If S is a compact subset of R and T is a closed subset of S, then T is compact. Prove this using the definition of compactness.

[tex]\it x^3+8=x^3+2^3=(x+2)(x^2-2x+4)\\ \\ \\ a^3+b^3=(a+b)(a^2-ab+b^2)[/tex]

Answer:

Step-by-step explanation:

[tex] \sf{x - 2} \: and \: { {x}^{2} + x - 6}[/tex]

To find the H.C.F of the algebraic expressions, they are to be factorised and a common factor or the product of common factors is obtained as their H.C.F

Let's solve

First expression = x - 2

Second expression = x + x - 6

Here, we have to find the two numbers which subtracts to 1 and multiplies to 6

= x + ( 3 - 2 ) x + 6

Distribute x through the parentheses

= x + 3x - 2x + 6

Factor out x from the expression

= x ( x + 3 ) - 2x + 6

Factor out -2 from the expression

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

Factor out x+3 from the expression

= ( x + 3 ) ( x - 2 )

Here, x - 2 is common in both expression.

Thus, H.C.F = x - 2

Hope I helped!

Best regards!!!

Answer:

x - 2

Step-by-step explanation:

by factorization method

1) x - 2

2) x^2 + x - 6

by splitting method

x^2 + 3x - 2x - 6

taking separate common from the first two terms and last two terms

x(x + 3) - 2(x + 3)

now writing x+3 once and the other term to get the right answer

(x + 3)(x - 2)

in both parts just see the similar term and write it as HCF

HCF= x - 2

and the second method by which you can get this answer is division method

Answer:

x = -7, x = 7

Step-by-step explanation:

Firstly, you are going to set the equation to 0, and then factor it.

Set equation to 0 -----> f(x)= x^2 - 49 will become x^2 - 49 = 0

Now, you're going to factor the equation.

You'll get (x-7) (x+7) upon factoring.

Thirdly, you will set (x-7)(x+7) equal to 0 and also solve for x.

Keep in mind that you'll be treating them as two separate equations

So, ----> (x-7) = 0 (x+7) = 0

When you solve for the x, you'll find out that x is equal to 7 and -7 ---> these are your zeros.

Answer:

Step-by-step explanation:

A = {p, q, r}

Subsets: {p,q,r}, {p,q}, {p,r}. {q,r}, {p}, {q}, {r}, ∅

::::

Slope of RT = (-7/2 - 0)/(-3/2 - 0) = 7/3

Point-slope equation for line of slope 7/3 that passes through (0,0):

y = (7/3)x

Answer:

The graph which shows the line y-1=2(x+2) is shown in the attachment

Answer:a

a

   [tex]336.04 < \mu < 443.96[/tex]

b

  The  margin of error will increase

c

The  margin of error will decreases

d

The 99% confidence interval is  [tex]0.4107 < p < 0.4293[/tex]

Step-by-step explanation:

From the question we are  told that

   The sample size  [tex]n = 19[/tex]

    The sample mean is  [tex]\= x = \$\ 390[/tex]

    The  standard deviation is  [tex]\sigma = \$ \ 120[/tex]

Given that the confidence level is  95% then the level of significance is mathematically represented as

           [tex]\alpha = 100 - 95[/tex]

          [tex]\alpha = 5 \%[/tex]

          [tex]\alpha = 0.05[/tex]

Next we obtain the critical value of [tex]\frac{\alpha }{2}[/tex] from the normal distribution table

    So  

         [tex]Z_{\frac{\alpha }{2} } = 1.96[/tex]

The  margin of error is mathematically represented as

      [tex]E = Z_{\frac{\alpha }{2} } * \frac{\sigma}{\sqrt{n} }[/tex]

=>    [tex]E = 1.96 * \frac{120}{\sqrt{19} }[/tex]

=>   [tex]E = 53.96[/tex]

The 95% confidence interval is  

     [tex]\= x - E < \mu < \= x + E[/tex]

=>   [tex]390 - 53.96 < \mu < 390 - 53.96[/tex]

=>  [tex]336.04 < \mu < 443.96[/tex]

When the confidence level increases the [tex]Z_{\frac{\alpha }{2} }[/tex] also increases which increases the margin of error hence the confidence level becomes wider

Generally the sample size mathematically varies with margin of error as follows

         [tex]n \ \ \alpha \ \ \frac{1}{E^2 }[/tex]

So if the sample size increases the margin of error decrease

The  sample proportion is mathematically represented as

       [tex]\r p = \frac{210}{500}[/tex]

       [tex]\r p = 0.42[/tex]

Given that the confidence level is 0.99 the level of significance is  [tex]\alpha = 0.01[/tex]

The critical value of [tex]\frac{\alpha }{2}[/tex] from the normal distribution table is  

      [tex]Z_{\frac{\alpha }{2} } = 2.58[/tex]

  Generally the margin of error is mathematically represented as

       [tex]E = Z_{\frac{\alpha }{2} }* \sqrt{ \frac{\r p (1- \r p )}{n} }[/tex]

=>   [tex]E = 0.42 * \sqrt{ \frac{0.42 (1- 0.42 )}{ 500} }[/tex]

=>     [tex]E = 0.0093[/tex]

The 99% confidence interval  is

     [tex]\r p - E < p < \r p + E[/tex]

     [tex]0.42 - 0.0093 < p < 0.42 + 0.0093[/tex]

     [tex]0.4107 < p < 0.4293[/tex]

Answer:

Table C

Step-by-step explanation:

For x and y to be proportional , then the values of

[tex]\frac{y}{x}[/tex] = constant k

Table B

[tex]\frac{y}{x}[/tex] = [tex]\frac{6}{3}[/tex] = 2

[tex]\frac{y}{x}[/tex] = [tex]\frac{24}{6}[/tex] = 4

[tex]\frac{y}{x}[/tex] = [tex]\frac{36}{9}[/tex] = 4

The values are not constant

Table C

[tex]\frac{y}{x}[/tex] = [tex]\frac{2}{3}[/tex]

[tex]\frac{y}{x}[/tex] = [tex]\frac{4}{6}[/tex] = [tex]\frac{2}{3}[/tex]

[tex]\frac{y}{x}[/tex] = [tex]\frac{6}{9}[/tex] = [tex]\frac{2}{3}[/tex]

These values are constant

Then Table C shows a proportional relationship between x and y

Answer:

28

Step-by-step explanation:

Find the gcf

Answer:

the person above is correct if i did this correct

Step-by-step explanation:

Answer:

$5

Step-by-step explanation:

Using Promotion A, Jane would buy the first pair for $30 and the second for 1/2 * 30 = $15 for a total of 30 + 15 = $45. Using Promotion B, she would buy the first pair for $30 and the second for 30 - 10 = $20 for a total of 30 + 20 = $50. Since 45 < 50, Promotion A is the better deal, so Jane would save 50 - 45 = $5.

eleven hours -  11 hours

Answer:

d the answer is d

Step-by-step explanation:

Answer:

As the calculated value of t =2.1698 is greater than t (0.05,11) = 1.796 reject H0 . It means  chosen bottles stored for a month convince you that the mean furfuryl ether content exceeds the taste threshhold.

Step-by-step explanation:

We formulate our null and alternative hypotheses as

H0 u≤ 6 ug     Ha : u > 6 ug

The significance level ∝ = 0.05

The test statistic used is

t = X` - u / s/ √n

which if H0 is true, has the students' t test with n-1 = 11 degrees of freedom.

The critical region t > t (0.05,11) = 1.796

We compute the t value from the data

Xi               Xi²

8.92         79.5664

6.99          48.8601

5.54          30.6916

5.73           32.8329

6.38           40.7044

5.51            30.3601

6.45           41.6025

7.50           56.25

8.48           71.9104

5.56          30.9136

6.90          47.61

6.46          41.7316          

80.42         553.0336      

Now x` = ∑x/ n = 80.42/12 = 6.70

S²= 1/n-1 ( ∑(xi- x`)²= 1/11 ( 553.034 - (80.42)²/12)

= 1/11 (553.034-538.948) = 1.2805

s= 1.1316

Putting the values in the test statistics

t = X` - u / s/ √n = 6.70- 6 / 1.1316 / √12

= 2.1698

The critical region t > t (0.05,11) = 1.796

As the calculated value of t =2.1698 is greater than t (0.05,11) = 1.796 reject H0 . It means  chosen bottles stored for a month convince you that the mean furfuryl ether content exceeds the taste threshhold.

Answer:

Step by step proof shown below.

Step-by-step explanation:

To prove the equation, you need to apply the Logarithm quotient rule and the Logarithm power rule. Here's how the quotient rule looks like.

[tex]log_b(x/y) = log_b(x) - log_b(y)[/tex]

And here's how the power rule looks like

[tex]log_a(x)^n = nlog_a(x)[/tex]

First let's apply the quotient rule.

[tex]\frac{f(x+h)-f(x)}{h} = \frac{log_a(x+h)-log_a(x) }{h} = \frac{log_a(\frac{x+h}{x} )}{h}[/tex]

Now we can do some quick simplification, and apply the power rule.

[tex]\frac{1}{h} log_a(1 + \frac{h}{x} ) = log_a(1+\frac{h}{x} )^\frac{1}{h}[/tex]

Answer:

0.801

Step-by-step explanation:

Answer:

0.801

Step-by-step explanation:

801/1000 = 0.801

Answer:

[tex]\bold{4\sqrt3 + i4}[/tex]

Step-by-step explanation:

Given complex number is:

[tex][2(cos10^\circ + i sin10^\circ)]^3[/tex]

To find:

Answer in rectangular form after using De Moivre's theorem = ?

i.e. the form [tex]a+ib[/tex] (not in forms of angles)

Solution:

De Moivre's theorem provides us a way of solving the powers of complex numbers written in polar form.

As per De Moivre's theorem:

[tex](cos\theta+isin\theta)^n = cos(n\theta)+i(sin(n\theta))[/tex]

So, the given complex number can be written as:

[tex][2(cos10^\circ + i sin10^\circ)]^3\\\Rightarrow 2^3 \times (cos10^\circ + i sin10^\circ)^3[/tex]

Now, using De Moivre's theorem:

[tex]\Rightarrow 2^3 \times (cos10^\circ + i sin10^\circ)^3\\\Rightarrow 8 \times [cos(3 \times10)^\circ + i sin(3 \times10^\circ)]\\\Rightarrow 8 \times (cos30^\circ + i sin30^\circ)\\\Rightarrow 8 \times (\dfrac{\sqrt3}2 + i \dfrac{1}{2})\\\Rightarrow \dfrac{\sqrt3}2\times 8 + i \dfrac{1}{2}\times 8\\\Rightarrow \bold{4\sqrt3 + i4}[/tex]

So, the answer in rectangular form is:

[tex]\bold{4\sqrt3 + i4}[/tex]

Answer:

Rate= 3 1/3%

Or Rate= 3.33%

Step-by-step explanation:

Final amount collected= $700

Initial amount given out= $600

Interest made= Final amount - initial amount

Interest made= $700-$600

Interest made= $100

Type of interest rate = simple

Number of years = 5

PRT/100= interest

R=(100*interest)/(PT)

R= (100*100)/(600*5)

R= 10000/3000

R= 10/3

R= 3 1/3%

Or R= 3.33%

A fixed value. In Algebra, a constant is a number on its own, or sometimes a letter such as a, b or c to stand for a fixed number. Example: in "x + 5 = 9", 5 and 9 are constants.

Answer:

The selling price after the markup is $1.82

Step-by-step explanation:

$1.40 * .30 =

Multiply $1.40 times .30 (which is same as 30%)

$1.40 * .30 = $0.42

Add $1.40 and $0.42

= $1.82

Hope this helps.

Answer:

There is a   min     value of   376     located at (x,y) =   (9,7)  

============================================================

Explanation:

Solve the second equation for y

x+y = 16

y = 16-x

Then plug it into the first equation

f(x,y) = 3x^2+4y^2 - xy

g(x) = 3x^2+4(16-x)^2 - x(16-x)

g(x) = 3x^2+4(256 - 32x + x^2) - 16x + x^2

g(x) = 3x^2+1024 - 128x + 4x^2 - 16x + x^2

g(x) = 8x^2-144x+1024

The positive leading coefficient 8 tells us we have a parabola that opens upward, and produces a minimum value (aka lowest point) at the vertex.

Let's compute the derivative and set it equal to zero to solve for x.

g(x) = 8x^2-144x+1024

g ' (x) = 16x-144

16x-144 = 0

16x = 144

x = 144/16

x = 9

The min value occurs when x = 9. Let's find its paired y value.

y = 16-x

y = 16-9

y = 7

The min value occurs at (x,y) = (9,7)

Lastly, let's find the actual min value of f(x,y).

f(x,y) = 3x^2+4y^2 - xy

f(9,7) = 3(9)^2+4(7)^2 - 9*7

f(9,7) = 376

The smallest f(x,y) value is 376.

Answer:

A. 17 years

Step-by-step explanation:

Use the simple interest equation, I = prt, where I is the interest money gained, p is the starting amount of money, r is the interest rate in decimal form, and t is the time in years.

Plug in the values to solve for t:

417.69 = (630)(0.039)(t)

417.69 = 24.57t

17 = t

= 17 years

So, the correct answer is A, 17 years

Answer:

58 cm

Step-by-step explanation:

Assuming that the squares’s sides are whole numbers, we can find the size of the squares by looking at numbers squared.  We find three that equal 153.

10²=10x10=100

7²=7x7=49

2²=2x2=4

100+49+4=153

Now we look at how they are put together to find the perimeter.

The 2x2 has 3 exposed sides totaling 6.

The 7x7 has a top and bottom of 7, and part of a third side of 7-2=5. 7+7+5=19

The 10x10 has 3 exposed sides of 10, and part of a third side of 10-7=3. 10+10+10+3=33

TOTAL Perimeter = 6+19+33=58 cm

Answer:

C

Step-by-step explanation:

The area (A) of a circle is calculated as

A = πr² ( r is the radius )

Here r = 18 cm , thus

A = π × 18² = 324π ≈ 1017.9 cm² → C

Answer:

C.) 1017.9 cm²

Step-by-step explanation:

For a given circle

radius (r) = 18 cm

Now,

Area of Circle

= πr²

= 3.14 × (18)² cm

= 3.14 × 324 cm

= 1017.9 cm²

Answer:

Hey there!

All of the values: 0, 3, and 4 are in the domain.

This is because h(x) = (x - 3)^2 is a parabola, or a quadratic. By definition, the domain, or the possible x values of a parabola are infinite.

Hope this helps :)

Answer:

4.5 cm

Step-by-step explanation:

a^2+b^2=c^2

A represents the leg we already know, which has a length of 4 cm. C represents the hypotenuse with a length of 6 cm:

4^2+b^2=6^2, simplified: 16+b^2=36

subtract 16 from both sides:

b^2=20

now find the square root of both sides and that is the length of the other leg.

sqrt20= 4.4721, which can be rounded to 4.5

Answer:

4.5 cm

Step-by-step explanation:

Since this is a right triangle, we can use the Pythagorean Theorem.

[tex]a^2+b^2=c^2[/tex]

where a and b are the legs and c is the hypotenuse.

One leg is unknown and the other is 4 cm. The hypotenuse is 6 cm.

[tex]a=a\\b=4\\c=6[/tex]

Substitute the values into the theorem.

[tex]a^2+4^2=6^2[/tex]

Evaluate the exponents first.

4^2= 4*4= 16

[tex]a^2+16=6^2[/tex]

6^2=6*6=36

[tex]a^2+16=36[/tex]

We want to find a, therefore we must get a by itself.

16 is being added on to a^2. The inverse of addition is subtraction. Subtract 16 from both sides of the equation.

[tex]a^2+16-16=36-16\\\\a^2=36-16\\\\a^2=20[/tex]

a is being squared. The inverse of a square is a square root. Take the square root of both sides.

[tex]\sqrt{a^2}=\sqrt{20} \\\\a=\sqrt{20} \\\\a=4.47213595[/tex]

Round to the nearest tenth. The 7 in the hundredth place tells us to round the 4 in the tenth place to a 5.

[tex]a=4.5[/tex]

Add appropriate units. In this case, centimeters.

a= 4.5 cm

The length of the other leg is about 4.5 centimeters.

Answer:

Part A:

0 = -16x^2 + 22x + 3

x intercepts = (-0.125, 0) and (1.5, 0)

Part B:

It's going to be a maximum. This is because, the graph opens to the bottom. The coordinates are approximately (0.69, 10.56).

Part C:

I would plug in various x values, then solve for the y value. Repeat the process, and get a good idea of the graph.

Hope this helps!

Answer:

Answer is (x - 1)² = 4

Step-by-step explanation:

[tex]{ \tt{ {x}^{2} - 2x - 3 = 0}} \\ [/tex]

By completing squares:

[tex]{ \tt{ {x}^{2} - 2x + { (\frac{2}{2} )}^{2} - 3 - {( \frac{2}{2}) }^{2} = 0 }} \\ { \tt{( {x}^{2} - 2x + 1) = 4}} \\ { \tt{(x - 1) {}^{2} = 4 }}[/tex]

[tex]{ \underline{ \sf{ \blue{christ \:† \: alone }}}}[/tex]