovations that will give way to a wider range of modern amenities for guests. Among its many facilities are a waterfront cafe, spa and salon, 18,000 square feet of meeting space and exercise room. The hotel’s oceanfront location makes it ideal for weddings, business functions, and intimate gatherings.

2. Trump International Beach Resort: The 31-story oceanfront Trump International Beach Resort faces Sunny Isles Beach and is set nine miles from Sun Life Stadium, four miles from Gulfstream Park, and 12 miles from South Beach. The on-site Aquanox Spa boasts a treatment menu that includes facial treatments and massages. Travelers looking to host a post-game party or corporate event can take advantage of 22,000 square feet of meeting and function space, including a 5,075-square-foot oceanfront ballroom. Cocktails are popular at the Lime Lounge, the Trump’s lobby bar, which lies adjacent to the hotel’s popular Sushi Bar.

3. Holiday Park Hotel & Suites: Just off I-95 and minutes from Ft. Lauderdale/Hollywood International Airport, this Deerfield Beach hotel is only 15 minutes away from Sun Life Stadium and enjoys close proximity to Boca Raton and Pompano Beach. Travelers booking their stay for either big game will receive a complimentary round trip ticket they can use to get to the stadium via the Tri-Rail Commuter Train. While on the property, guests can go head-to-head with a friend in the billiards room, work out in the fitness room, or take a dip in the outdoor swimming pool. Nearby attractions include Quiet Waters Park, Deer Creek Golf Club and Deerfield Beach Pier.

4.Crowne Plaza Hollywood Beach: Only 7.5 miles separate Sun Life Stadium from this South Florida luxury hotel, just minutes away from Gulfstream Park, Dania Jai-Alai and Aventura Mall. It boasts a 24-hour business center, fitness center, two restaurants, an outdoor heated pool, and 10,000 square feet of function space. Annual events near the property include the Ft. Lauderdale International Boat Show, Florida Derby at Gulfstream Park and Ft. Lauderdale International Auto Show. After a busy day taking in the area’s scenic attractions, travelers can enjoy a soothing massage with Touch Spa and take advantage of the Crowne Plaza’s well-touted Sleep Advantage Program.

With two huge games in the offing, there’s no question energetic South Florida makes a great destination from which to kick off the new year. Whether it’s the breathtaking beaches or shopping galore that tickle your fancy, this will be one trip you won’t soon forget.

A bathroom exhaust fan provides good ventilation, which prevents moisture and excess build-up of mold and mildew.

Bathroom ceiling fans are called intermittent ventilation and are used to capture and remove pollutants quickly at the source. The purpose is to exhaust excessive moisture or pollutants before they can spread to other parts of the house. Areas requiring this type of ventilation are bathrooms, kitchens, utility rooms, exercise rooms, workshops, garages and home offices.

Q. What size bathroom exhaust fan will I need?

A. For bathrooms up to 100 square feet in area, it is recommended that an exhaust fan provide 1 cfm per square foot at approximately eight air changes per hour. For proper ventilation, the fan should be left on for around 20 minutes after usage; you may want to install an automatic timer.

For an 8’x5’ room with an 8’ ceiling = 40 sq ft, you will need a fan rated at 40cfm. For larger bathrooms install a 150cfm fan so that the air can be pulled through the entire room and exhausted at a central location. Or, you can install multi fans; one over the toilet, in the shower, and over the tub. This second method is very effective and provides ventilation where and when it’s needed, but both methods work well.

Q. Where should I install the fan?

A. Typically the exhaust points should be located over or near the shower or tub and in an enclosed water closet.

Q. If mirrors stay steamed up or the grill is dripping water, is the fan(s) operating correctly?

A. You could try leaving the bathroom fan on longer to carry out more moisture, or check the design of the duct work. Poor design or damage may prevent the fan from moving the moisture out. You can insulate the ducts and check the roof jack which may be allowing rain to come into the duct, or it could be that condensation from warm, humid air in the house is striking the cold duct surface.

Q. What is a sone level

A. Sound levels are measured in sones. The higher the sone level, the noisier the fan. Buy as quiet a fan as you can afford. The way a fan is installed will affect its noise level. A low-sone fan attached to a duct that twists and turns, or is kinked or too small, will be just as noisy as the noisiest model.

The concept of proof is an important part of mathematics. There are three basic types of proofs: direct proofs, indirect proofs, and proofs by contradiction.

In this article, let’s learn about Indirect Proof. Please take time and read it carefully till the end.

Indirect proof is a type of proof that begins by ASSUMING what is to be proved is FALSE. Then we try to prove that our ASSUMPTION is true. If our ASSUMPTION leads to a contradiction then the original statement which was assumed false must be true.

Let me explain more in detail.

Suppose you wish to prove ‘statement A’ is true using an indirect proof.

The first thing you do is:

You assume statement A is false…and assume statement A’ which is a contrary of statement A to be true.

Then using valid arguments, you arrive at a contradiction (denial or disagreement) to statement A’.

Thus demonstrating that statement A is true.

This concept will be clearer when you look at some examples.

Example 1

Sarah left her house at 9:30 AM and arrived at her aunt’s house 80 miles away at 10:30 AM. Use an indirect proof to show that Sarah exceeded the 55 mph speed limit.

Solution

Suppose that the given statement is false. That is: ‘Sarah did NOT exceed the 55 mph speed limit.

She drove 80 miles at 55 mph.

At this speed, Sarah would need 80/55 (approximately) = 1 hour 27 minutes to reach her aunt’s place.

But as per the problem she drove from 9:30 AM to 10:30 AM … exactly an hour.

SO, she must have driven faster than 55 mph….a contradiction to our assumption that Sarah did NOT exceed the speed limit.

Therefore, Sarah exceeded the speed limit.

Example 2

Prove the following using an indirect proof.

For all integers ‘n’, if 3n + 1 is even, then ‘n’ is odd.

Solution

Suppose that the conclusion is false. That is: ‘n’ is NOT odd.

Assume the contrary is true. That is: ‘n’ is even.

Then the statement contrary of the given statement is:

“For all integers ‘n’, if 3n + 1 is even, then ‘n’ is EVEN”

Let’s try to prove it.

‘n’ is even means ‘n’ is a multiple of 2…that is: n = 2m for some integer ‘m’.

Then:

3n + 1 = 3(2m) + 1 = 6m + 1 — Call it Equation (1)

Well…6m is even. So, 6m + 1 is odd.

Therefore, 3n + 1 is ODD…because 3n + 1 = 6m + 1 from Equation (1).

By assuming ‘n’ is even, we’ve shown that 3n + 1 is ODD which is a contradiction to our assumption.

Therefore:

If ‘n’ is odd then 3n + 1 is even. This is the contrapositive of the statement to be proved.

Since the contrapositive is true, it follows that the original statement “if 3n + 1 is even, then ‘n’ is odd” is true.

The next example is a classic problem where an Indirect Proof is used.

Example 3

Prove that square root of 2 or SQRT (2) is irrational using an indirect proof.

Solution

ASSUME that the given statement is false.

That is:

SQRT(2) is NOT irrational.

Assume the contrary to be true…that is…SQRT(2) is RATIONAL.

Let’s try to prove it.

A rational number is a real number that can be expressed as a quotient of two integers a/b, where b does not equal 0.

We’ve assumed SQRT(2) to be a rational number.

So:

SQRT(2) = a/b. This fraction a/b is in lowest terms – that is, a and b have no common factors.

Multiply each side by ‘b’ to get rid of the fraction.

b × SQRT(2) = a

Square both sides.

SQR (b) × 2 = SQR (a) which is the same as:

2 SQR (b) = SQR (a) — call it Equation (2)

SQR(a) is even…because from Equation (2) above, we have, SQR(a) = 2 SQR(b)…a multiple of 2.

SQR(a) is even…implies…’a’ is even. Then, a = 2k for some integer ‘k’.

Substitute a = 2k in Equation (2). We get:

2 SQR (b) = SQR (a) — Equation (2)

2 SQR (b) = SQR (2k)

2 SQR (b) = 4 SQR (k)

Cancel ‘2’ on either side. We have:

SQR(b) = 2 SQR(k)

The above equation shows that ‘SQR(b)’ is even…because SQR(b) = 2 SQR(k).

Again, SQR(b) is even implies ‘b’ is even.

If ‘a’ and ‘b’ are both even, then they will have a common factor…

Then…how can the fraction a/b be in lowest terms?

A contradiction…

SO, SQRT (2) is IRRATIONAL.

Example 4

Prove that “For all integers ‘n’, if ‘n’ is odd then SQR(n) is odd” using an indirect proof.

Solution

Suppose the conclusion is false.

That is:

SQR(n) is NOT odd.

ASSUME the contrary… SQR (n) is even.

Then the statement contrary of the given statement is:

“For all integers ‘n’, if ‘n’ is odd then SQR(n) is even”

Let’s try to prove it.

If SQR(n) is even, then SQR(n) can be expressed as a multiple of 4.

So:

SQR (n) = 4k for some integer ‘k’.

Take square root on either sides of the equation. We get:

n = 2 SQRT (k)

The above equation shows that ‘n’ is even, because ‘n’ is a multiple of 2…

By assuming ‘SQR(n)’ is even, we’ve shown that ‘n’ is EVEN which is a contradiction to our assumption.

So:

If ‘SQR(n)’ is odd then ‘n’ is odd. This is the contrapositive of the statement to be proved.

Since the contrapositive is true, it follows that the original statement “If ‘n’ is odd then ‘SQR (n)’ is odd” is true.

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Math Dictionary