8.43 An advertising executive wants to estimate the mean amount of time that consumers spend with digital media daily. From past studies, the standard deviation is estimated as 45 minutes. a. What sample size is needed if the executive wants to be 90% confident of being correct to within {5 minutes

Answer:

To verify the Cauchy-Bunyakovsky-Schwarz Inequality, (p,q) must be less than (or equal to) ||p|| • ||q||

(1,1,1) is not equal to (-10,5)

Step-by-step explanation:

a°b° + a^1b^1 + a^2b^2 < 5x (-2x^2 + 1)

Any algebra raised to the power of zero is equal to 1.

a°b° = 1 × 1 = 1

1 + ab + a^2b^2 < -10x^3 + 5x

The vectors:

(1,1,1) < (-10,5)

This verifies the Cauchy-Schwarz Inequality

Triangle Inequality states that for any triangle, the sum of the lengths of two sides must be greater than or equal to the length of the third side.

Answer:

In Table C, y vary inversely with x.

1×18 = 18

2×9 = 18

3×6 = 18

18 = 18 = 18

Step-by-step explanation:

We are given four tables and asked to find out in which table y vary inversely with x.

We know that an inverse relation has a form given by

y = k/x

xy = k

where k must be a constant

Table A:

x     |      y

1     |      3

2     |     9

3     |    27

1×3 = 3

2×9 = 18

3×27 = 81

3 ≠ 18 ≠ 81

Hence y does not vary inversely with x.

Table B:

x     |      y

1     |     -5

2     |     5

3     |    15

1×-5 = -5

2×5 = 10

3×15 = 45

-5 ≠ 10 ≠ 45

Hence y does not vary inversely with x.

Table C:

x     |      y

1     |      18

2     |     9

3     |     6

1×18 = 18

2×9 = 18

3×6 = 18

18 = 18 = 18

Hence y vary inversely with x.

Table D:

x     |      y

1     |      4

2     |     8

3     |    12

1×4 = 4

2×8 = 16

3×12 = 36

4 ≠ 16 ≠ 36

Hence y does not vary inversely with x.

Answer:

The measure of each angle:

152.8°   and     27.2°

Step-by-step explanation:

Supplementary angles sum 180°

then:

a + b = 180°

a - b = 125.6°

then:

a = 180 - b

a = 125.6 + b

180 - b = 125.6 + b

180 - 125.6 = b + b

54.4 = 2b

b = 54.4/2

b = 27.2°

a = 180 - b

a = 180 - 27.2

a = 152.8°

Check:

152.8 + 27.2 = 180°

Answers:

Step-by-step explanation:

Let x and y be the measures of each angle.

x + y = 180°

x - y = 125.6°

180 - 125.6 = 54.4

Now we divide 54.4 evenly to get y.

y = 27.2°

To get x, we substitute y into the equation.

x = 27.2 + 125.6

x = 152.8°

To check, we plug these in to see if they equal 180°.

27.2 + 152.8 = 180° ✅

Answer:

The variance of the data is 15.

σ² = 15

Step-by-step explanation:

The mean is given as

X = 8

х        |    (x - X)    |    (x - X) ²

12       |        4         |    16

9        |        1         |     1    

11        |        3         |    9

5       |        -3        |    9

3       |        -5        |    25

The variance is given by

[tex]\sigma^2 = \frac{1}{n-1} \sum (x - X)^2[/tex]

[tex]\sigma^2 = \frac{1}{5 - 1} (16 + 1 + 9 + 9 +25) \\\\\sigma^2 = \frac{1}{4} ( 16 + 1 + 9 + 9 +25) \\\\\sigma^2 = \frac{1}{4} (60) \\\\\sigma^2 = 15[/tex]

Therefore, the variance of the data is 15.

Answer:

Following are the answer to this question:

Step-by-step explanation:

The principle vale of Arg(3)

[tex]Arg(3)=-\pi+\tan^{-1} (\frac{|Y|}{|x|})[/tex]

The principle value of the [tex]\logi= \log(0+i)\ \ \ \ \ _{where} \ \ \ x=0 \ \ y=1> 0[/tex]

So, the principle value:

a)

[tex]\to \log(i)=\log |i|+i Arg(i)\\\\[/tex]

             [tex]=\log \sqrt{0+1}+i \tan^{-1}(\frac{1}{0})\\\=\log 1 +i \tan^{-1}(\infty)\\\=0+i\frac{\pi}{2}\\\=i\frac{\pi}{2}[/tex]

b)

[tex]\to \log(-i)= \log(0-i ) \ \ \ x=0 \ \ \ y= -1<0\\[/tex]

Principle value:

[tex]\to \log(-i)= \log|-i|+iArg(-i) \\\\[/tex]

                 [tex]=\log \sqrt{0+1}+i(-\pi+\tan^{-1}(\infty))\\\\=\log1 + i(-\pi+\frac{\pi}{2})\\\\=-i\frac{\pi}{2}[/tex]

c)

[tex]\to \log(-1+i) \ \ \ \ x=-1, _{and} y=1 \ \ \ x<0 and y>0[/tex]

The principle value:

[tex]\to \log(-1+i)=\log |-1+i| + i Arg(-1+i)[/tex]

                     [tex]=\log \sqrt{1+1}+i(\pi+\tan^{-1}(\frac{1}{1}))\\\\=\log \sqrt{2} + i(\pi-\tan^{-1}\frac{\pi}{4})\\\\=\log \sqrt{2} + i\tan^{-1}\frac{3\pi}{4}\\\\[/tex]

d)

[tex]\to i^i=w\\\\w=e^{i\log i}[/tex]

The principle value:

[tex]\to \log i=i\frac{\pi}{2}\\\\\to w=e^{i(i \frac{\pi}{2})}\\\\=e^{-\frac{\pi}{2}}[/tex]

e)

[tex]\to (-i)^i\\\to w=(-i)^i\\\\w=e^{i \log (-i)}[/tex]

In this we calculate the principle value from b:

so, the final value is [tex]e^{\frac{\pi}{2}}[/tex]

f)

[tex]\to -1^i\\\\\to w=e^{i log(-1)}\\\\\ principle \ value: \\\\\to \log(-1)= \log |-1|+iArg(-i)[/tex]

                [tex]=\log \sqrt{1} + i(\pi-\tan^{-1}\frac{0}{-1})\\\\=\log \sqrt{1} + i(\pi-0)\\\\=\log \sqrt{1} + i\pi\\\\=0+i\pi\\=i\pi[/tex]

and the principle value of w is = [tex]e^{\pi}[/tex]

g)

[tex]\to -1^{-i}\\\\\to w=e^(-i \log (-1))\\\\[/tex]

from the point f the principle value is:

[tex]\to \log(-1)= i\pi\\\to w= e^{-i(i\pi)}\\\\\to w=e^{\pi}[/tex]

h)

[tex]\to \log(-1-i)\\\\\ Here x=-1 ,<0 \ \ y=-1<0\\\\ \ principle \ value \ is:\\\\ \to \log(-1-i)=\log\sqrt{1+1}+i(-\pi+\tan^{-1}(1))[/tex]

                    [tex]=\log\sqrt{2}+i(-\pi+\frac{\pi}{4})\\\\=\log\sqrt{2}+i(-\frac{3\pi}{4})\\\\=\log\sqrt{2}-i\frac{3\pi}{4})\\[/tex]

The average rate of change between two input values is the total change of the function values (output values) divided by the change in the input values.

Answer:

Step-by-step explanation:

From the summary of the given data;

After the second dose, 137 of 452 subjects in the experimental group (group 1) experienced drowsiness as a side effect.

Let consider [tex]p_1[/tex] to be the probability of those that experience the drowsiness in group 1

[tex]p_1[/tex] = [tex]\dfrac{137}{452}[/tex]

[tex]p_1[/tex] = 0.3031

After the second dose, 31 of 99 subjects in the control group (group 2) experienced drowsiness as a side effect.

Let consider [tex]p_2[/tex] to be the probability of those that experience the drowsiness in group 1

[tex]p_2[/tex] = [tex]\dfrac{31}{99}[/tex]

[tex]p_2[/tex] = 0.3131

The objective is to be able to determine if the evidence suggest that a lower proportion of subjects in group 1 experienced drowsiness as a side effect than subjects in group 2 at the α=0.05 level of significance.

In order to do that; we have to state the null and alternative hypothesis; carry out our test statistics and make conclusion based on it.

So; the null and the  alternative hypothesis can be computed as:

[tex]H_o :p_1 =p_2[/tex]

[tex]H_a= p_1<p_2[/tex]

The test statistics is computed as follows:

[tex]Z = \dfrac{p_1-p_2}{\sqrt{p_1 *\dfrac{1-p_1}{n_1} +p_2 *\dfrac{1-p_2}{n_2}} }[/tex]

[tex]Z = \dfrac{0.3031-0.3131}{\sqrt{0.3031 *\dfrac{1-0.3031}{452} +0.3131 *\dfrac{1-0.3131}{99}} }[/tex]

[tex]Z = \dfrac{-0.01}{\sqrt{0.3031 *\dfrac{0.6969}{452} +0.3131 *\dfrac{0.6869}{99}} }[/tex]

[tex]Z = \dfrac{-0.01}{\sqrt{0.3031 *0.0015418 +0.3131 *0.0069384} }[/tex]

[tex]Z = \dfrac{-0.01}{\sqrt{4.6731958*10^{-4}+0.00217241304} }[/tex]

[tex]Z = \dfrac{-0.01}{0.051378 }[/tex]

Z = - 0.1946

At the level of significance ∝ = 0.05

From the standard normal table;

the critical value for Z(0.05) = -1.645

Decision Rule: Reject the null hypothesis if Z-value is lesser than the critical value.

Conclusion: We do not reject the null hypothesis because the Z value is greater than the critical value. Therefore, we cannot conclude that a lower proportion of subjects in group 1 experienced drowsiness as a side effect than subjects in group 2

Answer: 1. [tex]-\dfrac{5}{6}[/tex]  2. [tex]-\dfrac{1}{3}[/tex] . 3. [tex]-3[/tex]

Step-by-step explanation:

Formula: Slope[tex]=\dfrac{y_2-y_1}{x_2-x_1}[/tex]

1. (-2,2) (3,-3)

Slope [tex]=\dfrac{-3-2}{3-(-2)}[/tex]

[tex]=\dfrac{-5}{3+2}\\\\=\dfrac{-5}{6}[/tex]

Hence, slope of line passing through  (-2,2) (3,-3) is [tex]-\dfrac{5}{6}[/tex] .

2. (-5,1) (4,-2)

Slope [tex]=\dfrac{-2-1}{4-(-5)}[/tex]

[tex]=\dfrac{-3}{4+5}\\\\=\dfrac{-3}{9}\\\\=-\dfrac{1}{3}[/tex]

Hence, slope of line passing through  (-2,2) and (3,-3) is [tex]-\dfrac{1}{3}[/tex] .

3. (-1,5) (2,-4)

Slope [tex]=\dfrac{-4-5}{2-(-1)}[/tex]

[tex]=\dfrac{-9}{2+1}\\\\=\dfrac{-9}{3}\\\\=-3[/tex]

Hence, slope of line passing through (-1,5) and (2,-4) is -3.

Answer:

z(c)  = - 1,64

We reject the null hypothesis

Step-by-step explanation:

We need to solve a proportion test ( one tail-test ) left test

Normal distribution

p₀ = 63 %

proportion size  p = 51 %

sample size  n = 114

At 5% level of significance   α = 0,05, and with this value we find in z- table z score of z(c) = 1,64  ( critical value )

Test of proportion:

H₀     Null Hypothesis                        p = p₀

Hₐ    Alternate Hypothesis                p < p₀

We now compute z(s) as:

z(s) =  ( p - p₀ ) / √ p₀q₀/n

z(s) =( 0,51 - 0,63) / √0,63*0,37/114

z(s) =  - 0,12 / 0,045

z(s) = - 2,66

We compare z(s) and z(c)

z(s) < z(c)      - 2,66 < -1,64

Therefore as z(s) < z(c)  z(s) is in the rejection zone we reject the null hypothesis

Answer:

the probability that one parachute of the  five parachute is damaged is 0.156

Step-by-step explanation:

From the given information;

Let consider X to be the altitude above the  ground that a parachute opens

Then; we can posit that the probability that the parachute is damaged is:

P(X ≤ 100 )

Given that the population mean μ = 155

the standard deviation σ = 30

Then;

[tex]P(X \leq 100 ) = ( \dfrac{X- \mu}{\sigma} \leq \dfrac{100- \mu}{\sigma})[/tex]

[tex]P(X \leq 100 ) = ( \dfrac{X- 155}{30} \leq \dfrac{100- 155}{30})[/tex]

[tex]P(X \leq 100 ) = (Z \leq \dfrac{- 55}{30})[/tex]

[tex]P(X \leq 100 ) = (Z \leq -1.8333)[/tex]

[tex]P(X \leq 100 ) = \Phi( -1.8333)[/tex]

From standard normal tables

[tex]P(X \leq 100 ) = 0.0334[/tex]

Hence; the probability of the given parachute damaged is 0.0334

Let consider Q to be the dropped parachute

Given that the number of parachute be n= 5

The probability that the parachute opens in each trail be  p = 0.0334

Now; the random variable Q follows the binomial distribution with parameters n= 5 and p = 0.0334

The probability mass function is:

Q [tex]\sim[/tex] B(5, 0.0334)

Similarly; the event that one parachute is damaged is :

Q ≥ 1

P( Q ≥ 1 ) = 1 - P( Q < 1 )

P( Q ≥ 1 ) = 1 - P( Y = 0 )

P( Q ≥ 1 ) = 1 - b(0;5; 0.0334 )

P( Q ≥ 1 ) = [tex]1 -(^5_0)* (0.0334)^0*(1-0.0334)^5[/tex]

P( Q ≥ 1 ) = [tex]1 -( \dfrac{5!}{(5-0)!}) * (0.0334)^0*(1-0.0334)^5[/tex]

P( Q ≥ 1 ) = 1 -  0.8437891838

P( Q ≥ 1 ) = 0.1562108162

P( Q ≥ 1 ) [tex]\approx[/tex] 0.156

Therefore; the probability that one parachute of the  five parachute is damaged is 0.156

Answer: 0.9964

Step-by-step explanation:

Consider,

P (disk failure) = 0.06

q = 0.06

p = 1- q

p = 1- 0.06,

p = 0.94

Step 2

Whereas p represents the probability that a disk does not fail. (i.e. working entire year).

a)

Step 3

a)

n = 2,

let x be a random variable for number...

Continuation in the attached document

Answer:

a. uses a related sample - repeated measures

c. uses a related sample (matched subjects)

Step-by-step explanation:

A) You are interested in a potential treatment for compulsive hoarding. You treat a group of 50 compulsive hoarders and compare their scores on the Hoarding Severity scale before and after the treatment. You want to see if the treatment will lead to lower hoarding scores.  

The design described uses a related sample - repeated measures because the scores were compared on the Hoarding Severity scale before and after the treatment.

B) John Caccioppo was interested in possible mechanisms by which loneliness may have deterious effects of health. He compared the sleep quality of a random sample of lonely people to the sleep quality of a random sample of nonlonely people.

The design described uses a related sample (matched subjects)

Answer:

Step-by-step explanation:

The class interval is simply the difference between the lower or upper class boundary or limit  of a class and the lower or upper class boundary or limit of the next class.

In this case for the class

$4 up to $7 18 and

$7 up to $10 36

The lower class boundary of the first class is $4 and the lower class boundary of the second class is $7

Answer:

The answer is 5th angle = [tex]\bold{42^\circ}[/tex]

Step-by-step explanation:

Given that pizza is divided into six unequal slices.

Largest slice has an angle of [tex]90^\circ[/tex].

He eats the pizza from largest to smallest.

Let the difference in angles in each slice = [tex]d^\circ[/tex]

1st angle = [tex]90^\circ[/tex]

2nd angle = 90-d

3rd angle = 90-d-d = 90 - 2d

4th angle = 90-2d-d = 90 - 3d

5th angle = 90-3d-d = 90 - 4d

6th angle = 90-4d -d = 90 - 5d

We know that the sum of all the angles will be equal to [tex]360^\circ[/tex] (The sum of all the angles subtended at the center).

i.e.

[tex]90+90-d+90-2d+90-3d+90-4d+90-5d=360\\\Rightarrow 540 - 15d = 360\\\Rightarrow 15d = 540 -360\\\Rightarrow 15d = 180\\\Rightarrow d = 12^\circ[/tex]

So, the angles will be:

1st angle = [tex]90^\circ[/tex]

2nd angle = 90- 12 = 78

3rd angle = 78-12 = 66

4th angle = 66-12 = 54

5th angle = 54-12 = 42

6th angle = 42 -12 = 30

So, the answer is 5th angle = [tex]\bold{42^\circ}[/tex]

Answer:

Hey there!

The common ratio is 5, because you multiply by 5 to get from one term to the next.

Hope this helps :)

Answer:

5

Step-by-step explanation:

To find the common ratio take the second term and divide by the first term

55/11 = 5

The common ratio would be 5

Answer:

11/36

Step-by-step explanation:

Find the probability that neither dice shows a 5 (also means the dice can show any number except 5- where there are 5 possible choices out of 6):

= 5/6 x 5/6

=25/36

If we subtract the probability that neither dice shows a 5, we can obtain the probability that at least 1 dice shows a 5- (either one of them is 5, or both of them is 5)

1- 25/36

=11/36

Answer:

1.9375.

Step-by-step explanation:

To solve this, we must use PEMDAS.

The first things we take care of are parentheses and exponents.

Since there are no parentheses, we do exponents.

4^-1+8^-1÷1/2/3^3​

= [tex]\frac{1}{4} +\frac{1}{8} / 1/ 2/ 27[/tex]

= 1/4 + (1/8) / 1 * (27 / 2)

= 1/4 + (27 / 8) / 2

= 1/4 + (27 / 8) * (1 / 2)

= 1/4 + (27 / 16)

= 4 / 16 + 27 / 16

= 31 / 16

= 1.9375.

Hope this helps!

Answer:

49.923 mph

Step-by-step explanation:

we know that the car traveled 133 miles in h hours at an average speed of x mph.

That is, xh = 133.

We can also write this in terms of hours driven: h = 133/x.

If x was 30 mph faster, then h would be one hour less.

That is, (x + 30)(h - 1) = 133, or h - 1 = 133/(x + 30).

We can rewrite the latter equation as h = 133/(x + 30) + 1

We can then make a system of equations using the formulas in terms of h to find x:

h = 133/x = 133/(x + 30) + 1

133/x = 133/(x + 30) + (x + 30)/(x + 30)

133/x = (133 + x + 30)/(x + 30)

133 = x*(133 + x + 30)/(x + 30)

133*(x + 30) =  x*(133 + x + 30)

133x + 3990 = 133x + x^2 + 30x

3990 = x^2 + 30x

x^2 + 30x - 3990 = 0

Using the quadratic formula:

x = [-b ± √(b^2 - 4ac)]/2a  

= [-30 ± √(30^2 - 4*1*(-3990))]/2(1)  

= [-30 ± √(900 + 15,960)]/2

= [-30 ± √(16,860)]/2

= [-30 ± 129.846]/2

= 99.846/2  -----------  x is miles per hour, and a negative value of x is neglected, so we'll use the positive value only)

= 49.923

Check if the answer is correct:

h = 133/49.923 = 2.664, so the car took 2.664 hours to drive 133 miles at an average speed of 49.923 mph.

If the car went 30 mph faster on average, then h = 133/(49.923 + 30) = 133/79.923 = 1.664, and 2.664 - 1 = 1.664.

Thus, we have confirmed that a car driving 133 miles at about 49.923 mph would have arrive precisely one hour earlier by going 30 mph faster

Answer:

Step-by-step explanation:

Let the measure of third angle be X

The sum of interior angle of triangle = X

Let's create an equation

[tex]x + 32 + 90 = 180[/tex]

Add the numbers

[tex]x + 122 = 180[/tex]

Move constant to R.H.S and change its sign

[tex]x = 180 - 122[/tex]

Subtract the numbers

[tex]x = 58[/tex] °

Hope this helps...

Best regards!!

Answer:

The answer is below

Step-by-step explanation:

A sugar refinery has three processing plants, all receiving raw sugar in bulk. The amount of raw sugar (in tons) that one plant can process in one day can be modelled using an exponential distribution with mean of 4 tons for each of three plants. If each plant operates independently,a.Find the probability that any given plant processes more than 5 tons of raw sugar on a given day.b.Find the probability that exactly two of the three plants process more than 5 tons of raw sugar on a given day.c.How much raw sugar should be stocked for the plant each day so that the chance of running out of the raw sugar is only 0.05?

Answer: The mean (μ) of the plants is 4 tons. The probability density function of an exponential distribution is given by:

[tex]f(x)=\lambda e^{-\lambda x}\\But\ \lambda= 1/\mu=1/4 = 0.25\\Therefore:\\f(x)=0.25e^{-0.25x}\\[/tex]

a) P(x > 5) = [tex]\int\limits^\infty_5 {f(x)} \, dx =\int\limits^\infty_5 {0.25e^{-0.25x}} \, dx =-e^{-0.25x}|^\infty_5=e^{-1.25}=0.2865[/tex]

b) Probability that exactly two of the three plants process more than 5 tons of raw sugar on a given day can be solved when considered as a binomial.

That is P(2 of the three plant use more than five tons) = C(3,2) × [P(x > 5)]² × (1-P(x > 5)) = 3(0.2865²)(1-0.2865) = 0.1757

c) Let b be the amount of raw sugar should be stocked for the plant each day.

P(x > a) = [tex]\int\limits^\infty_a {f(x)} \, dx =\int\limits^\infty_a {0.25e^{-0.25x}} \, dx =-e^{-0.25x}|^\infty_a=e^{-0.25a}[/tex]

But P(x > a) = 0.05

Therefore:

[tex]e^{-0.25a}=0.05\\ln[e^{-0.25a}]=ln(0.05)\\-0.25a=-2.9957\\a=11.98[/tex]

a  ≅ 12

Answer:

a=6, b=5.5

Step-by-step explanation:

By looking at the sides of the triangles it can easily be seen that some of the sides match up. Side b is similar to the side of 11 and same with side a and the side of 3. Since one side is 16 and the other side on the smaller triangle is 8, the bigger triangle is twice as large than the smaller one. So 3 x 2 = 6 and 11 / 2 = 5.5

Sam's weight to the nearest stone is equal to 8.0 stone.

Given the following data:

To determine Sam's weight to the nearest stone:

In this exercise, you're required to determine Sam's weight to the nearest stone. Thus, we would convert his weight in kilograms to pounds and lastly to stone as follows:

Conversion:

1 kg = 2.2 pounds.

51 kg = [tex]51 \times 2.2[/tex] = 112.2 pounds.

Next, we would convert the value in pounds to stone:

14 pounds = 1 stone.

112.2 pounds = X stone.

Cross-multiplying, we have:

[tex]14X = 112.2\\\\X=\frac{112.2}{14}[/tex]

X = 8.01 ≈ 8.0 stone.

Read more on weight here: brainly.com/question/13833323

Answer:

√3/2

Explanation:

The directional derivative at the given point is gotten using the formula;

∇f(x,y)•u where u is the unit vector in that direction.

∇f(x,y) = f/x i + f/y j

Given the function f(x, y) = y cos(xy),

f/x = -y²sin(xy) and

f/y = -xysin(xy)+cos(xy)

∇f(x,y) = -y²sin(xy) i + (cos(xy)-xysin(xy)) j

∇f(x,y) at (0,1) will give;

∇f(0,1) = -0sin0 i + cos0j

∇f(0,1) = 0i+j

The unit vector in the direction of angle θ is given as u = cosθ i + sinθ j

u = cos(π/3)i+ sin(π/3)j

u = 1/2 i + √3/2 j

Taking the dot product of both vectors;

∇f(x,y)•u = (0i+j)•(1/2 i + √3/2 j)

Note that i.i = j.j = 1 and i.j = 0

∇f(x,y)•u = 0 + √3/2

∇f(x,y)•u = √3/2

The directional derivative of [tex]f[/tex] at the given point in the direction indicated is [tex]\frac{\sqrt{3}}{2}[/tex].

The directional derivative is represented by the following formula:

[tex]\nabla_{\vec v} f = \nabla f(x_{o},y_{o}) \cdot \vec v[/tex]    (1)

Where:

The gradient of [tex]f[/tex] is calculated below:

[tex]\nabla f (x_{o},y_{o}) = \left[\begin{array}{cc}\frac{\partial f}{\partial x} (x_{o}, y_{o}) \\\frac{\partial f}{\partial y} (x_{o}, y_{o})\end{array}\right][/tex] (2)

Where [tex]\frac{\partial f}{\partial x}[/tex] and [tex]\frac{\partial f}{\partial y}[/tex] are the partial derivatives with respect to [tex]x[/tex] and [tex]y[/tex], respectively.

If we know that [tex](x_{o}, y_{o}) = (0, 1)[/tex], then the gradient is:

[tex]\nabla f(x_{o}, y_{o}) = \left[\begin{array}{cc}-y^{2}\cdot \sin xy\\\cos xy -x\cdot y\cdot \sin xy\end{array}\right][/tex]

[tex]\nabla f (x_{o}, y_{o}) = \left[\begin{array}{cc}-1^{2}\cdot \sin 0\\\cos 0-0\cdot 1\cdot \sin 0\end{array}\right][/tex]

[tex]\nabla f (x_{o}, y_{o}) = \left[\begin{array}{cc}0\\1\end{array}\right][/tex]

If we know that [tex]\vec v = \cos \frac{\pi}{3}\,\hat{i} + \sin \frac{\pi}{3} \,\hat{j}[/tex], then the directional derivative is:

[tex]\Delta_{\vec v} f = \left[\begin{array}{cc}0\\1\end{array}\right]\cdot \left[\begin{array}{cc}\cos \frac{\pi}{3} \\\sin \frac{\pi}{3} \end{array}\right][/tex]

[tex]\nabla_{\vec v} f = (0)\cdot \cos \frac{\pi}{3} + (1)\cdot \sin \frac{\pi}{3}[/tex]

[tex]\nabla_{\vec v} f = \frac{\sqrt{3}}{2}[/tex]

The directional derivative of [tex]f[/tex] at the given point in the direction indicated is [tex]\frac{\sqrt{3}}{2}[/tex]. [tex]\blacksquare[/tex]

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Answer:

The probability that all three people on the subcommittee are men

= 20%

Step-by-step explanation:

Number of members in the committee = 15

= 8 men + 7 women

The probability of selecting a man in the committee

= 8/15

= 53%

The probability of selecting three men from eight men

= 3/8

= 37.5%

The probability that all three people on the subcommittee are men

= probability of selecting a man multiplied by the probability of selecting three men from eight men

= 53% x 37.5%

= 19.875%

= 20% approx.

This is the same as:

The probability of selecting 3 men from the 15 member-committee

= 3/15

= 20%

Answer:

8lb of the cheaper Candy

17.5lb of the expensive candy

Step-by-step explanation:

Let the cheaper candy be x

let the costly candy be y

X+y = 25.5....equation one

2.2x +7.3y = 25.5(5.7)

2.2x +7.3y = 145.35.....equation two

X+y = 25.5

2.2x +7.3y = 145.35

Solving simultaneously

X= 25.5-y

Substituting value of X into equation two

2.2(25.5-y) + 7.3y = 145.35

56.1 -2.2y +7.3y = 145.35

5.1y = 145.35-56.1

5.1y = 89.25

Y= 89.25/5.1

Y= 17.5

X= 25.5-y

X= 25.5-17.5

X= 8

Answer:

The required probability = 0.144

Step-by-step explanation:

Since the probability of making money is 60%, then the probability of losing money will be 100-60% = 40%

Now the probability we want to calculate is the probability of making money in the first two days and losing money on the third day.

That would be;

P(making money) * P(making money) * P(losing money)

Kindly recollect;

P(making money) = 60% = 60/100 = 0.6

P(losing money) = 40% = 40/100 = 0.4

The probability we want to calculate is thus;

0.6 * 0.6 * 0.4 = 0.144

Answer:

1-{0.2(m-3)+¼}=0

1{0.2m-0.6+¼}=0

1-{(0.8m-2.4+1)/4}=0

1-(0.8m-1.4)/4=0

lcm

(4-0.8m-1.4)/4=0

(2.6-0.8m)/4=0

cross multiply

2.6-0.8m=0

m=2.6/0.8

m=3.25

The solution of the expression are,

⇒ m = 3.25

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

Given that;

Expression is,

⇒ 1 - | 0.2 (m - 3) + 1/4 | = 0

Now, We can simplify as;

⇒ 1 - | 0.2m - 0.6 + 1/4| = 0

⇒ 1 - |0.2m - 0.6 + 0.25| = 0

⇒ 1 - |0.2m - 0.35| = 0

⇒ 1 = 0.2m + 0.35

⇒ 1 - 0.35 = 0.2m

⇒ 0.2m = 0.65

⇒ m = 3.25

Thus, The solution of the expression are,

⇒ m = 3.25

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Answer:

A sample of at least 541 is needed if we wish to be 98% confident that our sample mean will be within 4 hours of the true mean.

Step-by-step explanation:

We are given that an electrical firm manufactures light bulbs that have a length of life that is approximately normally distributed with a standard deviation of 40 hours.

We have to find a sample such that we are 98% confident that our sample mean will be within 4 hours of the true mean.

As we know that the Margin of error formula is given by;

The margin of error =  [tex]Z_(_\frac{\alpha}{2}_) \times \frac{\sigma}{\sqrt{n} }[/tex]

where, [tex]\sigma[/tex] = standard deviation = 40 hours

            n = sample size

            [tex]\alpha[/tex] = level of significance = 1 - 0.98 = 0.02 or 2%

Now, the critical value of z at ([tex]\frac{0.02}{2}[/tex] = 1%) level of significance n the z table is given as 2.3263.

So, the margin of error =  [tex]Z_(_\frac{\alpha}{2}_) \times \frac{\sigma}{\sqrt{n} }[/tex]

                 [tex]4=2.3263 \times \frac{40}{\sqrt{n} }[/tex]

                 [tex]\sqrt{n}= \frac{40 \times 2.3263}{ 4}[/tex]

                  [tex]\sqrt{n}=23.26[/tex]

                   n = [tex]23.26^{2}[/tex] = 541.03 ≈ 541

Hence, a sample of at least 541 is needed if we wish to be 98% confident that our sample mean will be within 4 hours of the true mean.

Answer:

Step-by-step explanation:

The happiest used in a test in statistics are the null and the alternative hypothesis. The null hypothesis is usually the default statement while the alternative hypothesis is thevopposite of the null hypothesis.

In this case study, the null hypothesis is u1 = u2: the average mean time it takes to accelerate to 30 miles per hour for car 1 is the same as that for car 2.

The alternative hypothesis is u1 > u2: the mean time it takes to accelerate to 30 miles per hour is greater than that for car 2 thus car 1 is slower to accelerate as it takes more time.

Answer:

H0: μ=2; Ha: μ>2

Step-by-step explanation:

The null hypothesis is the default hypothesis while the alternative hypothesis is the opposite of the null and is always tested against the null hypothesis.

In this case study, the null hypothesis is that adults were eating, on average, the recommended 2 cups of vegetables a day: H0: μ=2 while the alternative hypothesis is adults were eating, on average, more than the recommended 2 cups of vegetables a day Ha: μ>2.