Stefani Germanotta (better known as Lady Gaga) was planning her post Grammy concert in New England. Her newly hired manager, the famous young math whiz named Lynnet did some statistical analysis and told Lady Gaga that in Boston, MA there are about 10,000 fans who are likely to attend her concert even if they need to drive for two hours, in Hartford, CT - 9,000 fans and in Providence, RI - 8,000 fans who are similarly so eager to see Gaga that they could drive almost anywhere in New England. Each of these cities have large concert halls capable to fit audience of up to 30,000. Distance between Boston and Providence is about 50 miles, Boston and Hartford is 100 miles and Hartford and Providence is approx. 80 miles. Lynnet and Lady Gaga met in the office and contemplated where to hold the concert in order to minimize the total amount of miles travelled by all the fans. Lady Gaga concluded: Ok, then Boston it is, as the city with the largest amount of fans. But Lynnet convinced her otherwise. How? And what city the concert is going to be?
Answers accepted all day long on Friday, on our Family Puzzle Marathon. They will be hidden until Saturday morning (EST) and everyone who contributed something reasonable will get a puzzle point. Please, explain your answer.
15 comments:
The concert should be held in Providence.
To be in Boston, the 9,000 Hartford fans would drive 100 miles each and the 8,000 Providence fans would drive 50 miles each. This is a total of 1.3 million fan-miles.
Having the concert in Providence, which is more centrally located between the other two cities, would have 10,000 Boston fans driving 50 miles each and 9,000 Hartford fans driving 80 miles each, for a total of only 1.22 million fan-miles.
Similarly, having everyone drive to Hartford would have the Boston and Providence fans driving a total of 1.64 million fan-miles.
So, having the concert in Providence would minimize the total number of miles traveled by all the fans.
The concert should be in Providence.
If in Boston total mileage for fans traveling would be 1,300,000:
(8000x50) + (9000x100)
If in Hartford total mileage would be 1,640,000:
(10,000x100) + (8,000x80)
If in Providence total mileage would be 1,220,000:
(10,000x50) + (9000x80)
Most artists would select the hall with the nicest hotel rooms nearby, the nicest backstage facilities, the best sound.light systems and crew. But I'm pretty sure this isn't the answer that's wanted.
Another unwanted answer is, ticket prices, which may very well not be the same in the 3 cities.
Providence. For Providence, the travel total is (50 miles)(10,000 people) from Boston, for 500,000 people miles PLUS (80 miles)(9000 people) from Hartford, for 720,000 people miles, TOTALLING 120,000 people miles. The same calculation for Boston gives you 130,000 people miles. For Hartford, 1,640,000.
The best location for the concert is Providence. The following matrix multiplication depicts the total number of miles the fans need to travel should the concert be in one of the 3 mentioned cities. The smallest number of total miles (1220k) corresponds to Providence.
| 0 100 50| |10k|
|100 0 80| x | 9k| = |1300k 1800k 1220k|
| 50 80 0| | 8k|
The concert would be in Providence.
If in Boston total number of miles traveled by fans would be 1,300,000:
(8000x50) + (9000x100)
If in Hartford the total mileage would be 1,640,000:
(10,000x100) + (8000x80)
If in Providence the total mileage would be 1,220,000:
(10,000x50) + (9000x80)
The answer is Providence, because the total mileage that will be needed to get there will be lowest.
Providence (10,000*50m + 9,000*80m) = 1,220,000m
Boston (9,000*100m + 8,000*50m) = 1,300,000m
Hartford (10,000*100m + 8,000*80m) = 1,640,000m
In order to minimize the total distance, we find out quite simply how many fans there are * how many miles they need to drive depending on which city the concert is held in.
Just looking at the map, my initial 'guess' would be that Lynnet thinks the concert should be held in Providence.
This would mean that 10000 fans from Boston would drive 50 miles (or 500,000 miles in total)
and 9000 fans from Hartford would drive 80 miles (or 720,000 miles in total).
Therefore, in total, it would be 1.22 million miles.
Just to make sure we're on the right track, I'll calculate the total miles if Hartford was the concert city.
8000 fans in Providence would drive 80 miles so 640,000 miles in total and 10000 fans from Boston would drive 100 miles or 1 million miles in total.
Therefore, in total, its 1.64 million miles.
Lastly, if the concert city was in Boston:
9000 * 100 = 900,000 miles
8000 * 50 = 400,000 miles
1.3 million miles.
Therefore, Lady Gaga should have the concert in Providence to minimize the total number of miles traveled for her fans.
I convinced her that she should hold the concert in Providence.
If the concert is held in Boston - fans from Hartford have to drive 900,000 mi
fans from Providence have to drive 400,000 mi
which totals 1,300,000 miles driven
If the concert is held in Hartford - fans from Boston have to drive 1,000,000 mi
fans from Providence must drive 640,000 mi
which totals 1,640,000 miles driven
If the concert is held in Providence - fans from Boston need to drive 500,000 mi
fans from Hartford drive 720,000 mi
which totals 1,220,000 miles driven
Providence is where the concert should be held to minimize distance driven
Lynnet is correct. They should hold the concert in Providence, RI where the average driving distance per fan is only about 45.1 miles. The average driving distance per fan if held in Boston would be about 48.1 miles, and if held in Hartford, about 60.7 miles.
Here are my calculations:
Average driving distance to Boston:
[(10,000x0)+(8000x50)+(9000x100)]/27,000 total fans = 48.1, rounded to nearest tenth.
Average driving distance to Hartford:
[(9000x0)+(10,000x100)+(8000x80)]/27,000 = 60.7
Average driving distance to Providence:
[(8000x0)+(10,000x50)+(9000x80)]/27,000 = 45.1
For Boston, we have: 10000*0+8000*50+9000*100=1300000
for Providence: 8000*0+10000*50+9000*80=122000
for Hartford: 9000*0+10000*100+8000*80=1640000
To minimize the amount of miles travelled by all the fans, Providence is a better choice than Boston.
Now, we do not know if the fans living in Hartford are willining to travel and to maximize the profit from the concert, Hartford will be a better choice.
Lynnet wants to accomodate evrybody so it is Providence.
The concert should be held in Providence. It's more central location outweighs its smaller fan base.
Here are the calculations:
If we hold the concert in Boston, we have 9000 fans traveling 100 miles (from Hartford) and 8000 fans traveling 50 miles (from Providence) for a total of 1.3 million miles
If we hold it in Hartford, we have 10,000 fans traveling 100 miles from Boston and 8000 fans going 80 miles from Providence for a total of 1.64 million miles
BUT if we hold it in Providence, we have 10,000 fans traveling 50 miles from Boston and 9000 fans traveling 80 miles from Hartford for a total of 1.22 million miles
(Assumption: The fans in the local area are traveling a small enough (and/or similar enough) number of miles as to not affect the outcome -- there is an 80,000 mile difference between the calculations for Boston and Providence. It could easily be the case that the Boston-based fans travel about 8 miles more on average than the Providence-based fans. But let's ignore this.)
I think the concert should be held in Hartford. Here's what I did.
I drew a circle around each city extending about 100 miles from the center of each city.(I figure that was about a 2 hour drive)
I then drew an obtuse triangle connecting the 3 cities and made a mark at the midpoint of each side of the triangle. From there I drew lines through the midpoints of the triangles.
Since about 1/3 of the circle surrounding Boston is the Atlantic Ocean, Boston's 10,000 fans come from the North,West And East. The fans from the South and West would be within 100 miles of Hartford.
Providence also is bordered by the Atlantic Ocean(looks like about 40 miles to the South)so the fans that hail from the West and South are easily within 2 hours of Hartford.
If the concert was held in Boston or Providence, more than half of Hartford's fans would have a greater than 2 hour drive.
Having it in Hartford would keep drive times less than 2 hours for the vast majority of Lady Gaga fans.
Anonymous above, please share with us your name or nick name so we can attribute all your insightful comments to someone specific and add up your puzzle points. You can get a prize after 5 and a puzzle made about you after 10. Your solution to this puzzle is very interesting as it brings the story even closer to reality. Everyone should read and visualize the Anonymous solution from above. A big fat puzzle point for A!
Tom's notes are very valuable as he is a musician and has a real showbiz experience. He throws in a few more variables: price of the tickets in each city, quality of the hotel rooms for the artist, sound system and crew availability. See, Lynnet, there is so much to learn for us. A puzzle point for Tom for mentoring.
The rest of you all correctly calculated what Lynnet had in mind - making the concert in Providence will minimize the total combined number of miles traveled by all the fans. Congratulations on another puzzle point for: SteveGoodman18, Annie, Tom, Bean, -lex-, Marina M, Wang, Donna, annie-marie, Kim and of course Lynnet.
It looks like this is 50th puzzle point for Kim. We have to celebrate next Friday.
And Tom just earned his 20th point. A double celebration!
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