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bazaltina [42]
3 years ago
11

Let $f(x)$ be a function defined for all positive real numbers satisfying the conditions $f(x) > 0$ for all $x > 0$ and $f

(x - y) = \sqrt{f(xy) + 1}$ for all $x > y > 0$. Determine $f(2009)$.
Mathematics
2 answers:
algol [13]3 years ago
3 0
Suppose we choose x=1 and y=\dfrac12. Then

f(x-y)=\sqrt{f(xy)+1}\implies f\left(\dfrac12\right)=\sqrt{f\left(\dfrac12\right)+1}\implies f\left(\dfrac12\right)=\dfrac{1+\sqrt5}2


Now suppose we choose x,y such that

\begin{cases}x-y=\dfrac12\\\\xy=2009\end{cases}


where we pick the solution for this system such that x>y>0. Then we find

\dfrac{1+\sqrt5}2=\sqrt{f(2009)+1}\implies f(2009)=\dfrac{1+\sqrt5}2

Note that you can always find a solution to the system above that satisfies x>y>0 as long as x>\dfrac12. What this means is that you can always find the value of f(x) as a (constant) function of f\left(\dfrac12\right).
Crazy boy [7]3 years ago
3 0

Solution:

 It is given that, f(x) is a function such that, defined for all positive real numbers satisfying the conditions ,f(x) > 0 ,for all x > 0 , and also

    f(x-y)=\sqrt{f(xy)+1}\\\\x>0,y>0\\\\for, x=1, \text{and} y=\frac{1}{2}\\\\f(1-\frac{1}{2})=\sqrt{f(1\times \frac{1}{2})+1}\\\\f(\frac{1}{2})^2=f(\frac{1}{2})+1\\\\f(\frac{1}{2})=\frac{1+\sqrt{5}}{2}

Now, suppose

x=2009, y=0

f(2009-0)=\sqrt{f(2009*0)+1}\\\\f(2009)=\sqrt{f(0)+1}\\\\f(2009)=\sqrt{f(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}})+1}\\\\f(2009)=\sqrt{\sqrt{f(\frac{1}{\sqrt{2}}*\frac{1}{\sqrt{2}})+1}+1}\\\\f(2009)=\sqrt{\sqrt{f(\frac{1}{2})+1}+1}\\\\f(2009)=\sqrt{\sqrt{\frac{1+\sqrt{5}}{2}+1}+1}\\\\f(2009)=\sqrt\sqrt{\frac{3+\sqrt{5}}{2}}+1}\\\\f(2009)=\sqrt \sqrt{{5.236}{2}}+1}\\\\=\sqrt{3.6180}\\\\=1.9021

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The first and second year’s sales for a company were $328,000 and $565,000. The expenses for the first year were $117,000. The c
Serhud [2]
As a disclaimer, I can't say I'm completely confident in this answer. Use at own risk.

Formulas:

Year 1: 328,000 (sales) - 117,000 (expense) = 211,000 (profit)

Year 2: 565,000 (sales) - x (expense) = y (profit)

Net Profit: 211,000 + y = 113,000

Math

211,000 (profit y1) + 565,000 (sales y2) = 776,000

776,000 - 113,000 (net profit) = -663,000 (expenses)

Confirm:

Net Profit: 211,000 + y = 113,000 (listed in formulas, just a reminder)

Plug in: 565,000 (y2 sales) - 663,000 (our solution) = -98,000

211,000 (y1 net) + -98,000 (our plug in) = 113,000 (2 year net profit given to us)
6 0
3 years ago
How many times larger is the broadcast area of channel 19 than the broadcast area of channel 36?
GaryK [48]

Answer:

17

Step-by-step explanation:

You will have to subtract 36 and 19 and than you will get 17

8 0
2 years ago
If you rearrange the equation so w is the independent variable, then what is u?<br><br> -2u+6w=9
Serjik [45]

Answer:

u  = - ( 9-6w)/2

Step-by-step explanation:

to  evaluate for u in the expression -2u+6w=9 is simply to look for a way such that we would express u in terms of w and other variables.

solution

-2u+6w=9

-2u = 9 - 6w

divide both sides by the coefficient of u which is -2

-2u/u = 9 - 6w/-2

u  = - ( 9-6w)/2

therefore the value of u when rearranged in the equation -2u+6w=9 is evaluated to be equals to

u  = - ( 9-6w)/2

7 0
2 years ago
Josie is picking out wallpaper for a 9 feet by 10 feet accent wall in her new apartment. She has narrowed her choices down to th
elena55 [62]

Answer:

Josie should purchase floral daisy wallpaper.

Step-by-step explanation:

Area = Length × Width

The area of the Josie's accent wall = 9 × 10 = 90 feet²

The cost of Floral Daisy per square feet =\frac{\text{Area of the wallpaper}}{price}

                                                                    = \frac{(1.475\times 9.8)}{14.67}

                                                                    = \frac{14.455}{14.67}

                                                                    = 0.98534 ≈ $0.99/ft²

The cost of total wallpaper for the wall = 90 × 0.99 = $89.10

The cost of Flower Power per square feet = \frac{(8.33\times 12)}{99}

                                                                      = \frac{99.96}{99}

                                                                      = 1.0096 ≈ $1.01/ft²

The cost of total wallpaper for the wall = 90 × 1.01 = $90.90

total savings = 90.90 - 89.10 = $1.80

∵ The floral daisy wallpaper comes in size = 1.45 ft × 9.8 ft

∴ Josie needs the number of rolls of floral daisy wallpaper to cover her wall = 7

The cost of 7 rolls = 14.67 × 7 = $102.69

To cover her wall with flower power wallpaper, she needs the number of rolls = 2

The cost of 2 rolls = 99 × 2 = $198

We can see the cost of floral daisy wallpaper is less than flower power.

Therefore, Josie should purchase floral daisy wallpaper.

8 0
3 years ago
A = 1011 + 337 + 337/2 +1011/10 + 337/5 + ... + 1/2021
egoroff_w [7]

The sum of the given series can be found by simplification of the number

of terms in the series.

  • A is approximately <u>2020.022</u>

Reasons:

The given sequence is presented as follows;

A = 1011 + 337 + 337/2 + 1011/10 + 337/5 + ... + 1/2021

Therefore;

  • \displaystyle A = \mathbf{1011 + \frac{1011}{3} + \frac{1011}{6} + \frac{1011}{10} + \frac{1011}{15} + ...+\frac{1}{2021}}

The n + 1 th term of the sequence, 1, 3, 6, 10, 15, ..., 2021 is given as follows;

  • \displaystyle a_{n+1} = \mathbf{\frac{n^2 + 3 \cdot n + 2}{2}}

Therefore, for the last term we have;

  • \displaystyle 2043231= \frac{n^2 + 3 \cdot n + 2}{2}

2 × 2043231 = n² + 3·n + 2

Which gives;

n² + 3·n + 2 - 2 × 2043231 = n² + 3·n - 4086460 = 0

Which gives, the number of terms, n = 2020

\displaystyle \frac{A}{2}  = \mathbf{ 1011 \cdot  \left(\frac{1}{2} +\frac{1}{6} + \frac{1}{12}+...+\frac{1}{4086460}  \right)}

\displaystyle \frac{A}{2}  = 1011 \cdot  \left(1 - \frac{1}{2} +\frac{1}{2} -  \frac{1}{3} + \frac{1}{3}- \frac{1}{4} +...+\frac{1}{2021}-\frac{1}{2022}  \right)

Which gives;

\displaystyle \frac{A}{2}  = 1011 \cdot  \left(1 - \frac{1}{2022}  \right)

\displaystyle  A = 2 \times 1011 \cdot  \left(1 - \frac{1}{2022}  \right) = \frac{1032231}{511} \approx \mathbf{2020.022}

  • A ≈ <u>2020.022</u>

Learn more about the sum of a series here:

brainly.com/question/190295

8 0
2 years ago
Read 2 more answers
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