A fuel oil company claims that one-fifth of the homes in a certain city are heated by oil. Do we have reason to believe that fewer than one-fifth are heated by oil if, in a random sample of 1000 homes in this city, 136 are heated by oil? Please show all 4 steps of the classical approach clearly using α = 0.05.

In polar coordinates, the inequality changes to

[tex]x^2+y^2\le4x\implies r^2\le4r\cos\theta\implies r\le4\cos\theta[/tex]

which is a circle of radius 2 and centered at (2, 0). The set D is then

[tex]D=\left\{(r,\theta)\mid0\le r\le4\cos\theta\land0\le\theta\le\pi\right\}[/tex]

The integral is then

[tex]\displaystyle\iint_D\frac{y^2}{x^2+y^2}\,\mathrm dx\,\mathrm dy=\int_0^\pi\int_0^{4\cos\theta}\frac{r^2\sin^2\theta}{r^2}r\,\mathrm dr\,\mathrm d\theta[/tex]

[tex]=\displaystyle\int_0^\pi\int_0^{4\cos\theta}r\sin^2\theta\,\mathrm dr\,\mathrm d\theta[/tex]

[tex]=\displaystyle\frac12\int_0^\pi((4\cos\theta)^2-0^2)\sin^2\theta\,\mathrm d\theta[/tex]

[tex]=\displaystyle8\int_0^\pi\cos^2\theta\sin^2\theta\,\mathrm d\theta[/tex]

There are several ways to compute the remaining integral; one would be to invoke the double-angle formula,

[tex]\sin(2\theta)=2\sin\theta\cos\theta[/tex]

so that the integral is

[tex]=\displaystyle8\int_0^\pi\frac{\sin^2(2\theta)}4\,\mathrm d\theta[/tex]

[tex]=\displaystyle2\int_0^\pi\sin^2(2\theta)\,\mathrm d\theta[/tex]

Then invoke another double-angle formula,

[tex]\sin^2\theta=\dfrac{1-\cos(2\theta)}2[/tex]

to change the integral to

[tex]=\displaystyle\int_0^\pi1-\cos(4\theta)\,\mathrm d\theta[/tex]

[tex]=(\pi-\cos(4\pi))-(0-\cos0)=\boxed{\pi}[/tex]

Answer:

[tex] y = \frac{4}{3} - 5[/tex]

Step-by-step explanation:

[tex] - 3y = 15 - 4x \\ - 3y = - 4x + 15 \\ \\ y = \frac{ - 4x + 15}{ - 3} \\ \\ y = \frac{ - 4}{ - 3} x + \frac{15}{ - 3} \\ \\ y = \frac{4}{3} x - 5 \\ which \: is \: in \: slope - intercept \: form.[/tex]

Answer:

3√2

Step-by-step explanation:

If you draw the diagonal, you have a 45°45°90° triangle.

The two legs are 3, so the hypotenuse is 3√2

Answer:

s = 4.43

Step-by-step explanation:

Using formula for bigger circle

s =r∅

Where s is the Arc length, r is rdius and ∅ is theta(angle)

8.84=5∅

∅= 8.84/5

Angle = 1.77 radians

So both angles equal to 1.77 radians

Now again

Using formula

s = r∅

Where s is the Arc length, r is rdius and ∅ is theta(angle)

s = (2.5)(1.77)

s ≈ 4.43

Answer:

20.75

Step-by-step explanation:

Answer:

C. -20.75

Step-by-step explanation:

Answer:

[tex] z= \frac{0.21- 0.39}{0.0373}= -4.829[/tex]

[tex] z= \frac{0.32- 0.39}{0.0373}=-1.877[/tex]

And we can find the probability with this difference:

[tex] P(-4.829<z< -1.877) = P(Z<-1.877) -P(Z<-4.829)=0.0303- 6.86x10^{-7}=0.0303[/tex]

Step-by-step explanation:

For this case we have the following info given:

[tex] n = 171[/tex] represent the sample size

[tex]p =0.39[/tex] the proportion of interest

We want to find the following probability:

[tex] P( 0.21 < \hat p < 0.32)[/tex]

We can use the normal approximation for this case since np >10 and n (1-p) >10

For this case we know that the distribution for the sample proportion is given by:

[tex]\hat p \sim N( p , \sqrt{\frac{p (1-p)}{n}} )[/tex]

And we can use the following parameters:

[tex] \mu_{\hat p}= 0.39[/tex]

[tex] \sigma_{\hat p} =\sqrt{\frac{0.39*(1-0.39)}{171}}= 0.0373[/tex]

And we can apply the z score formula given by:

[tex] z = \frac{p \\mu_{\hat p}}{\sigma_{\hat p}}[/tex]

And using this formula we got:

[tex] z= \frac{0.21- 0.39}{0.0373}= -4.829[/tex]

[tex] z= \frac{0.32- 0.39}{0.0373}=-1.877[/tex]

And we can find the probability with this difference:

[tex] P(-4.829<z< -1.877) = P(Z<-1.877) -P(Z<-4.829)=0.0303- 6.86x10^{-7}=0.0303[/tex]

Answer:

[tex]74.8m[/tex]

Step-by-step explanation:

[tex]A=a^2\\P=4a\\P=4\sqrt{A} \\=4*\sqrt{349.69} \\=74.8m[/tex]

Answer:

Four-fifths

Step-by-step explanation:

As we know that

By multiplying the function by a constant, we may expand or shrink the function in the y-direction.

Now we have

[tex]y=a(b^{x})[/tex]

if [tex]a> 1[/tex]  > the function would enlarge

if [tex]0< a < 1[/tex] > the function would shrinks

Now

For case A

[tex]a = \frac{4}{5}[/tex]

[tex]0 < (\frac{4}{5} )< 1[/tex] ....... > the function would shrinks

For case B

[tex]a = \frac{5}{4}[/tex]

[tex](\frac{5}{4} )>1[/tex] .......> the function would enlarge

For case C

[tex]a = \frac{3}{2}[/tex]

[tex](\frac{3}{2} )>1[/tex] .........> the function would enlarge

For case D

[tex]a = \frac{7}{4}[/tex]

[tex](\frac{7}{4} )>1[/tex] .........> the function would enlarge

Therefore the second option is correct

Answer: 4/5

Step-by-step explanation: E2020

Answer:

D

Step-by-step explanation:

In a 30-60-90 triangle, the longer leg is [tex]\sqrt{3}[/tex] times larger than the smaller leg. The length of the shorter leg is therefore:

[tex]\dfrac{18}{\sqrt{3}}= \\\\\\\dfrac{18\sqrt{3}}{3}= \\\\\\6\sqrt{3}[/tex]

Hope this helps!

Answer:

LIKE TERMS: 6 and 9, 0.5kt and -10kt, and -7y2 and y2. UNLIKE TERMS: 3ad and 2bd, 5x and 5, and the last one is -4p and p2

Step-by-step explanation:

Answer: like terms: 6&9 , 0.5kt&-10kt , -7y^2&y^2

unlike terms 3ad&2bd, 5x&5, -4p&p^2

Step-by-step explanation:

passed

Answer:  d) 67

Step-by-step explanation:

[tex]determinant\ \left[\begin{array}{ccc}a&b&c\\d&e&f\\g&h&j\end{array}\right] = a\cdot det\left[\begin{array}{cc}e&f\\h&j\end{array}\right] -\ b\cdot det\left[\begin{array}{cc}d&f\\g&j\end{array}\right] +\ c\cdot det\left[\begin{array}{cc}d&e\\g&h\end{array}\right][/tex]

[tex]determinant\ \left[\begin{array}{ccc}2&3&4\\-3&2&1\\5&-1&6\end{array}\right] \\\\\\= 2\cdot det\left[\begin{array}{cc}2&1\\-1&6\end{array}\right] -\ 3\cdot det\left[\begin{array}{cc}-3&1\\5&6\end{array}\right] +\ 4\cdot det\left[\begin{array}{cc}-3&2\\5&-1\end{array}\right]\\\\\\=2[2(6)-1(-1)]-3[-3(6)-1(5)]+4[3(-1)-2(5)]\\\\\\=2(13)-3(-23)+4(-7)\\\\\\=26+69-28\\\\\\=\large\boxed{67}[/tex]

Answer:

y = 1/4x + 2

Step-by-step explanation:

Since they gave you point slope form already, all you need to do is convert that into slope-intercept form. Just distribute the parenthesis and move the 4 over. Once you do so, you should get C/3rd option as your answer.

Answer:

y= 359 x+1500

Step-by-step explanation:

find the slope  m= (2577-1859)÷(3-1) = 359

y=mx+b

find b : substitute x ,y, and m

get b = 1857 - 359*1 = 1500

Answer:

y= 359 x+1500

Step-by-step explanation:

Answer:

The trip will take 36 minutes.

Step-by-step explanation:

This question can be solved using a rule of three.

The boat maintains a constant speed of 15 knots (nautical miles per hour). How many minutes it will take to return to its home port 9 nautical miles away?

So in 60 minutes, 15 nautical miles. How many minutes for 9 nautical miles?

60 minutes - 15 nautical miles

x minutes - 9 nautical miles

[tex]15x = 60*9[/tex]

[tex]x = \frac{60*9}{15}[/tex]

[tex]x = 36[/tex]

The trip will take 36 minutes.

Answer:

After 121 passes, there will be 11 cups facing up

Step-by-step explanation:

Given that:

Peter initially  lines up 121 cups facing down in a straight line from left to right and consecutively labels them from 1 to 121.

We can have an inequality ; i.e 1 ≤ n ≤  121; if n represents the divisor including n itself for which n  = odd number. Thus at the end of this claim, the cup will be facing up.

On the ith pass, he flips over cups i, 2i, 3i, 4i,.... (By "flip," we mean changing the cup from face down to face up or vice versa.)

For each divisor on the ith pass of n;

[tex]i \ th \ pass \ = \ n \ \to \ p |n[/tex]  since we are dealing with possibility of having an odds number:

Thus; [tex]p =i[/tex] and [tex]i^2 = n[/tex]  where ; n = perfect square.

Thus ; we will realize that between 1 to 121 ; there exist 11 perfect squares. Therefore; as a result of that ; 11 cups will definitely be facing up after 121 passes

Answer:

670

Step-by-step explanation:

2003-814=1189

1189-519=670

Answer: The third number is 670.

Step-by-step explanation:

The sum means three numbers being added up is equal to 2003 so give two of those numbers you have to add them up and subtract it from 2003 to find the third number.

814 + 519 + x = 2003   where x is the third number

1333 + x = 2003

-1333          -1333  

  x = 670   So the third number is 670

Check:  

 814 + 670 + 519 = 2003

     2003 = 2003   so yes again 670 is the third number.

Answer:

a) The value of absolute minimum value = - 0.3536  

b) which is attained at   [tex]x = \frac{1}{\sqrt{2} }[/tex]  

Step-by-step explanation:

Step(i):-

Given function

                       [tex]f(x) = \frac{-x}{2x^{2} +1}[/tex]     ...(i)

Differentiating equation (i) with respective to 'x'

                     [tex]f^{l} = \frac{2x^{2} +1(-1) - (-x) (4x)}{(2x^{2}+1)^{2} }[/tex]   ...(ii)

                    [tex]f^{l}(x) = \frac{2x^{2}-1}{(2x^{2}+1)^{2} }[/tex]

Equating Zero

                   [tex]f^{l}(x) = \frac{2x^{2}-1}{(2x^{2}+1)^{2} } = 0[/tex]

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

                [tex]2 x^{2}-1 = 0[/tex]

               [tex]2 x^{2} = 1[/tex]

             [tex]x^{2} = \frac{1}{2}[/tex]

             [tex]x = \frac{-1}{\sqrt{2} } , x = \frac{1}{\sqrt{2} }[/tex]

Step(ii):-

Again Differentiating equation (ii) with respective to 'x'

[tex]f^{ll}(x) = \frac{(2x^{2} +1)^{2} (4x) - 2(2x^{2} +1) (4x)(2x^{2}-1) }{(2x^{2}+1)^{4} }[/tex]

put

      [tex]x = \frac{1}{\sqrt{2} }[/tex]

[tex]f^{ll} (x) > 0[/tex]

The absolute minimum value at   [tex]x = \frac{1}{\sqrt{2} }[/tex]

Step(iii):-

The value of absolute minimum value

                         [tex]f(x) = \frac{-x}{2x^{2} +1}[/tex]

                       [tex]f(\frac{1}{\sqrt{2} } ) = \frac{-\frac{1}{\sqrt{2} } }{2(\frac{1}{\sqrt{2} } )^{2} +1}[/tex]

         on calculation we get

The value of absolute minimum value = - 0.3536      

Final answer:-

a) The value of absolute minimum value = - 0.3536  

b) which is attained at   [tex]x = \frac{1}{\sqrt{2} }[/tex]    

Answer:

The third degree polynomial function = x³ + 27x² + 200x + 300

Step-by-step explanation:

The third-degree polynomial function has zeros −2 and −15.

From the above, we have been given two factors of the polynomial function. Let's derive the factors from the two zeros of the polynomial given.

The two given zeros of the polynomial can be written as:

x= -2

x+2 = 0

(x+2) is a factor of the polynomial

x= -15

x+15 = 0

(x+15) is a factor of the polynomial

So we have two factors of the polynomial (x+2) and (x+15). But since it is a third degree polynomial, we have to find the third factor.

Let (x-b) be the third factor and f(x) represent the third degree polynomial

f(x) = (x-b) (x+2) (x+15)

Expanding (x+2) (x+15) = x² + 2x + 15x + 30

(x+2) (x+15) = x² + 17x + 30

f(x) = (x-b) (x² + 17x + 30)

From the given information, a value of 1,170 is obtained when x=3

f(3) = 1170

Insert 3 for x in f(x)

f(3) = (3-b) (3² + 17(3) + 30)

1170 = (3-b) (9 + 51 + 30)

1170 = (3-b) (90)

1170/90 = 3-b

3-b = 13

b = 3-13 = -10

Insert value of b in f(x)

f(x) = [x-(-10)] (x² + 17x + 30)

f(x) = (x+10) (x² + 17x + 30)

f(x) = x³ + 17x² + 30x + 10x² + 170x + 200x + 300

f(x) = x³ + 27x² + 200x + 300

The third degree polynomial function = x³ + 27x² + 200x + 300

Answer:

height = 6 mm

Step-by-step explanation:

The prism is a rectangular prism. The base area of the prism is 24 mm². The volume of the prism is given as 144 mm³.

The height of the prism can be solved as follows.

Volume of the rectangular prism = Bh

where

B = base area

h = height

Volume = 144 mm³

B = 24 mm²

volume = Bh

144 = 24 × h

144 = 24h

divide both sides by 24

h = 144/24

h = 6 mm

Answer:

c

Step-by-step explanation:

edg 2022

Answer:

Step-by-step explanation:

On the y-axis, graph the point on (0,4). Then from there, go up one, and to the right 4.

Answer: i did the question i told you the steps

Step-by-step explanation:

From the starting point move three to the left. Then move four down. Then move three times to the left. Lastly move two down.

Answer:

557

Step-by-step explanation:

l x w

13x24

13x7

22x7

557

Answer:

0.5 = 50% of bottles have volumes less than 1,007 mL

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this question, we have that:

[tex]\mu = 1007[/tex]

What proportion of bottles have volumes less than 1,007 mL?

This is the pvalue of Z when X = 1007. So

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{1007 - 1007}{\sigma}[/tex]

[tex]Z = 0[/tex]

[tex]Z = 0[/tex] has a pvalue of 0.5

0.5 = 50% of bottles have volumes less than 1,007 mL

Answer:

There is no correlation between age and purchase price

Step-by-step explanation:

In the survey, the researcher found out that a younger customer will not necessarily purchase a lower or higher priced computer thing it is likely true that there might be no correlation between purchase price and age.

It assumes that a younger customer can buy either buy a lower priced computer or can also buy a higher priced if he or she has the money for it.

Answer:

Option A.

The heartbeat has a pattern of 60 + (5 x minutes) and linear graphs are straight. The only way the linear graph is straight if there is a pattern.

Step-by-step explanation:

Answer:

E:

Step-by-step explanation:

The equation of circles is

(x-a)²+(y-b)²=r²

Where

Center = (a,b) = (-6,-3) and r = 12

Now

The equation becomes

(x+6)²+(x+3)²=144

Answer:

Step-by-step explanation:

3/4=24 so 1/4= 24÷3= 8

1/4=8

So to get 4/4 or Cameron's age it is 8×4=32yrs

[tex]answer \\ 32 \: years \: old \\ solution \\ mary's \: age = 24 \\ let \: cameron's \: age \: be \: x \\ given \\ \frac{3}{4} x = 24 \\ or \: x = 24 \times \frac{4}{3} \\ x = 32 \\ hope \: it \: helps[/tex]

Answer:

Area of the figure = 254.5 cm²

Step-by-step explanation:

Area of rectangle = Length × Width

Area of triangle = 1/2(base × Height)

Dividing the figure into parts for convenience

So,

Rectangle 1  (the uppermost):

4 × 6 = 24 cm²

Rectangle 2 (below rectangle 1):

15 × 8 = 120 cm²

Rectangle 3 (with rectangle 2):

11 × 4 = 44 cm²

Triangle 1 :

1/2(7 × 19) = 133/2 = 66.5 cm²

Now adding up all to get the area of the figure:

Area of the figure = 24 + 120 + 44 + 66.5

Area of the figure = 254.5 cm²

Answer:

Step-by-step explanation:

The answer is 3x 987 colunm 2