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The answer is 68.

Here's the second one :

"y can exceed x by no more than 200 units"......this is a fancy way of saying that

y - x  ≤ 200   

and  we know that

x + 2y ≤ 1600

And we wish to maximize this

14x + 22y - 900

Here's a graph of the constraints : https://www.desmos.com/calculator/dgtafe0gd7

Note that the max profit is found at the corner point (400, 600) ....this will always be true.....the max - or min - will always occur at a corner point....!!!!!!

abc eee

P.S. The Guest is correct in adjusting the formula and the result he/she obtained. However, the interpretation is wrong!. This formula calculates the monthly payment on a loan of $153,000 @4% compounded monthly over a period of 15 years, or 180 months.

These are always a little difficult, NSS....

Note that we make a $22 profit on the Type A printer and a $19 profit on the Type  B printer

And we can order no more than 120 total in one one month

Let x  be the number of Type A printers and y be the number of Type B printers

So.....we have these constraints

22x+ 19y ≥  2400  and

x + y  ≤ 120

And we wish to minimize the cost function given by

237x +  122y

Here's a graph of the constraints : https://www.desmos.com/calculator/qz01ip2whh

Note that the minimum cost occurs at the corner point corner point (40,80)  

And that cost is    237(40) + 122(80)  = $19240

So.....we should order 40 Type A printers and 80 Type B printers

Create a table for the function  y  =  x² + 7x + y     for y values from -7 to 0.

This seems to be a strange question because:  y  =  x² + 7x + y 

                          subtracting y from both sides:  0  =  x² + 7x

and the y-term disappears!

So, solving for x:                                                0  =  x(x + 7)

      this means that either  x = 0  or  x = -7.

Thus, for every y-value, x has two possible values, either 0 or -7.

To change 14 fluid ounces per minute into gallons per hour:

Since there are 128 fluid ounces in one gallon, we need to divide 14 by 128 to change ounces into gallons.

Since there are 60 minutes per hour, we need to multiply 14 by 60 to change per minute into per hour.

Combining these:  14 fl oz / min x (1 gal / 128 fl oz) x (60 min / hr)  =  6.5625 gal per hour.

Written as a mixed fraction:  6 9/16 gal per hour.

If x is the radius of the semicircle, 

1250π = (1/2) πr^2

r = 50 

Length of rectangle = 2r = 100 

Width of rectangle = r

Area = 100 * 50 = 5000

Solve for x:
4 x^2 + 12 x - 12 = 0

Divide both sides by 4:
x^2 + 3 x - 3 = 0

Add 3 to both sides:
x^2 + 3 x = 3

Add 9/4 to both sides:
x^2 + 3 x + 9/4 = 21/4

Write the left hand side as a square:
(x + 3/2)^2 = 21/4

Take the square root of both sides:
x + 3/2 = sqrt(21)/2 or x + 3/2 = -sqrt(21)/2

Subtract 3/2 from both sides:
x = sqrt(21)/2 - 3/2 or x + 3/2 = -sqrt(21)/2

Subtract 3/2 from both sides:
x = sqrt(21)/2 - 3/2       or         x = -3/2 - sqrt(21)/2

NSS.....the first two contraints, x ≥ 0 and y ≥ 0, just tell us that the portion of the graph that we are interested in lies in the frist quadrant........so.....I won't graph those

See the others, here :  https://www.desmos.com/calculator/dr3w5xlcw1

Any maximum  - or minimum - occurs at a "corner point" ......there is only one corner point on this graph at (2.5, 2.5).....and this is the point that maximizes 7x - 3y

Thank you for your "thank you".

  !

The semicircle of area 1250 pi centimeters is inscribed inside a rectangle. The diameter of the semicircle coincides with the length of the rectangle. Find the area of the rectangle.

1.7028239429265474453 * 10^0

I'm having a great day...

And you?

can u factorise x^2-4x+8

It does not have rational or even real factors.     (The discriminant is negative. 16=32=-32)

So you can only factor it using complex numbers.

mx2 + 2mx = 2x − 1 − m

$mx^2 + 2mx = 2x − 1 − m\\ mx^2 + 2mx - 2x +1+ m=0\\ mx^2 + (2m - 2) x+(1+ m)=0\\ \text{This will have at least one solution when }\triangle\ge 0\\ \triangle=b^2-4ac\\ (2m-2)^2-4m(1+m)\ge 0\\ (4m^2-8m+4)-4m-4m^2\ge 0\\ -12m+4\ge 0\\ -12m\ge -4\\ m\le \frac{1}{3}$

[2nd] is in the top left corner.

153000(.04/12)/[1-(1+.04/12)-12(15)]

This the TVM formula to calculate the monthly payment required to save $153,000 over a period of 15 years in the future. However, you have made a small mistake in writing it. It should be written like this:

$153,000*(0.04/12) / [1 -(1+0.04/12)^(-12*15)]

$153,000*(0.0033333) / [1 - (1.0033333)^(-180)]

$510 / [0.45063721031272090137130624075444]

=$1,131.72 - The monthly payment required to save $153,000 in the future.

how do i get the answer to: 5.478 to the power 967?

$5.478^{967} = 1.7680762752393208561357751278000043204596021262381226... \times 10^{714}$

$\begin{array}{|rcll|} \hline z &=& 5.478^{967} \quad & | \quad \log_{10} \\ \log_{10}(z) &=& \log_{10} ( 5.478^{967} ) \\ \log_{10}(z) &=& 967 \cdot \log_{10} ( 5.478 ) \\ \log_{10}(z) &=& 967 \cdot 0.73862202792 \\ \log_{10}(z) &=& 714.247500997 \quad & | \quad 10^{()} \\ z &=& 10^{714.247500997} \\ &=& 10^{714+0.247500997} \\ &=& 10^{714}\cdot 10^{0.247500997} \\ &=& 10^{0.247500997} \cdot 10^{714}\\ z &=& 1.768076275239\ldots \cdot 10^{714}\\\\ \mathbf{5.478^{967}} &\mathbf{=}& \mathbf{1.768076275239\ldots \cdot 10^{714}} \\ \hline \end{array}$

What is the answer to P versus NP?

... there is no answer.

The sequence x_1, x_2, x_3, . . ., has the property that x_n = x_{n - 1} + x_{n - 2}

for all n \ge 3.

If x_{11} - x_1 = 99, then determine x_6.

$\begin{array}{|rcll|} \hline x_3 &=& x_2 + x_1 &=& 1\cdot x_2 + 1\cdot x_1 \\ x_4 &=& x_3 + x_2 &=& 2\cdot x_2 + 1\cdot x_1 \\ x_5 &=& x_4 + x_3 &=& 3\cdot x_2 + 2\cdot x_1 \\ \color{red}x_6 &\color{red}=& x_5 + x_4 &=& \color{red}5\cdot x_2 + 3\cdot x_1 \\ x_7 &=& x_6 + x_5 &=& 8\cdot x_2 + 5\cdot x_1 \\ x_8 &=& x_7 + x_6 &=& 13\cdot x_2 + 8\cdot x_1 \\ x_9 &=& x_8 + x_7 &=& 21\cdot x_2 + 13\cdot x_1 \\ x_{10} &=& x_9 + x_8 &=& 34\cdot x_2 + 21\cdot x_1 \\\\ x_{11} &=& x_{10} + x_9 &=& 55\cdot x_2 + 34\cdot x_1 \quad & | \quad x_{11} = 99 + x_1 \\ 99 + x_1 &=& 55\cdot x_2 + 34\cdot x_1 && \quad & | \quad - x_1 \\ 99 &=& 55\cdot x_2 + 33\cdot x_1 && \quad & | \quad : 11 \\ \color{red}9 &\color{red}=& \color{red}5\cdot x_2 + 3\cdot x_1 \\\\ \hline \color{red}9 &\color{red}=& \color{red}5\cdot x_2 + 3\cdot x_1 && \quad & | \quad \color{red}x_6 = 5\cdot x_2 + 3\cdot x_1\\ \mathbf{9} &\mathbf{=}& \mathbf{x_6} \\ \hline \end{array}$

Note

(1 - p)  ( 1 - p + p^2 + p^3 + p^4 + p^5 + p^6)  =

( 1 - p + p^2 + p^3 + p^4 + p^5 + p^6)

(     -p + p^2 -  p^3 - p^4  - p^5  - P^6  - p^7  )  =

1 - 2p  + 2p^2  - p^7   

$$x_n = x_{n - 1} + x_{n - 2}$ for all $n \ge 3$$

x₁₁  =  99 + x₁

x₉ + x₁₀  =  99 + (x₃ - x₂)

(x₇ + x₈) + (x₈ + x₉)  =  99 + (x₅ - x₄) - (x₃ - x₁)

(x₅ + x₆) + (x₆ + x₇) + (x₆ + x₇) + (x₇ +  x₈)  =  99 + x₅ - x₄ - x₃ + x₁

3x₆ + 3x₇ + x₈  =  99  - x₄ - x₃ + x₁

3x₆  +  3(x₅ + x₆)  + (x₆ + x₇)  = 99 - (x₆ - x₅) - ( x₅ - x₄) + (x₃ - x₂)

7x₆  + 3x₅ + x₇  = 99  -x₆ +x₄ + x₃ - x₂

8x₆ + 3(x₇ - x₆) + x₇  = 99  + x₄ +( x₅ - x₄) - (x₄ - x₃)

5x₆ + 4x₇  =  99 + x₅ - x₄ + x₃

5x₆ + 4(x₅ + x₆)  = 99 + x₅ - (x₆ - x₅) + (x₅ - x₄)

10x₆ + 4x₅  =  99 + 3x₅ -x₄

10x₆ + x₅ = 99 - x₄

10x₆ + x₅ + x₄  = 99

10x₆  + x₆  =  99

11x₆  = 99             divide both sides by 11

x₆  = 9

It could be any letter grade

x^3  = 20x + 7y       (1)

y^3  = 7x + 20y       (2)

Add (1)  and (2)

x^3 + y^3  =  27x + 27y

Factor both sides

(x + y) (x^2 - xy + y^2)  =  27 (x + y)

(x^2 - xy + y^2 )  = 27          (3)

Subtract  (1) and (2)

x^3 - y^3  = 13x - 13y

Factor both sides, again

(x - y) (x^2 + xy  + y^2)  =  13 ( x - y)

(x^2 + xy + y^2)  =  13         (4)

Subtract   (3)  from  (4)

2xy  =  -14       divide by 2

xy  = -7