Bosco

You lost me at the third line, i.e.,  

2⁰ + 2¹ + 2² + 2³ + ... + 2¹⁰⁰  =  1 + 2 + 4 + 8 + ... + 1024   

2¹⁰⁰ is more than 1024, a lot more.  But 2¹⁰ is 1024.  

Is it possible that a zero was dropped during the calculation?   

Or am I misreading something?  

_.

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 

_.

In a store window, there was a box of berries having a total weight of 200 kg.  The berries were 99% water, by weight.  After two days in the sun, the water content of the berries was only 90%, by weight.  What was the total weight of the berries after two days, in kg?  

You're starting with 200 kg of berries, and 99% of that is water.  

Since they're 99% water, that means that there's 1% of solid stuff.  

1% of 200 kg means that there are 2 kg of solid stuff.   

The water evaporates in the sun, but the solid stuff doesn't.  

So there still remains 2 kg of solid stuff.  

Since the new water content is 90%, that means that now the solid stuff is 10%.  

So 2 kg (the solid stuff) is 10% of the total weight.  

That means the new total weight is 20 kg.  

_.

if the car is going 40mph and it range it 135 miles how long will it drives  

start by considering the basic relationship   

Distance equals Velocity times Time                          D  =  V • T   

divide both sides by V to solve for T                          T  =  D / V   

                                                                                             135 miles  

                                                                                  T  =  ––—–––––––   

                                                                                           40 miles/hour  

                                                                                  T  =  3.375 hours  

 converting 0.375 hr to minutes                                 T  =  3 hrs, 22 mins, 30 seconds  

_.

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

The 375 in Answer #1 is correct.  

Check it at  https://www.symbolab.com/solver/expand-calculator  

_.

Find the number of ways of arranging the numbers 1,2 ,3, 4, 5, 6, 7, 8, 9   

in a row so that the product of any two adjacent numbers is even.  

For two numbers to have an even product, at least one of them must be even.  

That is, an odd number can never be adjacent to another odd number.  

So, odds and evens have to alternate, such as the example shown in the problem.  

Furthermore, since there are more odd numbers, the first number has to be odd.  

There are five odd numbers, and five positions into which they can be situated.  

So, there are 5 • 4 • 3 • 2 • 1 = 120 ways the odd numbers can be arranged.  

And, there are four even numbers, and four positions where they can be located.  

So, there are 4 • 3 • 2 • 1 = 24 ways that the even numbers can be arranged.  

Any even number arrangement can be combined with any odd number arrangement.  

Therefore, the number of possible mixes is 120 • 24 = 2880.   

_.

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  

_.

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 to 3/6 from 1/6.  

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

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

_.

A candy bar and a piece of bubble gum together cost 1.50.  

The candy bar costs a dollar more than the bubble gum.  

How much does the piece of bubble gum cost, in cents?  

Call the cost of a candy bar "C"  

Call the cost of a bubble gum "B"  

                                                                      C + B  =  1.50  

                                                                      C – B  =  1.00  

Just subtract the second from the first               2B  =  0.50  

                                                                             B  =  0.25  =  25¢  

_.

I drove to the beach at a rate of 40 miles per hour.  If I had driven at a rate of 42 miles per hour instead, then I would have arrived 45 minutes later.  How many miles did I drive?  

You mean 45 minutes earlier.  

                                                        D  =  V * T  

case 1                                              D  =  40 * T  

case 2                                              D  =  42 * (T – 45)  

Set the two equal                             42 * (T – 45)  =  40 * T  

                                                         42T – 1890  =  40T  

                                                            2T  =  1890  

                                                              T  =  945 minutes  =  15.75 hours  

Plug T back into original D = V * T        D  =  40 mi/hr * 15.75 hr  =  630 miles  

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