) Consider the following probability density function of the random variable X. f(x) = k(2x + 3x2 ), 0 ≤ x ≤ 2. (i) Determine the value of the constant k.

Answer: the solutions to the equation are x = √(10/3) and x = -√(10/3).

Step-by-step explanation:

To solve the equation 10 - 9x^2 + 4x = -6x^2 + 4x, we can simplify it and then solve for x.

Rearranging the equation, we have:

10 - 9x^2 + 4x = -6x^2 + 4x

Combining like terms, we get:

10 - 9x^2 = -6x^2

Subtracting -6x^2 from both sides, we have:

10 - 9x^2 + 6x^2 = 0

Simplifying further, we get:

10 - 3x^2 = 0

To solve for x, we can isolate the term with x^2:

-3x^2 = -10

Dividing both sides by -3, we have:

x^2 = 10/3

Taking the square root of both sides, we get:

x = ±√(10/3)

Therefore, the solutions to the equation are x = √(10/3) and x = -√(10/3).

The total number of pawpaws was 160, which can be found by using simple equations considering the amount taken by Eric and the proportion.

Let's start by finding out how many pawpaws are left after Mark takes 1/4 of the total. Let T be the total number of pawpaws. Mark gets 1/4 of T, which means there are 3/4 of T left:

3/4T = Total number of pawpaws - Mark's share

3/4T = T - 1/4T

3/4T = 3/4T

So, 3/4T is the amount shared between Eric and Francis. The ratio of Eric's share to Francis' share is 2:3. Let's call Eric's share E and Francis's share F.

E/F = 2/3

We can use this ratio to write an equation for Eric's share in terms of F:

E/F = 2/3

E = (2/3)F

We know Eric's share is 48, so we can substitute that into the equation and solve for F:

E = (2/3)F

48 = (2/3)F

F = (48*3)/2

F = 72

So, Francis has 72 pawpaws.

We can now find out how many pawpaws Mark got:

Mark's share + Eric's share + Francis's share = Total number of pawpaws

1/4T + E + F = T

1/4T + 48 + 72 = T

T = 480/3

T = 160

So, the total number of pawpaws is 160. Mark got 1/4 of the total, which is:

Mark's share = (1/4)*160 = 40

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The surface area for each of the cylinders is given as follows:

13. 126 yd².

14. 490 m².

15. 283 mm².

16. 297 cm².

The surface area of a cylinder of radius r and height h is given by the equation presented as follows, which combines the base area with the lateral area:

S = 2πrh + 2πr²

S = 2πr(h + r)

Item 13:

r = 2 yd and h = 8 yd, hence the surface area is given as follows:

S = 2π x 2(2 + 8)

S = 126 yd².

Item 14:

r = 6 m and h = 7 m, hence the surface area is given as follows:

S = 2π x 6(6 + 7)

S = 490 m².

Item 15:

r = 3 mm and h = 12 mm, hence the surface area is given as follows:

S = 2π x 3(3 + 12)

S = 283 mm².

Item 16:

r = 3.5 mm and h = 10 mm, hence the surface area is given as follows:

S = 2π x 3.5(3.5 + 10)

S = 297 cm².

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The product of powers of exponents cannot be used to simplify the binomial expansion

Given data ,

Let the binomial expansion be represented as A

A = ( 3z + y )³

According to the property of products of powers, exponents can be multiplied when a power is increased to a higher power.

The product of powers characteristic cannot be applied to the equation (3z + y)³. This is due to the fact that (3z + y)³ is a binomial raised to the power of 3, not just a power of a single word.

These terms cannot be simplified further using the product of powers property because they involve different variables or variable combinations.

In this case, it is more appropriate to expand the expression using the binomial expansion

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The solution of the equation is x=1/2 and y=1/3

The given equations are

18x+12y=13...(1)

30x+24y=23...(2)

Let us form the matrix form to solve the equation

[tex]\left[\begin{array}{ccc}18&12\\30&24\end{array}\right] \left[\begin{array}{ccc}x\\y\end{array}\right] =\left[\begin{array}{ccc}13\\23\end{array}\right][/tex]

Multiply equation (1) with 2

36x+24y=26..(3)

Subtract equation 3 from equation 2

30x+24y-36x-24y=23-26

-6x=-3

x=1/2

Now plug in the value of x in equation 1

18(1/2)+12y=13

9+12y=13

12y=4

y=4/12

y=1/3

Hence, the solution of the equation is x=1/2 and y=1/3

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A) One way to write mixed fractions [tex]6\frac{2}{4} \\[/tex]  is 3 + 3 + [tex]\frac{1}{4}[/tex] +[tex]\frac{1}{4}\\[/tex]

B) Saturday: 3 to 4 hours work = trim bushes and weed garden

Sunday: 4 to 5 hour work = paint sheet and mow lawn

A) [tex]6\frac{2}{4} \\[/tex]  can be written as a simple fraction 6 + [tex]\frac{2}{4}[/tex]

This can be further broken into and written as

3 + 3 +  [tex]\frac{1}{4}[/tex] +[tex]\frac{1}{4}\\[/tex]

B) Saturday : 3 to 4 hours of work

Trim bushes + Weed garden

[tex]1\frac{1}{6} +2\frac{2}{6}[/tex]

1 + [tex]\frac{1}{6}[/tex] + 2+ [tex]\frac{2}{6}\\[/tex]

3 + [tex]\frac{3}{6}[/tex]

3 + [tex]\frac{1}{2}[/tex]

[tex]3\frac{1}{2}[/tex]

Sunday: 4 to 5 hours of work

Paint sheet + mow lawn

[tex]1\frac{3}{6} +3 \frac{4}{6}[/tex]

1 + 3 + [tex]\frac{3}{6} +\frac{4}{6}[/tex]

4 + [tex]\frac{8}{6}[/tex]

[tex]4\frac{8}{6}[/tex]

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The Given question is incomplete the complete question is :

Mr. mason likes to work around the yard during the weekends

This is a ratio problem and Sophia is expected to simplify the ratio by finding the smallest possible values and not compounding them by multiplying them by some values.

We can represent the given ratio as 60:36,

60/36

We proceed to reduce the fraction by dividing both the numerator and the denominator by a common factor say 6,

10/6

We can further reduce this with a common factor of 2

5/2

Thus, the ratio we have 5:2

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The solutions to the quadratic equation 6x² = -3x + 1 to the nearest hundredth are -0.73 and 0.23.

Given the quadratic equation in the question:

6x² = -3x + 1

To solve the quadratic equation 6x² = -3x + 1, we can rearrange it into standard form, where one side is set to zero:

6x² + 3x - 1 = 0

Now we can solve the equation using the quadratic formula, which states that for an equation in the form ax² + bx + c = 0, the solutions for x are given by:

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

Here; a = 6, b = 3, and c = -1.

Let's substitute these values into the quadratic formula:

[tex]x = \frac{-b \±\sqrt{b^2-4ac} }{2a}\\\\ x= \frac{-3 \±\sqrt{3^2-4\ *\ 6\ *\ -1} }{2*6}\\\\x = \frac{-3 \±\sqrt{9+24} }{12}\\\\x = \frac{-3 \±\sqrt{33} }{12}\\\\x = -0.73, \ x=0.23[/tex]

Therefore, the values of x are -0.73 and 0.23.

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The expected value for a person buying a ticket is $0.35 rounded to the nearest cent.

The expected value is calculated considering the probabilities of winning each prize and the corresponding values of each prize.

Assuming:

P($25) as the probability of winning the $25 prize

P($10) as the probability of winning the $10 prize

There are 100 tickets sold, therefore, the probabilities can be found as follows:

P($25) = 1/100 (since there is only 1 $25 prize)

P($10) = 1/100 (since there is only 1 $10 prize)

The expected value (E), will then be:

E = P($25) * $25 + P($10) * $10

E = (1/100) * $25 + (1/100) * $10

E = $0.25 + $0.10

E = $0.35

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The number of lego planes that can be build in 40 minutes is A = 2.33

Given data ,

Johnny can build in 3 1/2 lego planes in 60 minutes

On dividing the number of planes he can build in 60 minutes (3 1/2) by 60:

From the proportion , we get

To find out how many planes he can build in 40 minutes, we can multiply the amount he can build in one minute by 40:

3.5 / 60 = A / 40

Multiply by 40 on both sides , we get

A = 2.33

Hence , Johnny can build approximately 2.33 lego planes in 40 minutes

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The evaluation of the geometric series is found as:

[tex]5(1- \frac{2}{3}^{8} / 1- \frac{2}{3} )[/tex]

The given series is

[tex]8∑k=1 5(2/3)^k-1[/tex]

A  geometric series is described as the sum of an infinite number of terms that have a constant ratio between successive terms.

Since this is a geometric series, we apply  formula for the sum of the first k terms.

[tex]S= a( 1- r^{k} / 1-r )[/tex]

From the series, first term  a = 5, common ratio  r = 2/3 , k = 8

We substitute the values to obtain:

he evaluation of the series is found as:

[tex]5(1- \frac{2}{3}^{8} / 1- \frac{2}{3} )[/tex]

Geometric series finds its applications in Physics, engineering, biology, economics, computer science, queueing theory, and finance.

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complete question is attached in image

Answer:

[tex](x-14)^2 +(y-22)^2 = (\sqrt{65})^2[/tex]

Step-by-step explanation:

If the two given points are the extremes of the diameter, the center of the circle has to be its middle point - that we can find by taking the average of the coordinates. The center thus sits in

[tex](\frac{22+6}2; \frac{21+23}2)[/tex] or [tex](14; 22)[/tex]. At this point we either find the length of the diameter and halve it, or the distance between the center and either point. Let's go for the diameter.

[tex]r=\sqrt{(22-6)^2+(21-23)^2}=\sqrt{16^2+2^2} = \sqrt {260}=2\sqrt{65}[/tex]. That makes our radius half of that. We can easily write the equation of the circle now:

[tex](x-14)^2 +(y-22)^2 = (\sqrt{65})^2[/tex]

Now, in theory you can improve it by multiplying it out and taking every term to the LHS, but I think it's good enough like that.

Answer:

Step-by-step explanation:

$100x0.5x1=$5

Step-by-step explanation:

3. Let P(t) be the number of people infected by the virus at time t (in days). We can model the situation with the following exponential function:

P(t) = 400 * 1.25^t

Here, 400 represents the initial number of infected people, and 1.25 represents the growth factor, since the virus is spreading to 25% more people each day.

To find the number of people infected after t days, we can substitute t = (log(3000) - log(400)) / log(1.25) into the equation:

P(t) = 400 * 1.25^t

P(t) = 400 * 1.25^((log(3000) - log(400)) / log(1.25))

P(t) ≈ 2,343

Therefore, approximately 2,343 people are infected when the total number of infections reaches 3000.

4. Let P(t) be the population of the town at time t (in years). We can model the situation with the following exponential function:

P(t) = 10,800 * 0.975^t

Here, 10,800 represents the initial population in 2002, and 0.975 represents the decay factor, since the population is decreasing at a rate of 2.5% each year.

To find when the population reaches half the 2002 value, we can set P(t) = 5,400 and solve for t:

5,400 = 10,800 * 0.975^t

0.5 = 0.975^t

log(0.5) = t * log(0.975)

t ≈ 28.2

Therefore, the population will reach half the 2002 value in approximately 28.2 years, which corresponds to the year 2030.

Answer:

3) 9.03 days

4)  27.38 years

Step-by-step explanation:

To model the spread of the virus over time, we can use an exponential function in the form:

[tex]\large\boxed{P(t) = P_0(1 + r)^t}[/tex]

where:

Given the virus has infected 400 people in the town and is spreading to 25% more people each day:

Substitute these values into the formula to create a function for P in terms of t:

[tex]P(t) = 400(1 + 0.25)^t[/tex]

[tex]P(t) = 400(1.25)^t[/tex]

To find how many days it will take for 3000 people to be infected, set P(t) equal to 3000 and solve for t:

[tex]\begin{aligned}P(t)&=3000\\\implies 400(1.25)^t&=3000\\(1.25)^t&=7.5 \\\ln (1.25)^t&=\ln(7.5)\\t \ln (1.25)&=\ln(7.5)\\t &=\dfrac{\ln(7.5)}{\ln (1.25)}\\t&=9.02962693...\end{aligned}[/tex]

Therefore, it will take approximately 9.03 days for the virus to infect 3000 people, assuming the daily growth rate remains constant at 25%.

Note: After 9 days, 2980 people would be infected. After 10 days, 3725 people would be infected.

[tex]\hrulefill[/tex]

To model the population of the town over time, we can use an exponential function in the form:

[tex]\large\boxed{P(t) = P_0(1 - r)^t}[/tex]

where:

Given the initial population was 10,800 and the population has decreased at a rate of 2.5% each year:

Substitute these values into the formula to create a function for P in terms of t:

[tex]P(t) = 10800(1 -0.025)^t[/tex]

[tex]P(t) = 10800(0.975)^t[/tex]

To find how many days it will take for the population to halve, set P(t) equal to 5400 and solve for t:

[tex]\begin{aligned}P(t)&=5400\\\implies 10800(0.975)^t&=5400\\(0.975)^t&=0.5 \\\ln (0.975)^t&=\ln(0.5)\\t \ln (0.975)&=\ln(0.5)\\t &=\dfrac{\ln(0.5)}{\ln (0.975)}\\t&=27.3778512...\end{aligned}[/tex]

Therefore, it will take approximately 27.38 years for the population to reach half the 2002 value, assuming the annual decay rate remains constant at 2.5%.

ANSWER: 100% of 2000000000 is 2000000000

What is 100 Percent of 2000000000?

100 percent *2000000000

= (100/100)*2000000000

= (100*2000000000)/100

= 200000000000/100 = 2000000000

Now we have: 100 percent of 2000000000 = 2000000000

Question: What is 100 percent of 2000000000?

We need to determine 100% of 2000000000 now and the procedure explaining it as such

Step 1: In the given case Output Value is 2000000000.

Step 2: Let us consider the unknown value as x.

Step 3: Consider the output value of 2000000000 = 100%.

Step 4: In the Same way, x = 100%.

Step 5: On dividing the pair of simple equations we got the equation as under

2000000000 = 100% (1).

x = 100% (2).

(2000000000%)/(x%) = 100/100

Step 6: Reciprocal of both the sides results in the following equation

x%/2000000000% = 100/100

Step 7: Simplifying the above obtained equation further will tell what is 100% of 2000000000

x = 2000000000%

Therefore, 100% of 2000000000 is 2000000000

Using similar side theorem, the side with equivalent proportion to the given side is RQ/SQ

Similar triangles are triangles that have the same shape, but their sizes may vary. All equilateral triangles, squares of any side lengths are examples of similar objects. In other words, if two triangles are similar, then their corresponding angles are congruent and corresponding sides are in equal proportion.

In this problem, we can use this same theory to find the equivalent side of the given proportion.

OQ / PQ = RQ / SQ

The equivalent side is RQ/SQ

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The question that Diondra would ask that does not allow for variability is A. Which movie did her best friend like the most?

The other three questions ("How many awards did each movie win?", "What is the total dollar amount that each movie made from ticket sales?", and "How many weeks did each movie play in the theater?") all offer metrics by which a movie's popularity could be measured, and those measurements could vary from movie to movie.

However, the question about her best friend's favorite movie depends solely on the opinion of one individual, not a variable measure, and therefore doesn't allow for variability.

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Answer:

Step-by-step explanation:

y=mx+b

m= slope and b= y intercept

you'll start with a point on the y axis on -1

and the slope will go up so towards the right corner up but 2/1 points

Based on the given polynomial function, as x approaches positive or negative infinity, the function f(x) = 2x³ - 3x - 4x⁴ + 1 approaches negative infinity.

The end behavior of a polynomial function is determined as follows:

Examine the leading term of the function f(x) = -4x⁴ + 2x³ - 3x + 1

The leading term of the function is -4x⁴.

For large positive values of x, as x approaches positive infinity, the leading term -4x⁴ becomes increasingly negative since x⁴ grows faster than any positive constant.

Therefore, the end behavior of the function as x approaches positive infinity is that f(x) approaches negative infinity.

Similarly, for large negative values of x, as x approaches negative infinity, the leading term -4x⁴ becomes increasingly negative since x⁴ grows faster than any positive constant.

Therefore, the end behavior of the function as x approaches negative infinity is that f(x) approaches negative infinity.

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Answer:

B. 1 tick mark represents 0.2 unit

Step-by-step explanation:

Looking at the axis and going over 2 tick marks is .4

.4 /2 = .2

Each tick mark is .2 units

Answer:

20

Step-by-step explanation:

She biked an equal amount each day for 8 days to a total of 32 miles. We can write that as 8x = 32. 32/8 = 4 so x = 4. To find how much shell bike in 5 days, we multiply it by x(4). 5*4 = 20.

The transformations are:

Shift up 3 units.

Shift left 5 units.

Vertical compression of scale factor 2.

The correct option is the first one.

Here, we have,

Here we start with the original function:

y = (x - 1)^2 - 3

And it will be transformed into the new function:

y = (1/2)*(x + 4)^2

So let's start with the first function, and apply transformations until we reach the new one.

y = (x - 1)^2 - 3

If we shift it up by 3 units, then we get:

y = [ (x - 1)^2 - 3 ] + 3 = (x - 1)^2

If now we shift it to the left 5 units, then we get:

y = ( (x + 5) - 1)^2 = (x + 4)^2

finally, we compress the graph by a factor of 2, so we get:

y = (1/2)*(x + 4)^2

The order may be different, but from that we conclude that the correct option is the first option.

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complete question:

graph of y = (x- 1)2 -3 transformed to produce the graph of y = - 1/2(X+-4)2?

O The graph is translated left 5 units, compressed vertically by a factor a of 2, and translated up 3 units.

The graph is stretched vertically by a factor of 2, translated left 5 units, and translated up 3 units.

• The graph is translated left 5 units, compressed horizontally by a factor of 2, and translated down 3 units.

• The graph is stretched horizontally by a factor of 7, translated left 5 units, and translated down 3 units:

The range of possible values for the population parameter can be estimated using the margin of error, which is calculated as the critical value times the standard error.

Assuming a 95% confidence level, the critical value is approximately 1.96. The standard error for a sample proportion can be calculated as:

SE = sqrt[(p * (1 - p)) / n]

Where p is the sample proportion and n is the sample size. Substituting the values given in the question, we get:

SE = sqrt[(0.65 * 0.35) / n]

We do not know the sample size, so we cannot calculate the standard error exactly. However, we can use a rule of thumb that states that if the sample size is at least 30, we can use the normal distribution to estimate the margin of error.

With a sample proportion of 0.65, the margin of error can be estimated as:

ME = 1.96 * sqrt[(0.65 * 0.35) / n]

We do not know the sample size, so we cannot calculate the margin of error exactly. However, we can use the rule of thumb that a margin of error of about ±5% is typical for a 95% confidence level.

Using this margin of error, we can construct the following range of possible values for the population parameter:

0.65 ± 0.05

This range can be expressed as (0.6, 0.7), which corresponds to option A.

Therefore, the correct answer is option A) (0.6, 0.69).

The probability that Anita gets a purple tulip and Lois gets a pink rose is 1/9.

Total number of flowers in the bouquet = 7 purple tulips + 9 yellow daisies + 12 pink roses = 28 flowers.

P(Anita gets a purple tulip) = 7 purple tulips / 28 total flowers = 7/28 = 1/4.

P(Lois gets a pink rose) = 12 pink roses / 27 remaining flowers = 12/27 = 4/9.

P(Anita gets a purple tulip and Lois gets a pink rose) = P(Anita gets a purple tulip) * P(Lois gets a pink rose)

= (1/4) * (4/9) = 1/9.

Therefore, the probability that Anita gets a purple tulip and Lois gets a pink rose is 1/9.

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Answer:

The answer is 176

Step-by-step explanation:

First lets find the area of rectangle by multiplying 16 by 8 to get 128.

Now lets find the area of each triangle. We know that one side is 6 cm and the base of both is 16.

So to find the base of one triangle we divide 16 by 2 to get 8.

Now we know that the base of one triangle is 8, so lets sub. that into the equation for a triangle. 1/2(b)(h) to get

1/2(6)(8)=24

Since there are two triangle 24+24=48 for area of both triangles

Now lets add that to the answer for the rectangle to get area for whole structure

128+48 = 176

So the answer is 176

HOPE THAT HELPS :)

If Ivy had given 3 more people bonuses, Ivy would have rewarded 13 of the workforce, Ivy Corporation's workforce has 592 employees.

Let's assume that the total workforce at Ivy Corporation is represented by "x".

According to the problem statement, Ivy Corporation gave a bonus to 74 people. Therefore, the remaining non-bonus-receiving employees would be (x-74).

If Ivy had given 3 more people bonuses, then the number of employees that would receive the bonus would be (74+3)=77.

According to the problem, 77 is equal to 13% of the total workforce (x):

77 = 0.13x

We can solve for x by dividing both sides by 0.13:

x = 592

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If the initial velocity is 75 feet per second, it will take approximately 5.125 seconds for the ball to hit the ground.

The given formula h= -16t²+vt+s represents the height (h) of an object thrown vertically in the air at time (t), with initial velocity (v) and initial height (s). In this case, we are given that the initial height of the softball is 4 feet and the initial velocity is 75 feet per second.

We want to find out how long it will take for the ball to hit the ground, which means we want to find the time (t) when the height (h) is 0.

Substituting the given values into the formula, we get:

0 = -16t² + 75t + 4

This is a quadratic equation in standard form, which we can solve using the quadratic formula:

t = (-b ± √(b² - 4ac)) / 2a

Where a=-16, b=75, and c=4. Substituting these values into the formula, we get:

t = (-75 ± √(75² - 4(-16)(4))) / 2(-16)

t = (-75 ± √(5625 + 256)) / (-32)

t = (-75 ± √(5881)) / (-32)

We can simplify the expression under the square root as follows:

√(5881) = √(49121) = 711 = 77

So we have:

t = (-75 ± 77) / (-32)

Simplifying further, we get two possible solutions:

t = 0.5 seconds or t = 5.125 seconds

Since the softball player hits the ball when it is 4 feet above the ground, we can disregard the solution t=0.5 seconds (which corresponds to when the ball is at its maximum height) and conclude that it will take approximately 5.125 seconds for the ball to hit the ground.

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On rounding the given number to nearest hundred miles will give the result -8.92.

Rounding to nearest hundred miles refers to changing the number present on hundreds place. We see that the number 8 is at unit's place, 9 is at tenth's place and 2 is at hundred's place.

Based on the rules of rounding, we will check the number to the right of number on hundred's place. Since the next number is 5, the number we will round the number 1 present at hundred's place. Hence, the final number will be -8.92.

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