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If  \(a + b = 7\) and \(a^3 + b^3 = 44\),
what is the value of the sum \(\dfrac{1}{a} + \dfrac{1}{b}\)?
Express your answer as a common fraction.

\(\begin{array}{|rcll|} \hline (a+b)^3 &=& a^3+b^3+3ab(a+b) \quad | \quad a + b = 7,~a^3 + b^3 = 44 \\ 7^3 &=& 44 +21ab \\ 21ab &=& 343 - 44 \\ \mathbf{ ab } &=& \mathbf{ \dfrac{299}{21} } \\ \hline \dfrac{1}{a} + \dfrac{1}{b} &=& \dfrac{a+b}{ab} \quad | \quad a + b = 7,~ ab=\dfrac{299}{21} \\\\ &=& \dfrac{7*21}{299} \\\\ \mathbf{ \dfrac{1}{a} + \dfrac{1}{b} } &=& \mathbf{ \dfrac{147}{299} } \\ \hline \end{array}\)

I pick two whole numbers x and y between 1 and 10 inclusive (not necessarily distinct). My friend picks two numbers x - 4 and 2y + 1. If the product of my friend's numbers is one greater than the product of my numbers, then what is the product of my numbers?

Here's a diagram.

D is on side AC of Triangle ABC. We know [ABC] = 120, AD = 8, DC = 3. What is [ABD]?

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AC = 11       [ABC] = 120      AD = 8

AB = 240 / 11 = 21 ⁹/₁₁

[ABD] = 1/2(8 * 21 ⁹/₁₁) = 87 ³/₁₁

YEP...I missed that !   THANX     I edited my answer .....   ~ EP

Arrange in standard circle form to find r^2     first divide by 2

x^2 + 5x     + y^2  - 3y   = 24        complete the square for x and y

(x+ 2.5)^2   + ( y - 1.5)^2   = 24 + 6.25 + 2.25

                                              r^2 = 32.5                        pi r^2 = area = 32.5 pi units2       *** edited ***   to fix error noted by  catmg !

(1) x^3    + 5x^3   =   6 x³

16/56 = 2/7         x⁷ / x⁴      =  x^7-4 = x³

2x³ / 7

(x+3)^2 =(x+3)(x+3)  =  x^2 + 6x + 9   

You don't have the diagram included.....    Generally a square will enclose the max area.... if one side is a building, then a square with 148/3 = 49 1/3 ft sides would be the biggest .....   area would be     49 1/3 * 49 1/3   ft²

You're welcome. :))

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Nice. :))

I got the same answer, but I think you forgot to divide the 48 by 2. 

x^2 + 5x + y^2 - 3y = 24

(x + 2.5)^2 + (y - 1.5)^2 = 24 + 1.5^2 + 2.5^2

(x + 2.5)^2 + (y - 1.5)^2 = 32.5

area = 32.5pi

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Every $N$ possible is divisible by 3 bc every number that has digits adding to a multiple of three is divisible by 3. so the answer is 24 bc 24 is four factroial which is the number of possiblities there are.

Expanding and simplifying results in   5 x^2 + 20 x + 282         min occurs at x = - b/2a = -2 

   sub in that value for 'x' to calcualte the min value of the fxn = 262   

Setting the derivative to 0, we get minimum value of 284.

Arrange in standard circle form to find r^2     first divide by 2

x^2 + 5x     + y^2  - 3y   = 48      complete the square for x and y

(x+ 2.5)^2   + ( y - 1.5)^2   = 48 + 6.25 + 2.25

                                              r^2 = 56.5                        pi r^2 = area = 56.5 pi units²

Thank you both very much, 6th grade math is kind of hard.

hm, incorrect

Thank you very much.

The maximum value of x^2 + 5x + 6 is 182.

how to rewite 10x + 4 - 3 + 8x + 2, x2 + 4 + 2(x - 1) + 3 + 2x, and 3x2 - 48x?   

First, group terms of the same power    

10x + 8x + 4 – 3 + 2                18x + 3              3 • (6x + 1)  

x² + 4 + 2x + 2x – 2 + 3                                    x² + 4x + 5  

3x² – 48x                                                         3x • (x – 16)    

Volume, just multiply all 3 lengths together. 8 1/2 * 8 1/2 * 12 2/3

Surface area ad all the combinations of multipling 2 terms times 2. 2((8 1/2 * 8 1/2) + (8 1/2 * 12 2/3) + (8 1/2 * 12 2/3))

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I'll show you 1, try to do the other one by yourself. :))

I'm also assuming you meant. 

3x^2 - 48x

3x(x - 16)   # factor 3x from both sides

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Answer:

32

Step-by-step explanation:

p=sample proportion

p=number of defective parts/size of sample = 4/25

Estimate:

(sample proportion)x(size of population) = (4/25)x(200)=32

fish tested = 5x5=25

fish population = 40x5=200

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There are 35 possible values of N.