CPhill

x^4-x^3=500

The answer that Anonymous gave, (5), IS correct. The question is....how did we arrive at that ??

Rearranging by subtracting 500 from both sides, we have

x^4-x^3 - 500 = 0

We have a theorem in math known as the "rational zeros" theorem that says that IF we have any "rational" solutions to the above (solutions that can be expressed as a fraction), these solutions come from all the possible factors of 500 divided by all the possible factors of the coefficient in front of the x^4 term, i.e., "1."

Of course, listing all the possible factors of 500 would be lengthy, but we can list the first few...we have ±1, ±2, ±4, ±5, etc. and all the possible factors of 1 are just ±1.

So..our list of (possble factors of 500)/(possible factors of 1) would include ±1, ±2, ±4, ±5, etc.

And we would find, by substituting these for "x' - one at a time, of course - into the above equation would provide us with our (only) "rational" solution of x = +5. Note that +5 is "rational," because we can express it as (+5)/(+1).

Note one thing......the "rational zeros" theorem doesn't guarantee  that we have any "rational" solutions....it just says that, if we do, they will come from the list of (all possible factors of 500)/(all possible factors of 1).

(Another theorem called the "bounds on zeros" theorem will tell us when we can stop searching for any additional zeros.)

whats the volume of a cirlcle with the diameter of 25.4cm

Well.....we don't have a "volume" for a circle...."volume" ony applies to three-dimensional objects. In a circle, we can find its "area" - a two-dimensional measure.

The area is given by (pi) * (radius)^2   ...Note that, since our diameter is 25.4cm, the radius is half of that, i.e., 12.7cm.....and using (3.14) for (pi), we have

(3.14) * (12.7cm)^2 =  506.45cm²

If we wanted to, we could "round" this to either 506.5cm² or just 506cm²

That's it....

A. Cube Net has been buying 5-pound bags of coffee from Jamaica for $0.25 each. You know a farm in Costa Rica that sells 8-pound bags for $0.32 each. Which is the best company to use and why?

B. Both farms charge 6.5% sales tax for buying their products. How much would 700 pounds of coffee cost you from only the company you selected?

C. The farm you chose also charges $6 to ship a 100-pound package to California. What is your total cost for 700 pounds of coffee now?

Well...let's see

The coffee from Jamaica must cost 5 cents per pound and the coffee from Costa Rica must cost 4 cents per pound. All things being equal, the Costa Rican coffee is the better deal.

To answer the second part....... we have..... 700lbs.*(.04 per pound) *(1 + .065) = $29.82

Since it costs $6 to ship 100lbs.....we're shipping 7 times that much, so our cost must be 7 * $6 = $42

So....adding the cost with the additional sales tax plus the shipping cost, we get $29.82 + $42  = $71.82

And that's it.....

Mmmm....let's look at this one again....

We need to solve 

(A− I)u=0

1 -2i    -5                      x       =          0

1        -1 - 2i                 y       =          0

This generates the following system

(1 - 2i) x - (5)y          = 0

1x           -(1 + 2i)y    = 0

Using the second equation, we have

x = (1 + 2i)y

And if we let y = some convenient value, say, 1....then x  = (1 + 2i)  and we have the following eigenvector associated with   lambda = (1 + 2i)

(1 + 2i)

     1

BTW....you can check this solution in the 1st equation of the system.....I believe it "works"

To find the eigenvector associated with the second lambda, (1 - 2i), we follow a similar process and we have

1 +2i    -5                      x       =          0

1        -1 + 2i                 y       =          0

This generates the following system

(1 + 2i) x - (5)y       = 0

1x           -1y + 2iy  = 0

And using the second equation, we have

x = (1 - 2i)y

And if we let y = some convenient value, say, 1....then x  = (1 - 2i)  and we have the following eigenvector associated with   lambda = (1 - 2i)

(1 - 2i)

     1

As before....you can check this solution in the 1st equation of the system.....

I think this is correct, but my Linear Algebra skills have been sitting in the closet for awhile...

Thanks for that answer, Melody.....Note something interesting.....The time it takes to double his money under these conditions is independent of what he started with...In other words, it would take him as long to double $1M as it would $500......... ( )

The break-even markup was 40 percent of the selling price. If the total income for a day on which the store broke even was $6400, what did the store owner pay for the items sold?

Let me take a stab at this one. Let's suppose we had a store and that every item sold for $1. Then...40% of the selling price must just be 40 cents. Then, we must have paid 60 cents for each item because our cost (per item) - 60 cents - plus the markup (per item) - 40 cents = selling price = $1.

Then we must have sold 6400 items, and since we paid 60 cents for each, our total cost would have been 6400*(.60) = $3840. Note that we must have fixed costs of $(6400 - 3840) = $(2560) per day. This makes sense because our cost function is .60(x) + 2560  and our revenue function is just 1(x). If x represents the number of items sold in a day, these "curves" are equal when x = 6400 !!! Thus we "break even" at that point. Also note that our fixed cost is 40% of our break-even revenue. The more items that we sell above 6400, the less our fixed cost is as a % of our revenue. For example, at 10,000 items, our fixed cost is only 25.6% of the revenue.

I think that's it......but I'll consider constructive criticism, too......

Find x when -1/2(2x-3)+1/3(4-6x)=1/4(7-2x)-3

OK...to make things a little easier, let's multiply through by the common denominator of 2, 3 and 4, namely - 12. Doing this will get rid of those pesky fractions !!

So we have

(12)*(-1/2)(2x-3)+(12)*(1/3)(4-6x)=(12)*(1/4)(7-2x)-3     =

-6(2x-3) + 4(4-6x) = 3(7-2x) -3  ......  Now, let's "distribute" the "coefficients" outside each parenthesis "across" the terms inside.....this gives....

-12x + 18 + 16 - 24x = 21 -6x - 3  Now.....combining "like" terms, we have

-36x + 34 = 18 - 6x   ......add  36x to each side

 34 = 18  + 30x  .......subtract 18 from both sides....this gives

16 = 30x

Now dividing by 30 on both sides, we have

x = 16/30  which reduces to 8/15

Now, let's check our answer using this....-6(2x-3) + 4(4-6x) = 3(7-2x) -3   so we have

-6(2(8/15) -3) + 4 (4 -6(8/15)) = 3(7- 2(8/15)) - 3   ???

-96/15 + 18 + 16 - 192/15 = 21 - 48/15 - 3 ???

-288/15 + 34 = 18 - 48/15   ???

16 = 240/15  ???

16 = 16  .......√ √ √

And we're finished !!!

Twenty-five minus four b equals negative six b plus forty-seven

Ok......let's write this in "math" fashion....so we have

25 - 4b = 6b + 47    Since, I assume, we want to "solve" for b, let's add 4b to both sides...this gives

25 = 10b + 47  ....Now, subtract 47 from both sides...we have

-22 = 10b  ......Now divide by 10.... we have

-22/10 = b....let's "reduce" this to....... b = -11/5

Now...let's check our answer

25 - 4(-11/5) = 6(-11/5) + 47   ???

25 + 44/5 = -66/5 + 47  ???

110/5 = 22   ???

22 = 22

And our job is done......don't forget to tip your waitress!!!  

Just "distribute" the "4" across the "x" and the "-5"

I know "zip" about Physics, but even I learned something here!!

Thanks, Rom !!!