A book sold 34,100 copies in the first month the release suppose this represents 7.9% of the number of copies sold to date how many copies have you sold to date

Answer:

[tex]z=-2.32[/tex]  

[tex]p_v =P(Z<-2.32)= 0.010[/tex]  

If we compare the p value and using any significance level for example [tex]\alpha=0.05[/tex] we see that [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can say the the proportion 1 is significant lower than the proportion 2 at 5% of significance.  

Step-by-step explanation:

1) Data given and notation  

[tex]X_{1}=37[/tex] represent the number of people with characteristic 1

[tex]X_{2}=58[/tex] represent the number of people with characteristic 2

[tex]n_{1}=139[/tex] sample 1 selected

[tex]n_{2}=147[/tex] sample 2 selected

[tex]p_{1}=\frac{37}{139}=0.266[/tex] represent the proportion of people with characteristic 1

[tex]p_{2}=\frac{58}{147}=0.395[/tex] represent the proportion of people with characteristic 2

z would represent the statistic (variable of interest)  

[tex]p_v[/tex] represent the value for the test (variable of interest)

2) Concepts and formulas to use  

We need to conduct a hypothesis in order to check if the proportion 1 is less than the proportion 2, the system of hypothesis would be:  

Null hypothesis:[tex]p_{1} \geq p_{2}[/tex]  

Alternative hypothesis:[tex]p_{1} < p_{2}[/tex]  

We need to apply a z test to compare proportions, and the statistic is given by:  

[tex]z=\frac{p_{1}-p_{2}}{\sqrt{\hat p (1-\hat p)(\frac{1}{n_{1}}+\frac{1}{n_{2}})}}[/tex]   (1)

Where [tex]\hat p=\frac{X_{1}+X_{2}}{n_{1}+n_{2}}=\frac{37+58}{139+147}=0.332[/tex]

3) Calculate the statistic

Replacing in formula (1) the values obtained we got this:  

[tex]z=\frac{0.266-0.395}{\sqrt{0.332(1-0.332)(\frac{1}{139}+\frac{1}{147})}}=-2.32[/tex]  

4) Statistical decision

For this case we don't have a significance level provided [tex]\alpha[/tex] we can assuem it 0.05, and we can calculate the p value for this test.  

Since is a one left tailed test the p value would be:  

[tex]p_v =P(Z<-2.32)= 0.010[/tex]  

So if we compare the p value and using any significance level for example [tex]\alpha=0.05[/tex] we see that [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can say the the proportion 1 is significant lower than the proportion 2 at 5% of significance.  

Answer:

4y/(y+4)

Step-by-step explanation:

2y/(y-3) x [(4y -12) /(2y+8)]

To determine this, at first we have to break the parentheses. Since there is no matching values, we have to multiply the numerators and denominators.

[2y x (4y - 12)] / (y-3) x (2y + 8)

or, [(2y*4y) - (2y*12)]/[(y*2y) + (y*8) - (3*2y) - (3*8)]

(using algebraic equation)

or, (8y^2 - 24y)/(2y^2 + 8y - 6y - 24)

or, (8y^2 - 24y)/(2y^2 + 2y - 24)

or, 8y(y - 3)/2(y^2 + y - 12) (taking common)

or, 4y(y - 3)/(y^2 + 4y - 3y - 12)

or, 4y(y - 3)/[y(y + 4) - 3 (y + 4)] (Using factorization or Middle-Term factor)

or, 4y (y - 3)/(y + 4)(y - 3)

or, 4y/(y + 4) [as (y-3)/(y-3) = 1, we have dropped the part]

The answer is = 4y/(y+4)

Answer: Triangle, Square, Rectangle, Trapezium

Step-by-step explanation:

Cutting the cube from above, in a way that the slice is diagonal, making the slice touches two points that's almost at the edges diagonally facing each other of the cube will give a Trapezium (A)

Cutting the cube from above, in a way that the slice cuts exactly through the edges diagonally facing each other will give a Triangle (B)

Cutting the cube from above perpendicularly to the length, the two new faces made from the cube are squares (C)

Cutting the cube from above perpendicularly too will give two rectangles from the above face (D)

Sum of Option 2 does not equal to others

Step-by-step explanation:

We have to find each sum to check which is a outlier.

so,

Option 1:

∑i^2 where i = 1 to  3

[tex]Sum = 1^2+2^2+3^2\\=1+4+9\\=14[/tex]

Option 2:

∑i^3 where i = 1 to 2

So,

[tex]Sum = 1^3+2^3\\= 1 +8\\=9[/tex]

Option 3:

∑(i+1) where i = 1 to 4

[tex]Sum = (1+1) + (2+1) +(3+1) +(4+1)\\=2+3+4+5\\=14[/tex]

Option 4:

∑(2i-2) where i = 4 to 5

[tex]Sum = [2(4)-2]+[2(5)-2]\\=(8-2)+(10-2)\\=6+8\\=14[/tex]

Hence,

Sum of Option 2 does not equal to others

Keywords: Sum, Formulas

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The sum from i equals 1 to 2 of i cubed does not equal the others.

To determine which sum does not equal the others, we need to evaluate each sum.

From the evaluations, we can see that the sum from i equals 1 to 2 of i cubed does not equal the others.

Answer:

5000(1+0.0245) raise to 5

$5643.26

Step-by-step explanation:

The amount of money that will received after 5 years is $5643.256

Compound Interest

The compound interest of a primary money P with rate of interest r for time t is the total money that include interest and primary as well and can be calculated with the formula

[tex]A=P(1+\frac{r}{100})^t[/tex]

Here we have given

Primary money = P = $5000

Rate of interest = r = 2.45 %

Time = 5 year

Substitute these values into above formula and we get

[tex]A=5000(1+\frac{2.45}{100})^5[/tex]

[tex]A=5000(1.0245)^5[/tex]

A = $5643.256

Therefore the total amount that will received after 5 year is $5643.256

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Answer:

y = 6

Step-by-step explanation:

Its going by 6's

Answer: The correct option is C

Step-by-step explanation:

Looking at the right angle triangle ABC, three sides are known and the angles are unknown. To find sin A, we will take A to be our reference angle, we will have the following

Hypotenuse = AB = 26

Opposite side = BC = 24

Adjacent side = AC = 10

Applying trigonometric ratio

SinA = opposite/hypotenuse

SineA = 24/26

To find cos A, A remains our reference angle, we will have the following

Hypotenuse = AB = 26

Opposite side = BC = 24

Adjacent side = AC = 10

Applying trigonometric ratio

CosA = adjacent/hypotenuse

CosA = 10/26

The correct option is C

Answer:

[tex]\left\{\begin{array}{l}y\ge 2x+4\\ \\y<-x+2\end{array}\right.[/tex]

Step-by-step explanation:

1. The solid line passes trough the points (0,4) and (-2,0). The equation of this line is:

[tex]\dfrac{x-0}{-2-0}=\dfrac{y-4}{0-4}\\ \\y-4=2x\\ \\y=2x+4[/tex]

The origin doesn't belong to the shaded region, so its coordinates do not satisfy the inequality. Thus,

[tex]y\ge 2x+4[/tex]

2. The dotted line passes trough the points (0,2) and (2,0). The equation of this line is:

[tex]\dfrac{x-0}{2-0}=\dfrac{y-2}{0-2}\\ \\y-2=-x\\ \\y=-x+2[/tex]

The origin belongs to the shaded region, so its coordinates  satisfy the inequality. Thus,

[tex]y< -x+2[/tex]

Hence, the system of two inequalities is

Answer:

[tex]x_{1}=\frac{-13+\sqrt{153}}{2}\\x_{2}=\frac{-13-\sqrt{153}}{2}[/tex]

Step-by-step explanation:

The given expression is

[tex]x^{2}=-13x-4[/tex]

To solve this quadratic equation, we first need to place all terms in one side of the equation sign

[tex]x^{2} +13x+4=0[/tex]

Now, to find all solutions of this expression, we have to use the quadratic formula

[tex]x_{1,2}=\frac{-b\±\sqrt{b^{2}-4ac}}{2a}[/tex]

Where [tex]a=1[/tex], [tex]b=13[/tex] and [tex]c=4[/tex]

Replacing these values in the formula, we have

[tex]x_{1,2}=\frac{-13\±\sqrt{(13)^{2}-4(1)(4)}}{2(1)}\\x_{1,2}=\frac{-13\±\sqrt{169-16}}{2}=\frac{-13\±\sqrt{153}}{2}[/tex]

So, the solutions are

[tex]x_{1}=\frac{-13+\sqrt{153}}{2}\\x_{2}=\frac{-13-\sqrt{153}}{2}[/tex]

If we approximate each solution, it would be

[tex]x_{1}=\frac{-13+\sqrt{153}}{2}\approx -0.32\\\\x_{2}=\frac{-13-\sqrt{153}}{2} \approx -12.68[/tex]

Answer:

D on Edge

Step-by-step explanation:

The person that traveled the most distance is Julie.

An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.

Example: 2 + 3x + 4y = 7 is an expression.

We have,

From the figure,

Julie:

Total distance covered.

= sports complex to school + school to mall + mall + library

= 2/3 + 2/5 + 1(1/3)

= 2/3 + 2/5 + 4/3

= (10 + 6 + 20)/15

= 36/15

= 12/5

= 2(2/5) miles

= 2.4 miles

Kyle:

Total distance covered.

= house to mall + mall to library

= 4/5 + 1(1/3)

= 4/5 + 4/3

= (12 + 20)/15

= 32/15

= 2(2/15) miles

= 2.13 miles

Thus,

Julie has traveled more distance than Kyle.

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Answer: There are 50 ways to select in this way and there is only 1 combination is possible i.e. 3 students and 4 professors.

Step-by-step explanation:

Since we have given that

Number of professors = 5

Number of students = 5

We need to find the number of ways of 7 members in such that number of professors must be one more than students.

So, if we select 3 students, then there will be 4 professors.

So, Number of ways would be

[tex]^5C_3\times ^5C_4\\\\=10\times 5\\\\=50[/tex]

Hence, there are 50 ways to select in this way and there is only 1 combination is possible i.e. 3 students and 4 professors.

Answer:

Range is 57.

Variance is 375.143.

standard deviation is 19.37.

Step-by-step explanation:

Consider the provided information.

Range is the difference between highest and lowest data value.

The highest data value is 57 and lowest is 0.

Thus the range is 57-0=57

Range is 57.

Mean is the sum of data value divided by the number of data value:

[tex]\bar x=\frac{31+52+35+57+0+32+35+52+46+41+30+41+0+0}{14}\approx32.286[/tex]

The variance is the sum of squared deviation from the mean divided by n-1.

[tex]s^2 =\frac{\sum(x_i -\bar x)^2}{n - 1}[/tex]

Substitute the respective values in the above formula we get:

[tex]s^2=\frac{(31 - 32.286)^2 +(52-32.286)^2+ ... + (0 -32.286)^2}{14 - 1}\approx 375.143[/tex]

Hence, the variance is 375.143.

Standard deviation is square root of variance.

standard deviation = [tex]\sqrt{375.143}[/tex]

standard deviation ≈ 19.37

Hence, standard deviation is 19.37.

For all cans consumed, the statistics are not representative of the population because in the calculations each brand is weighted equally. Each of the 14 brands of soda is unlikely to be consumed in the same way.

It is very unlikely that all 14 drinks are consumed equally. So,given data is not representative of population

Answer:

E

Step-by-step explanation:

For a function to be differentiable at a point, it must be continuous at that point [ f(x⁻) = f(x⁺) ], and smooth at that point [ f'(x⁻) = f'(x⁺) ].

f(-1⁻) = 3(-1) + 5 = 2

f(-1⁺) = -(-1)² + 3 = 2

So the function is continuous.

f'(-1⁻) = 3

f'(-1⁺) = -2(-1) = 2

So the function is not smooth.

Therefore, the derivative f'(-1) does not exist.

4 inches of rain would fall in 6 hours

Given that, A rain storm came through Clifton park  

And it was accumulating [tex]\frac{2}{3}[/tex] inches of rain/hour

So amount of rain accumulated in 1 hour = [tex]\frac{2}{3}[/tex]

Thus amount of rain accumulated in six hours is calculated by multiplying the amount of water accumulating per hour and 6

Amount of water accumulated in 6 hours = Amount of water accumulated in 1 hour [tex]\times[/tex] 6

[tex]\text { Amount of water accumaulated in 6 hours }=\frac{2}{3} \times 6=4[/tex]

Another way:

Let "n" be the amount of rain accumulated in 6 hours

1 hour ⇒ [tex]\frac{2}{3}[/tex] rain accumulated

6 hours ⇒ "n"

By cross multiplication, we get

[tex]6 \times \frac{2}{3} = 1 \times n\\\\n = \frac{2}{3} \times 6 = 4[/tex]

Hence, 4 inches of rain would fall in 6 hours.

If the rain fell at a constant rate of 2/3 inch per hour, then 4 inches of rain would fall in a total of 6 hours. This calculation is made by multiplying the rate of rainfall by the total time.

The question asks how many inches of rain would fall in Clifton park in 6 hours if the rate was consistently 2/3 inch per hour. Given the constant rate of rainfall, we can calculate the total inches of rain that fell in 6 hours by multiplying the rate (2/3 inches/hour) by the total time in hours (6 hours).

So, doing the multiplication:

(2/3 inch/hour) * (6 hours) = 4 inches of rain.

This means that if the rain continued to fall at the same rate, we would expect 4 inches of rain to accumulate in Clifton park over 6 hours.

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Erica will take 4.5 minutes to run from home to school

Given that , Arica can run [tex]\frac{1}{6}[/tex] of a kilometer in a minute  

Her school is [tex]\frac{3}{4}[/tex] th of a kilometer away from her home  

We have to find at this speed how long will it take Erica to run from home to school

The relation between speed distance and time is given as:

[tex]\text { Distance }=\text { speed } \times \text { time }[/tex]

Plugging in values, we get

[tex]\frac{3}{4}=\frac{1}{6} \times \text { time taken }[/tex]

[tex]\begin{array}{l}{\text { Time taken to reach school }=\frac{3}{4} \times 6} \\\\ {\text { Time taken to reach home }=\frac{3}{2} \times 3} \\\\ {\text { Time taken to reach home }=\frac{9}{2}=4.5}\end{array}[/tex]

Hence, she takes 4.5 minutes to reach school from her home

Answer:

The third graph from left to right

Step-by-step explanation:

The function [tex]f(x)=\left [ x \right ][/tex] is called Greatest Integer Function of x is such that it returns the largest integer  less than or equal to x

Some examples of points are (0,0),(0.5,0),(1,1),(1.9,1),(-0.7,-1)

Since our function is

[tex]f(x)=\left [ x \right ]-2[/tex]

We must subtract 2 to the points above like

(0,-2),(0.5,-2),(1,-1),(1.9,-1),(-0.7,-3)

The only graph that complies with such requirements is the third one

Answer:

The solutions for 3 questions are explained one after the other below.

Step-by-step explanation:

1).Let x be the number of paperback books that she buys,  y be the number of hardback books that she buys.

for the first condition, i.e, she has decided to spend at most $150.00 on books,the required inequality will be :

[tex]8x+12y\leq 150[/tex]

for the second condition , i.e, she wants to purchase at least 12 books,

the required inequality will be:

[tex]x+y\geq 12[/tex]

2). the graph is in the attachment..

3). x,y are the two required solutions. where,

x =number of paperback books she buys.

y=number of hardback books she buys.

Answer:

1) equations 1, 2, 3  and 4

2)  see picture attached (the region of the solutions is in yellow)

3) x = 18.75 and y =0

   y = 12.5 and x =0

Step-by-step explanation:

Let's call x the number of paperback books bought and y the number of  hardback books bought.

She decided to spend at most $150.00. All paperback books are $8.00 and all hardback books are $12.00. Combining this information we get:

x*8 + y*12 ≤ 150 (eq.  1)

Anna wants to purchase at least 12 books. Mathematically:

x + y ≥ 12 (eq. 2)

On the other hand,  both the number of paperback books bought and the number of  hardback books bought must be positive, that is:

x ≥ 0 (eq. 3)

y ≥ 0 (eq. 4)

3) One possible solution is got if we make y = 0 and to take eq. 1 as an equality, then:

From eq. 1: x*8 = 150

x = 150/8 = 18.75

equations 2 and 3 are also satisfied

Another option is to make x = 0 and to take eq. 1 as an equality, then:

From eq. 1: y*12 = 150

y = 150/12 = 12.5

equations 2 and 4 are also satisfied

Answer:

Step-by-step explanation:

volume of water

[tex]=\frac{1}{2}*50*40*60=60000 ~cm^3[/tex]

when the base is 100×60

let h be depth of water.

100×60×h=60000

h=60000/6000=10 cm.

Aa. 8x2 - 2

i

Step-by-step explanation:

4x2 = 8

8-3 = 5

5 + 4 =

Answer:

c. 2x2+6x−2

Step-by-step explanation:

Answer: B

Step-by-step explanation:

Running at the same constant rate, 6 identical machines can produce a total of 270 bottles per minute. Since the machines are identical and running at the same constant rate, it means each of them as the same rate. The rate of each machine can produce would be determined by dividing the combined unit rate by 6. It becomes

270/6 = 45 bottles per minutes

The rate for 10 machines running at the same constant rate would be

10 × 45 = 450 bottles per minutes.

If the 10 machines produce 450 bottles per minutes, then,

In 4 minutes, the 10 machines will produce 4 × 450 = 1800 bottles

Answer:

  k = 4

Step-by-step explanation:

The remainder theorem tells you that the remainder from division of f(x) by (x-1) is f(1). Evaluating the expression for x=1 gives ...

  2(1³) -3(1²) +k(1) -1 = 2 -3 +k -1 = k -2

We want this to be equal to 2, so ...

  k -2 = 2

  k = 4

Applying the Remainder Theorem to the given polynomial, we can substitute x = 1 into the polynomial equation and solve for k, which gives us k = 4.

The question asks to find the value of k when given polynomial 2x^3 - 3x^2 + kx - 1 is divided by x - 1 and the remainder is 2. We utilize the Remainder Theorem for this, which states that when a polynomial f(x) is divided by x-c, the remainder is equal to f(c).

So, by substituting x = 1 in the given polynomial as per the Remainder Theorem, we have: 2(1)^3 - 3(1)^2 + k(1) - 1 = 2. Simplifying this equation leads us to: 2 - 3 + k -1 = 2, which can further be simplified to k - 2 = 2. Thereby, solving for k gives us k = 4.

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Answer:

55 trees per hectare.

Step-by-step explanation:

An apple orchard contains 50 trees per hector. The average yield per tree is 600 apples.

If the trees are spaced more closely, when being planted, the yield per tree drops by 10 apples for each extra tree.

Let x extra tree is planted and then the average yield per tree reduces by 10x.

Therefore, yield as a function of x can be written as  

Y(x) = (50 + x)(600 - 10x) = 30000 + 100x - 10x²

Therefore, condition for maximum yield is [tex]\frac{dY(x)}{dx} = 0[/tex]

So, 100 - 20x = 0

⇒ x = 5

So, when the number of trees that should be planted per hectare is (50 + 5) = 55, then only the yield will be maximum. (Answer)

Answer: 2w

Step-by-step explanation: if the garden is shaped like a square, then all the sides are equal,

length = breadth, and the Area of a square or rectangle is the length multiplied by the breadth

and to find the length and breadth, we find the square root of the area

The area is 4w2

We know that 4 is the perfect square of 2, making 2 the square root of 4

And w2 is the square of w

This is elementary algebra, a x a = a2

b x b = b2, w x w = w2

So adding both together, the square root of 4w2 = 2w

The length of each side of the square-shaped garden with area 4w2 is 2w.

The student has given the area of a square-shaped garden as 4w2. Since the area of a square is calculated by squaring the length of one of its sides (side2), to find the length of each side, we need to find the square root of the area. The square root of 4w2 is 2w, because (2w)2 equals 4w2. Therefore, the length of each side of the garden is 2w.

Answer:

E. 12 days

Step-by-step explanation:

So first, we need to find the rates at which each machine will produce w widgets. For machine y, its rate would be:

[tex]y=\frac{W}{T}[/tex]

where W is the number of widgets produced and T is the time it takes to produce them.

We know that x takes 2 more days to produce the same amount of widgets, so the time it takes machine x to produce them can be written as T+2. This will give us the following rate for machine x:

[tex]x=\frac{W}{T+2}[/tex]

the problem also tells us that the two machines working together will produce 5W/4 widgets in 3 days, so if we add the rates for x and y, we will get the total rate which would be:

[tex]x+y=\frac{5W/4}{3}[/tex]

which can be simplified to:

[tex]x+y=\frac{5W}{12}[/tex]

we can now substitute the rates for x and y in the equation so we get:

[tex]\frac{W}{T+2}+\frac{W}{T}=\frac{5W}{12}[/tex]

we can simplify this equation by dividing everything into W, so we get:

[tex]\frac{1}{T+2}+\frac{1}{T}=\frac{5}{12}[/tex]

and we can multiply everything by the LCD. In this case the LCD is 12T(T+2) so we get:

[tex]\frac{1}{T+2}(12T)(T+2)+\frac{1}{T}(12T)(T+2)=\frac{5}{12}(12T)(T+2)[/tex]

which simplifies to:

12T+12(T+2)=5T(T+2)

we can do the respective multiplications so we get:

[tex]12T+12T+24=5T^{2}+10T[/tex]

which simplifies to:

[tex]24T+24=5T^{2}+10T[/tex]

and now we can set the equation equal to zero so we end up with:

[tex]5T^{2}+10T-24T-24=0[/tex]

which simplifies to:

[tex]5t^{2}+14T-24=0[/tex]

now we can solve this by any of the available methods there are to solve quadratic equations. I will solve it by factoring, so we get:

(5T+6)(T-4)=0

so we can set each of the factors equal to zero so we get:

5T+6=0

[tex]T=-\frac{6}{5}[/tex]

this answer isn't valid because there is no such thing as a negative time. So we find the next time then:

T-4=0

T=4

So it takes 4 days for machine x to produce W widgets. We can now rewrite x's rate like this:

[tex]x=\frac{W}{T+2}[/tex]

so

[tex]x=\frac{W}{4+2}[/tex]

[tex]x=\frac{W}{6}[/tex]

With this information, we know that the number of wigets produced can be found by using the following formula:

W=xd

in this case d is the number of days (this is for us not to confuse the previous T with the new time)

so when solving for d we get that:

[tex]d=\frac{W}{x}[/tex]

so when substituting we get that:

[tex]d=\frac{2W}{W/6}[/tex]

when simplifying we get that:

d=12

Answer:

Axis of symmetry.

Step-by-step explanation:

We have been given an incomplete statement. We are supposed to complete the given statement.

Given statement: The vertical line passing through the vertex of a parabola is called the ________.

We know that a parabola is symmetric about axis of symmetry . The line passing through the vertex of parabola divides the parabola into two mirror images.

Therefore, the vertical line passing through the vertex of a parabola is called the the axis of symmetry.

Answer:

V'(t) = [tex]-250(1 - \frac{1}{40}t)[/tex]

If we know the time, we can plug in the value for "t" in the above derivative and find how much water drained for the given point of t.

Step-by-step explanation:

Given:

V = [tex]5000(1 - \frac{1}{40}t )^2[/tex]  , where 0≤t≤40.

Here we have to find the derivative with respect to "t"

We have to use the chain rule to find the derivative.

V'(t) = [tex]2(5000)(1 - \frac{1}{40} t)d/dt (1 - \frac{1}{40}t )[/tex]

V'(t) = [tex]2(5000)(1 - \frac{1}{40} t)(-\frac{1}{40} )[/tex]

When we simplify the above, we get

V'(t) = [tex]-250(1 - \frac{1}{40}t)[/tex]

If we know the time, we can plug in the value for "t" and find how much water drained for the given point of t.

Answer:

Solutions of the equation are 22.5°, 30°.

Step-by-step explanation:

The given equation is sin(5θ) - sin(3θ) = cos(4θ)

We take left side of the equation

sin(5θ) - sin(3θ) = [tex]2cos(\frac{5\theta+3\theta}{2})sin(\frac{5\theta-3\theta}{2})[/tex]

= [tex]2cos(4\theta)sin(\theta)[/tex] [From sum-product identity]

Now we can write the equation as

2cos(4θ)sin(θ) = cos(4θ)

2cos(4θ)sinθ - cos(4θ) = 0

cos(4θ)[2sinθ - 1] = 0

cos(4θ) = 0

4θ = 90°

θ = [tex]\frac{90}{4}[/tex]

θ = 22.5°

and (2sinθ - 1) = 0

sinθ = [tex]\frac{1}{2}[/tex]

θ = 30°

Therefore, solutions of the equation are 22.5°, 30°

Answer:

The volume of the solid is [tex]\frac{48\pi}{5}[/tex]

Step-by-step explanation:

Consider the provided information.

The​ cross-sections perpendicular to the​ y-axis are circular disks with diameters running from the​ y-axis to the parabola [tex]x=\sqrt6y^2[/tex]

Therefore, diameter is [tex]d=\sqrt6y^2[/tex]

Radius will be [tex]r=\frac{\sqrt6y^2}{2}[/tex]

We can calculate the area of circular disk as: πr²

Substitute the respective values we get:

[tex]A=\pi(\frac{\sqrt6y^2}{2})^2[/tex]

[tex]A=\pi(\frac{6y^4}{4})=\frac{3\pi y^4}{2}[/tex]

Thus the volume of the solid is:

[tex]V=\int\limits^2_0 {\frac{3\pi y^4}{2}} \, dy[/tex]

[tex]V=[{\frac{3\pi y^5}{2\times 5}}]^2_0[/tex]

[tex]V=\frac{48\pi}{5}[/tex]

Hence, the volume of the solid is [tex]\frac{48\pi}{5}[/tex]

The volume of solid represent the how much space an object occupied. In the given problem volume can be determine by taking the integration of Area of solid.

The volume of solid is [tex]\frac{48\pi }{5}[/tex].

Given:

The​ cross-sections perpendicular to the​ y-axis are circular disks with diameters running from the​ y-axis to the parabola is [tex]x=\sqrt{6}y^2[/tex].

The diameter of the solid is [tex]d=\sqrt{6}y^2[/tex].

Calculate the radius of the solid.

[tex]r=\frac{d}{2}\\r=\frac{\sqrt{6}y^2}{2}[/tex]

Write the expression for area of circular disk.

[tex]A=\pi r^2\\A=\pi (\frac{\sqrt{6}y^2}{2})^2\\A=\frac{3\pi y^4}{2}[/tex]

Calculate the volume of solid.

[tex]V=\int\limits^2_0 {\frac{3\pi y^4 }{2} } \, dy\\V=[\frac{3\pi y^5}{2\times 5}]_{0}^{2}\\V=\frac{48\pi }{5}[/tex]

Thus, the volume of solid is [tex]\frac{48\pi }{5}[/tex] .

Learn more about volume here:

https://brainly.com/question/13569823

Answer:

A) First number: 13

B) Second number: 21

C) Third number: 63

Step-by-step explanation:

Let x, y and z be 1st, 2nd and 3rd numbers respectively.

We have been given that sum of three numbers is 97. We can represent this information in an equation as:

[tex]x+y+z=97...(1)[/tex]

The 3rd number is 3 times the second. We can represent this information in an equation as:

[tex]z=3y...(2)[/tex]

The second number is 8 more than the first. We can represent this information in an equation as:

[tex]y=x+8...(3)[/tex]

Substituting equation (3) in equation (2), we will get:

[tex]z=3(x+8)[/tex]

Substituting [tex]z=3(x+8)[/tex] and [tex]y=x+8[/tex] in equation (1), we will get:

[tex]x+x+8+3(x+8)=97[/tex]

[tex]x+x+8+3x+24=97[/tex]

[tex]5x+32=97[/tex]

[tex]5x+32-32=97-32[/tex]

[tex]5x=65[/tex]

[tex]\frac{5x}{5}=\frac{65}{5}[/tex]

[tex]x=13[/tex]

Therefore, the first number is 13.

Now, we will substitute [tex]x=13[/tex] in equation (3) as:

[tex]y=21[/tex]

Therefore, the second number is 21.

Now, we will substitute [tex]y=21[/tex] in equation (2) as:

[tex]z=3(21)[/tex]

[tex]z=63[/tex]

Therefore, the third number is 63.