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Use the Cauchy Schwarz inequality!

Florist Florence buys flowers wholesale at $40 a bouquet. If she sells them at $50 a bouquet, she makes a $10 profit for each sale but she can only sell 500 bouquets. For every $1 she raises the price above $50, the number of bouquets she can sell will reduce by 10. To maximize her profit, what should be her selling price per bouquet? 

Her purchase price is the same irrespective of the selling price,

so let's just see what will maximize the total selling price. 

Price   Number Sold   Income  

  50           500            25,000 

  51           490            24,990 

  52           480            24,960 

  53           470            24,910 

I think I'm seeing a trend here.  A couple more... 

  60           400            24,000 

  70           300            21,000 

I've seen enough.  Have you?  She comes out best selling 500 for $50 each.  

_.

-8+-15 = -8-15 = -(8+15) = -23

So the ans is -23

f(x) = x^3 + 1

f(20)  = 8001

Top row : 1 2 -1 0 -2

Second : 2 -2 0 -1 1

Third : 0 -1 -2 1 2

Fourth : -2 0 1 2 -1

Fifth : -1 1 2 -2 0

P(x) = x^3/2 - 3x/2.

P(12) = 848

Use Pythagoras's Theorem:

Let the diagonal ==D

D^2 ==10^2  +  9^2

D^2==181    Take the sqrt of both sides

D ==Sqrt(181) ==13.4536 ==~13.5 inches - longest diagonal.

Don't undestand this:  1\tfrac{1}{2}1 2 1     and this:   \frac{2}{3} 3 2 ​

2x=14 x=7

1. Put it in a equation: 3n-x = n+x=8

2. Add them up:4n = 16 <=> n =4      Add x to both sides : 3n = n+2x <=> 2x=2n x =n
3. Since x=n and n=4 we have the answer: x=4

[1, 2, 3, 7, 8, 9, 10, 15, 16, 17, 22, 23, 24, 29, 30, 31,  36, 37, 38, 43, 44, 45]==22 such integers

They are ALL < 50

No two of them sum up to a multiple of 7

They form: 22 C 2 ==231 sums and non of them is a multiple of 7.

1000/19 =52 and a bit

9999/19=526 and a bit

abba  9*10 = 90 Possible palindromes.

1881, 3553, 5225, and one more. I will let you find it.

I looked at algebraic ways to do it but altimately it was just as easy (for me) to look at the possibilites prettuy much one by one.

One question per post, thanks.

Nathan starts with N toys 

Which means that Ricky starts with 3N toys

3N-k = 8        N+k=8

Solve simultaneously.

Let's use the hint, \(\text{If} \space \space t=\sqrt{x} \iff t^2=x\)

We get: \(lim_{t->3}\dfrac{sin(t-3)}{t^2-9}\)

Factor the denominator:

\(lim_{t->3}\dfrac{sin(t-3)}{(t-3)(t+3)}\)

Then, let's rewrite this as:

\(lim_{t->3}[\dfrac{sin(t-3)}{(t-3)}*\dfrac{1}{t+3}]\)

Using the following law of limits: \(lim_{x->a}[g(x)f(x)]=(lim_{x->a}g(x))*(lim_{x->a}f(x))\)

So we get:

\(lim_{t->3}[\dfrac{sin(t-3)}{(t-3)}*\dfrac{1}{t+3}]=[lim_{t->3}\dfrac{sin(t-3)}{(t-3)}]*[lim_{t->3}\dfrac{1}{t+3}]\)

Recall: \(lim_{y->0}\frac{sin(y)}{y}=1\)  (You could find the proof of this on youtube, but this is usually a standard result; that is, when you are solving a question, you do not need to prove this.)

Now, (t-3) as t-->3 will go to 0 and (t-3) as t goes to 3 will go to 0 as well, so basically this is the same as: \(lim_{y->0}\frac{sin(y)}{y}=1\) 

(Where y=t-3). Hence, the limit in the first bracket is just 1, and the second limit in the other bracket could be done by direct substitution.

So:

\(1*(\dfrac{1}{3+3})=\dfrac{1}{6}\) which is our answer! :)
I hope this helps!

hector spent a total of exactly $200 on shirts and pants. The shirts cost $!5 each and the pants cost $22 each. how many shirts did hector buy? 

Look at the total, it's 200.  It ends with a zero.  

So added together, the total of the costs of the shirts and the pants has to end with a zero. 

The only digit you can get at the end of the shirts total is a zero or a five. 

Five is no good because you can't get the other five as a multiple of 22. 

So, the number of shirts times 15 has to end with a zero. 

And the number of pants times 22 has to end with a zero, also. 

For the pants total to end with zero, it will have to be multiplied by either a 5 or a 0. 

Ten is no good because the total would be more than 200 so it has to be 5. 

That means Hector bought 5 pants for a total of 110.  That leaves 90 for the shirts.  

Shirts  =  90/15  =  6 shirts.  

_.

A chicago subway card is worth $11. For each ride on the subway, either $1.50 or $1.80 is deducted from the value remaining on the card. what is the least number of cents that could be left on the card after any number of rides?   

Assume you spend 1.50 as many times as you can. 

You ride 7 times for a total of 10.50 so you have 0.50 left over. 

You can use up another 0.30 of that 0.50 by substituting one 1.80 ride. 

Six rides at 1.50 plus one ride at 1.80 costs you 9.00 + 1.80 = 10.80.       20¢ left on the card.  

Let's put the totals in a tabulation, using a decreasing number of 1.50 rides. 

(7)(1.50) + (0)(1.80) = 10.50 +        0  = 10.50    0.50 left 

(6)(1.50) + (1)(1.80) =   9.00 +   1.80  = 10.80    0.20 left  

(5)(1.50) + (1)(1.80) =   7.50 +   1.80  =   9.30    1.70 left 

(4)(1.50) + (2)(1.80) =   6.00 +   3.60  =   9.60    1.40 left 

(3)(1.50) + (3)(1.80) =   4.50 +   5.40  =   9.90    1.10 left 

(2)(1.50) + (4)(1.80) =   3.00 +   7.20  = 10.20    0.80 left 

(1)(1.50) + (5)(1.80) =   1.50 +   9.00  = 10.50    0.50 left 

(0)(1.50) + (6)(1.80) =        0 + 10.80  = 10.80    0.20 left 

The table also contains an increasing number of 1.80 rides, so

all combinations of rides totaling nearly 11.00 are accounted for. 

So the least amount that can be left on the card is determined to be 20¢.  
_.

f(20) = 11.

Look at the first and the last numbers

Then look at the 2nd and the 2nd last numbers..........what do you see and what do you do?

the product of two consecutive negative integers is 272. find the larger of the two integers. 

Ignoring the polarity for the moment. 

Since they're consecutive, they're close to each other, but more important,

they're close to the square root.  So what's the square root of 272? 

If you don't know, you can use the calculator in your phone to find it.  It's about 16.49. 

So what two consecutive numbers close to 16 multiply to 272? 

Note the final digits of those two numbers will have to multiply to a number that ends in 2. 

How about 6 and 7, that's 42 so it sounds like a winner.  So are the numbers 16 and 17? 

Does 16 x 17 = 272?  Try it and see.  YES. 

Now going back, we have to remember that the polarity is negative. 

So, the larger number is the one that's less negative, so our answer is  –16. 

_.

Thank you!

D = x - 4

R = D + 2x

S = R - 5

R + 2S + 3D = D + 2x + 2(D + 2x - 5) + 3D   =  6D + 6x - 10 = ... can you finish from here?

Hello Keihaku!

\((4x^{9/2})(\frac{1}{2}x^{1/2})=2x^{\frac{9+1}{2}}=\color{blue}2x^5\)

  !