Bosco

I drove to the beach at a rate of 40 miles per hour.  If I had driven at a rate of 50 miles per hour instead, then I would have arrived 45 minutes later.  How many miles did I drive?  

You mean 45 minutes earlier.  Obviously, if you drive faster, you get there faster.  

This problem makes use of

the following relationship:                   Distance = Velocity x Time 

                                                           D  =  V • T  

case 1                                                 D  =  (40) • (T)  

case 2                                                 D  =  (50) • (T – 45)  

Since the Distance, D, is the  

same for both cases, let's set       

the "V•T"s equal to each other.             (50)(T – 45)  =  (40)(T)  

                                                               50T – 2250  =  40T  

Subtract 40T from both sides                  10T – 2250  =  0  

Add 2250 to both sides                                       10T  =  2250  

Divide both sides by 10                                           T  =  225   (this is in minutes)  

Divide minutes by 60 to get hours          225 minutes  =  3.75 hours  

Plug this T back into original equation                     D  =  (40 mi/hr) • (3.75 hr)  =  150 miles  

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Find the coefficient of y^4 in the expansion of (2y-5+3y^2)^6.  

I'm just copying/pasting the answer to this same question  

that I answered 6 hours ago.  

That's too much multiplication to try to do it manually.  

So, I used a handy tool that has been put at our disposal,  

i.e, https://www.symbolab.com/solver/expand-calculator    

I found that the coefficient is 375.  Go to there and see for yourself.  

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The Tasty as Pi Bakery has two locations! Each location offers the same flavors. Both locations display one pie of each flavor in a rectangle.  

The downtown location has 4 rows in its display. The uptown location has 6 rows in its display. If one of the locations has a square display, how many pies are in each of the other location's rows?  

1.  Each location offers the same flavors.  

2.  Both locations display one pie of each flavor. 

3.  So, both locations display the same number of pies. 

4.  Both locations displays are rectangles.  

5.  One location has a square display.  

If the downtown location has the square display,  

then it has a four by four display, total 16 pies.  

Then, the uptown display must have the 6 rows. 

Have to fit 16 pies into 6 rows?  It doesn't come  

out even, so this one cannot be a rectangle..  

Leave that for now, while we look at the other option.  

If the uptown location has the square display,  

then it has a six by six display, total 36 pies.  

Then the downtown display is the one with 4 rows. 

If they put 9 pies in each row, that's 36 pies, too. 

I think this makes a case for each location displaying 36 pies each.  

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Find the coefficient of y^4 in the expansion of (2y-5+3y^2)^6.  

That's too much multiplication to try to do it manually.  

So, I used a handy tool that has been put at our disposal,  

i.e, https://www.symbolab.com/solver/expand-calculator    

I found that the coefficient is 375. Go to there and see for yourself.  

_.

An aquarium tank is 1/6 full of water. When 2 gallons of water are added, the tank becomes 1/2 full. What is the total capacity of the aquarium tank, in gallons?  

Note that 1/2 is the same as 3/6.  

So 2 gallons took the tank from 1/6 to 3/6. 

So 2 gallons added 2/6.  That means 1 gallon is 1/6.  

Therefore, the total capacity of the aquarium is 6 gallons.  

_.

In Linguistics 101, the ratio of the number of juniors to the number of seniors is 3:2.  When six more juniors join the class, and one senior drops the class, the ratio of the number of juniors to the number of seniors becomes 2:3.  How many students are in the class after these changes?  

                                                        J          3  

                                                       ––   =   ––  

                                                        S         2  

                                                       2J  =  3S  

                                                      (J + 6)         2  

                                                      –––––   =   ––   

                                                      (S – 1)        3  

                                                      3J + 18  =  2S – 2  

                                                        J  =  3S/2  

                                                       (3)(3S/2) + 18  =  2S – 2  

                                                        9S/2 + 18  =  2S – 2        

Multiply everything by 2 to    

get rid of the denominator                   9S + 36  =  4S – 4  

 I saw here that the number of Seniors is going to be negative  

I went back and considered the original proposal.    

If you've got Juniors to Seniors = 3 to 2  then add more Juniors  

than Seniors, the proportion could change, but it can not invert. 

This problem cannot be solved.  

_.

3 is immediately seen obviously incorrect because  

the sqrt(12) by itself is already more than 3, before  

you start adding all those other things to it.    

(e^2+1+2e/4) *   e^-1  

This needs some more parentheses to clarify exactly what is wanted.  

The problem I'm having specifically is the 2e/4 term.  

Why wouldn't that have just been called e/2 instead.  

The only reason I can think of for writing 2e/4 instead of simply e/2   

is that the denominator 4 applies to everything that comes before.  

Thusly                                         e² + 1 + 2e         e^–1  

                                                  —–————  •  ———  

                                                           4                  1  

That can be written as                 (e + 1)²   

                                                   –––––––  

                                                        4e  

I don't know if this is correct ... "correct" meaning what was  

actually wanted, so I'm not going to paint it red like I usually do.  

_.

Find the ordered triple (p,q,r) that satisfies the following system:  

p - 2q = 3  

q - 2r = -2 + q  

p + r = 9 + p  

Per the second line       –2r = –2  therefore  r = 1  

Per the third line,                                          r = 9  

You can't have r equal two different amounts,  

so the system as stated can not be solved.   

_.

Will and Grace are canoeing on a lake.  Will rows at 50 meters per minute and Grace rows at 20 meters per minute. Will starts rowing at 2 p.m. from the west end of the lake, and Grace starts rowing from the east end of the lake at 2 p.m. If they always row directly towards each other, and the lake is 3800 meters across from the west side of the lake to the east side, at what time will the two meet?  

Will's rate is 50 m/min.  

Grace's rate is 20 m/min.  

They're rowing toward each other,  

so their rate of closure is 70 m/min.  

The lake is 3800 meters across.  

The time it will take them to meet is given by  

                                                                                    Distance  

                                                                            T  =  –––––––  

                                                                                       Rate  

                                                                                    3800 m  

                                                                            T  =  –––––––  

                                                                                    70 m/min  

The time it takes them to meet                             T  =  54.29 min  

Note that (0.29 min • 60 sec/min)  =  17 seconds

The time on the clock is                                             2:54:17 pm 

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