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pychu [463]
2 years ago
11

Prove 2√(x) + 1/√(x + 1) <= 2√(x+1) for all x in [0,inf)

Mathematics
1 answer:
EleoNora [17]2 years ago
3 0
Start by multiplying each side of the inequality by \sqrt{x + 1} and simplifying:

2 \sqrt x + \frac{1}{\sqrt{x+1}}  \leq  2 \sqrt{x + 1}
(2 \sqrt x + \frac{1}{\sqrt{x+1}})(\sqrt{x + 1}) \leq (2 \sqrt{x + 1})(\sqrt{x + 1})
2 \sqrt{x(x + 1)} + 1 \leq 2(x + 1)
2 \sqrt{x^2 + x} + 1 \leq 2x + 2
2 \sqrt{x^2 + x} \leq 2x + 1
\sqrt{x^2 + x} \leq x + \frac{1}{2}

From here, we can square both sides to get

x^2 + x \leq (x + \frac{1}{2})^2
x^2 + x \leq x^2 + x + \frac{1}{4}
0  \leq \frac{1}{4}, which is always true.
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Answer:

Yes, you can use this inequality to find the numbers of cars required.

Step-by-step explanation:

12  + 3n > 28     where n = the number of cars required

3n > 28 -12

3n > 16

n > 5 1/3

Greater than 5 1/3 gives 6 cars.

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3 years ago
The perimeter of the entire garden.
Leno4ka [110]
Add up all the lengths of the garden and you will be able to get the perimeter.
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1 year ago
It would take Delia 3 hours longer to re tile their bathroom by herself than it would for kari to retile on her own. If they wor
max2010maxim [7]

Answer:

6hours

Step-by-step explanation:

From the given information:

Suppose it took Karl x hours to retile her bathroom,

Then it will take Della 3 hours longer i.e (3+x) hours

If Della and Karl work together or will take them 2 hours

The objective is to determine how long it will take Delia to retile the bathroom alone?

∴

\dfrac{1}{x}+\dfrac{1}{3+x}= \dfrac{1}{2}

\dfrac{x+3+x}{x(3+x)}= \dfrac{1}{2}

\dfrac{2x+3}{3x+x^2}= \dfrac{1}{2}

By cross multiplying, we have:

2(2x+3) = 3x+x²

4x + 6 = 3x + x²

3x + x² - 4x - 6

x² - x - 6 = 0

Using quadratic equation

x² -3x +2x - 6 = 0

x(x - 3) + 2(x - 3) = 0

(x +2) (x - 3) = 0

x + 2 = 0 or x - 3 = 0

x = -2 or x = 3

Since we are concerned about the positive integer,

Then, Karl = x = 3 hours while Della which is (3+x) = 3+3 = 6hours

6 0
2 years ago
Evan wants to build a rectangular enclosure for his animals. One side of the pen will be against the barn, so he needs no fence
pentagon [3]

Answer:  A.   A=(1000-2w)*w      B. 250 feet

C.  125 000 square feet

Step-by-step explanation:

The area of rectangular is A=l*w    (1)

From another hand the length of the fence is 2*w+l=1000        (2)

L is not multiplied by 2, because the opposite side of the l is the barn,- we don't need in fence on that side.

Express l from (2):

l=1000-2w

Substitude l in (1) by 1000-2w

A=(1000-2w)*w        (3)   ( Part A. is done !)

Part B.

To find the width w  (Wmax) that corresponds to max of area A   we have to dind the roots of equation (1000-2w)w=0  ( we get it from (3))

w1=0  1000-2*w2=0

w2=500

Wmax= (w1+w2)/2=(0+500)/2=250 feet

The width that maximize area A is Wmax=250 feet

Part C.   Using (3) and the value of Wmax=250 we can write the following:

A(Wmax)=250*(1000-2*250)=250*500=125 000 square feets

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Answer:
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