The principal randomly selected six students to take an aptitude test. Their scores were: 81.6 72.0 81.1 86.4 70.2 83.1 Determine a 90% confidence interval for the mean score for all students

Answer:

see attachment

Step-by-step explanation:

The nurse will set the infusion of oxytocin at approximately 117 mL/hr.

To calculate the rate at which the nurse will set the infusion of oxytocin, we can use the formula:

Rate (mL/hr) = (Order dose × Volume ÷ Time)

Substituting the given values:

After calculating, we find that the nurse will set the infusion at approximately 117 mL/hr.

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Answer:

They have 45 quarters and 16 nickels.

Step-by-step explanation:

Let x be the number of quarters and y be the number of nickels in the jar,

Since, the jar contains 61 coins,

⇒ x + y = 61 ------(1)

Also, 1 quart = $ 0.25 and 1 nickel = $ 0.05,

So, the total cost = ( 0.25x + 0.05y ) dollars,

According to the question,

0.25x + 0.05y = 12.05,

⇒ 25x + 5y = 1205 -----(2),

Equation (2) - 5 × Equation (1),

20x = 1205 - 305

20x = 900

⇒ x = 45,

From equation (1)m,

y = 61 - 45 = 16,

Hence, they have 45 quarters and 16 nickels.

Let's solve this problem step by step.
We have two types of coins: quarters and nickels. Let's use two variables to represent the number of each type of coin in the jar.
Let \( Q \) represent the number of quarters and \( N \) represent the number of nickels. Since we have two unknowns, we'll need two equations to solve for them.
1. The total number of coins is 61:
\[ Q + N = 61 \]   (Equation 1)
2. The total value of the coins is $12.05. Since quarters are worth 25 cents each and nickels are worth 5 cents each, we can convert this total value into cents to avoid dealing with dollars and make the calculation easier.
\[ 12.05 dollars = 1205 cents \]
Now we set up an equation based on the value of the coins:
\[ 25Q + 5N = 1205 \]   (Equation 2)
These are our two equations:
\[ Q + N = 61 \]  
\[ 25Q + 5N = 1205 \]
Let's solve this system of linear equations.
First, we can simplify the second equation by dividing by 5 to make the numbers smaller and easier to work with:
\[ 5Q + N = 241 \]   (Simplified Equation 2)
Now, let's subtract Equation 1 from the Simplified Equation 2 to eliminate \( N \):
\[ (5Q + N) - (Q + N) = 241 - 61 \]
\[ 5Q + N - Q - N = 241 - 61 \]
\[ 4Q = 180 \]
Divide both sides by 4 to solve for \( Q \):
\[ Q = \frac{180}{4} \]
\[ Q = 45 \]
Now we know there are 45 quarters. To find the number of nickels, we plug the value of \( Q \) back into Equation 1:
\[ Q + N = 61 \]
\[ 45 + N = 61 \]
\[ N = 61 - 45 \]
\[ N = 16 \]
Therefore, there are 45 quarters and 16 nickels in the jar.

Answer:

27 degrees

Step-by-step explanation:

The standard deviation means it can be up or down by that many degrees. The temperatures can be between 20-28 degrees. 27 degrees is the only option in this set of numbers.

Answer:

your answer is C 27

Step-by-step explanation:

Final answer:

The true probability of Cody winning at least one more soda is approximately 0.67.

Explanation:

Cody buys six sodas, each with a 1/6 probability of winning another soda. To find the probability of winning at least one more soda, we need to find the probability of not winning any sodas and subtract it from 1. The probability of not winning a soda with each individual purchase is 5/6. Since the purchases are independent events, we can multiply the probabilities together to find the probability of not winning any sodas in all six purchases: (5/6) * (5/6) * (5/6) * (5/6) * (5/6) * (5/6) = (5/6)⁶ = 0.3349. Finally, we subtract this probability from 1 to find the probability of winning at least one more soda: 1 - 0.3349 = 0.6651 (rounded to the nearest hundredth).

Answer:

The null hypothesis can't be rejected.

Step-by-step explanation:

Given information:

Null hypothesis:

H₀ : π ≤ 0.70

Alternative hypothesis:

H₁ : π > 0.70

We need to check whether the null hypothesis is rejected or accepted.

If P-value < α, then we reject the null hypothesis H₀.

If P-value ≥ α, then we accept the null hypothesis H₀.

A sample of 100 observations revealed that p = 0.75 at the 0.05 significance level.

Here 0.75>0.05, it means p > α, therefore we can not reject the null hypothesis.

Final answer:

Explaining the rejection of a null hypothesis at a 0.05 significance level based on a sample proportion of 0.75.

Explanation:

The question:
The hypotheses given are H0: π ≤ 0.70 and H1: π > 0.70. A sample of 100 observations resulted in p = 0.75. At the 0.05 significance level, can the null hypothesis be rejected?

Step 1: Calculate the z-score for the given sample proportion.

Step 2: Find the p-value associated with the z-score.

Step 3: Compare the p-value to the significance level α (0.05) to decide whether to reject the null hypothesis or not.

Conclusion:
At the 0.05 significance level, the null hypothesis can be rejected because the p-value is less than 0.05, indicating sufficient evidence to conclude that the proportion is indeed greater than 0.70.

Answer:

Total deposit in each month is $ 932.84

Step-by-step explanation:

Step 1:  Monthly interest rate.

Monthly Rate = (1+annual rate)112−1

we have  Annual rate = 3.5%

Monthly Rate = (1+0.035)112−1 = 0.0028709

Step 2:  monthly payment

Monthly payment can be determined by using below formula:

A=P×i×1−(1+i)−n

A = monthly payment amount

P = total pay amount

i = monthly interest rate

n = total number of payments

In this example we have

P=$150000 ,

i=0.0028709  and

n=12×18=216

Monthly payment = P×i×1−(1+i)−n

= 150000×0.00287091−(1+0.0028709)−216 = 430.6351−(1.0028709)−216

Monthly payment  =$ 932.84

To have $150,000 in 18 years with a 3.5% APR compounded monthly, you need to deposit approximately $499.58 each month. The formula for the future value of an annuity and the power of compound interest are key to understanding how regular savings can grow over time.

To determine how much you have to deposit each month to have $150,000 in 18 years with an account paying 3.5% APR, you'll need to use the formula for the future value of an annuity. The future value of an annuity formula is:

Future Value = [tex]Pmt * ((1 + r/n)^{(nt)} - 1) / (r/n)[/tex]

where:

Pmt = monthly payment

r = annual interest rate (decimal)

n = number of times the interest is compounded per year

t = number of years.

First, convert the APR to a monthly interest rate by dividing by 12, since there are 12 months in a year. The monthly interest rate is 0.035 / 12. The interest is compounded monthly, so n is 12. Let's solve for Pmt (monthly deposit) using the formula:

[tex]150,000 = Pmt * ((1 + 0.035/12)^{(12*18)} - 1) / (0.035/12)[/tex]

By using a financial calculator or algebra, you can find the monthly deposit required:

Pmt ≈ $499.58

Therefore, you would need to deposit approximately $499.58 each month into your savings account to have $150,000 in 18 years.

It's important to start saving money early and to let the power of compound interest work in your favor, as seen in the provided examples where initial investments grow significantly over time due to compound interest.

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Answer:  The required probability of event B is P(B) = 0.37.

Step-by-step explanation:  For two events A and B, we are given the following probabilities :

P(A) = 0.34,    P(A ∩ B) = 0.27   and   P(A ∪ B) = 0.44.

We are to find the probability of event B, P(B) = ?

From the laws of probability, we have

[tex]P(A\cup B)=P(A)+P(B)-P(A\cap B)\\\\\Rightarrow 0.44=0.34+P(B)-0.27\\\\\Rightarrow 0.44=0.07+P(B)\\\\\Rightarrow P(B)=0.44-0.07\\\\\Rightarrow P(B)=0.37.[/tex]

Thus, the required probability of event B is P(B) = 0.37.

Answer:

No solution. Inconsistent.

Step-by-step explanation:

The equations are

[tex]2x+4y\leq 16\\x\leq 5\\y\leq 2\\x=6[/tex]

As we can see here that x=6 and x<=5 the solution area will not be bounded i.e., there will be no common area.

The lines do not intersect

Therefore there will be no solution.

Plotting the equations we get the graph below.

A strategy to avoid the question would be by just looking at the linear equations. It can be clearly seen that there are two lines that will never intersect and hence will have no solution.

Answer: second option.

Step-by-step explanation:

By definition, the measure of any interior angle of an equilateral triangle is 60 degrees.

Based in this, we know that the measusre of the angle ∠SRT is:

[tex]\angle SRT=\frac{60\°}{2}=30\°[/tex]

The, we can find the value of "y":

[tex]y+12=30\\y=30-12\\y=18[/tex]

To find the value of "x", we must use this identity:

[tex]cos\alpha=\frac{adjacent}{hypotenuse}[/tex]

In this case:

[tex]\alpha=30\°\\adjacent=x\\hypotenuse=RU=RS=4[/tex]

Substituting values and solving for "x", we get:

[tex]cos(30\°)=\frac{x}{4}\\\\4*cos(30\°)=x\\\\x=\sqrt{12}[/tex]

Answer:

The lines L1 and L2 neither parallel nor perpendicular

Step-by-step explanation:

* Lets revise how to find a slope of a line

- If a line passes through points (x1 , y1) and (x2 , y2), then the slope

 of the line is [tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]

- Parallel lines have same slopes

- Perpendicular lines have additive, multiplicative slopes

 ( the product of their slopes is -1)

* Lets solve the problem

∵ L1 passes through the point (1 , 10) and (-1 , 7)

- Let (1 , 10) is (x1 , y1) and (-1 , 7) is (x2 , y2)

∴ x1 = 1 , x2 = -1 and y1 = 10 , y2 = 7

∴ The slope of L1 is [tex]m1 = \frac{7-10}{-1-1}=\frac{-3}{-2}=\frac{3}{2}[/tex]

∵ L2 passes through the point (0 , 3) and (1 , 5)

- Let (0 , 3) is (x1 , y1) and (1 , 5) is (x2 , y2)

∴ x1 = 0 , x2 = 1 and y1 = 3 , y2 = 5

∴ The slope of L2 is [tex]m2=\frac{5-3}{1-0}=\frac{2}{1}=2[/tex]

∵ m1 = 3/2 and m2 = 2

- The two lines have different slopes and their product not equal -1

∴ The lines L1 and L2 neither parallel nor perpendicular

By calculating the slopes of L1 and L2, we find that they are 1.5 and 2 respectively. Since they are neither the same nor negative reciprocals, L1 and L2 are neither parallel nor perpendicular.

To determine if lines L1 and L2 are parallel, perpendicular, or neither, we need to calculate the slopes of both lines using the slope formula:

Slope formula: (y2 - y1) / (x2 - x1)

Calculating the slope of L1:

Points on L1: (1, 10) and (-1, 7)

Slope of L1 = (7 - 10) / (-1 - 1) = (-3) / (-2) = 1.5

Calculating the slope of L2:

Points on L2: (0, 3) and (1, 5)

Slope of L2 = (5 - 3) / (1 - 0) = 2 / 1 = 2

Since the slopes of L1 (1.5) and L2 (2) are neither the same nor negative reciprocals of each other, the lines L1 and L2 are neither parallel nor perpendicular.

Answer:

expected value of x  is 1 and standard deviation of x is 0.7071

Step-by-step explanation:

Given data

coin flip = 40 time

heads noted

to find out

the expected value and the standard deviation

Solution

we know coin is flipped 40 times and the number of heads is noted only

so we find standard deviation for x i.e.

standard deviation of  x = [tex]\sqrt{variance x}[/tex]   .......1

first we calculate variance x .i.e.

variance x = expected value of x² - (expected value of x)²   .............2

so now we calculate expected value of x

we know when x = 0 , p(x) = 1/4

x = 1 , p(x) = 1/2

x = 2 , p(x) = 1/4

so  expected value of x  =  [tex]\sum_{0}^{2}[/tex] x p(x)

i.e.   expected value of x  =  0 P(x) + 1 p(x) + 2 p(x)

expected value of x  = 0 (1/4) + 1 (1/2) + 2 (1/4)

expected value of x  =  0  +  (1/2) + (1/2)

expected value of x  =    1             ....................3

now calculate expected value of x²

so so  expected value of x²  =  [tex]\sum_{0}^{2}[/tex] x²p(x)

i.e.  expected value of x²  =   x² P(x) + 1² p(x) + 2² p(x)

expected value of x²  =    0 (1/4) + 1 (1/2) + 4 (1/4)

expected value of x² =   0  +  (1/2) + (1)

expected value of x²  = 1.5         .........................4

now put equation 3 and 4 value in equation 2 we get

variance x =  expected value of x² - (expected value of x)²

variance x =  1.5 - (1)²

variance x =  0.5

now put variance value in equation 1 we get

standard deviation of  x = [tex]\sqrt{variance x}[/tex]  

standard deviation of  x = [tex]\sqrt{0.5}[/tex]  

standard deviation of  x = 0.7071

so standard deviation of x is 0.7071

The expected value is 20 and the standard deviation is 3.16

The expected value

The given parameters are:

n = 40 ---- the number of flips

p = 0.5 --- the probability of obtaining a head

The expected value is calculated as:

[tex]E(x) = np[/tex]

This gives

[tex]E(x) = 40 * 0.5[/tex]

[tex]E(x) = 20[/tex]

Hence, the expected value is 20

The standard deviation

This is calculated as:

[tex]\sigma = \sqrt{np(1 - p)[/tex]

So, we have:

[tex]\sigma = \sqrt{40 * 0.5 * (1 - 0.5)[/tex]

[tex]\sigma = 3.16[/tex]

Hence, the standard deviation is 3.16

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Answer:

[tex]p=9900\\[/tex] bacterias in the initial two hours

Step-by-step explanation:

the growing rate is given by the ecuation

[tex]p(t)=0.8(t)^{3} +3.5 [/tex] thousand per hour

for t=2 we have

[tex]p(2)=0.8(2)^{3} +3.5 = 9.9[/tex] thousand

[tex]p=9900\\[/tex] bacterias

In two hours we have 9900 bacterias

Answer:

68 chickens and 12 cows.

Step-by-step explanation:

Let x represents the number of chicken and y represents the number of cows in the barnyard,

Given,

Total heads = 80

⇒ x + y = 80 ------(1),

Also, total legs = 184,

Since, a chicken has two legs and cow has 4 legs,

⇒ 2x + 4y = 184 -----(2),

Equation (2) - 2 × equation (1),

We get,

4y - 2y = 184 - 160

2y = 24

y = 12

From equation (1),

x + 12 = 80 ⇒ x = 80 - 12 = 68

Hence, the number of chicken = 68,

And, the number of cows = 12

Answer:

Simplest function is f(x) = 2x - 1

Step-by-step explanation:

When a function has a zero, simply put, that point is where the graph goes through the x-axis.  At this point, the y value is 0.

If a function has a 0 of 1/2, that translates to x = 1/2 and also (1/2, 0).  That means that y = 0 when x = 1/2.

If x = 1/2 and is a zero, then x - 1/2 = 0 and 2x - 1 = 0 (notice that y = 0 here and solving for x would get you right back to x = 1/2).

The simplest function with a zero of 1/2 is

f(x) = 2x - 1

You know it is true that the zero is 1/2 for two reasons.  First one is to set the right side equal to 0 and solve for x.

The second one is to graph the line y = 2x - 1 and see that it goes through the x axis at 1/2 where y = 0.

Answer:

Given:

(a.) ㏑( x + 5 ) − ㏑( x − 5 )

(b.) ㏑( x + 5 ) + ㏑( x − 5 )

To compute the above expression, we'll use the properties of natural logarithm. i.e.

㏑(a) − ㏑(b) =  ㏑[tex]\frac{a}{b}[/tex]

∴ ㏑( x + 5 ) − ㏑( x − 5 ) = ㏑[tex]\frac{x+5}{x-5}[/tex]

Similarly

㏑(a) + ㏑(b) =  ㏑[tex](a\times b)[/tex]

∴ ㏑( x + 5 ) + ㏑( x − 5 ) = ㏑([tex]x^{2}[/tex]-25)

Answer:

a. 13%

b. 20%

c. 2%

Step-by-step explanation:

The best way to solve this problem is by drawing a Venn diagram.  Draw a rectangle representing all the fourth-graders.  Draw two overlapping circles inside the rectangle.  Let one circle represent proficiency in reading.  This circle is 85% of the total area (including the overlap).  And let the other circle represent proficiency in math.  This circle is 78% of the total area (including the overlap).  The overlap is 65% of the total area.

a. Since the overlap is 65%, and 78% are proficient in math, then the percent of all students who are proficient in math but not reading is the difference:

78% − 65% = 13%

b. Since the overlap is 65%, and 85% are proficient in reading, then the percent of all students who are proficient in reading but not math is the difference:

85% − 65% = 20%

c. The percent of students not proficient in reading or math is 100% minus the percent proficient in only reading minus the percent proficient in only math minus the percent proficient in both.

100% − 20% − 13% − 65% = 2%

See attached illustration (not to scale).

The probability of a student being proficient in mathematics but not in reading is 13%, in reading but not in mathematics is 20%, and in neither reading nor mathematics is 2%.

To solve the student's query, we'll use the principle that the probability of an event is the number of favorable outcomes divided by the total number of outcomes. We can apply the addition rule for probabilities, which states that the probability of either of two mutually exclusive events occurring is the sum of their individual probabilities, minus the probability of both events happening.

Let P(M) be the probability the student is proficient in mathematics, P(R) be the probability the student is proficient in reading, and P(M & R) be the probability the student is proficient in both. The question asks for P(M) - P(M & R), the probability of being proficient in mathematics but not in reading. That is 78% - 65% = 13%.

Similarly, the probability of a student being proficient in reading but not mathematics is P(R) - P(M & R), which equals 85% - 65% = 20%.

To find the probability of a student being proficient in neither subject, we can find the probability of a student being proficient in at least one subject and then subtracting this from 100%. The probability of being proficient in at least one subject is P(R) + P(M) - P(M & R), or 85% + 78% - 65% = 98%. Thus, the probability of being proficient in neither is 100% - 98% = 2%.

Hence, we have:

[tex]proj_W(v)=[\dfrac{464}{21},\dfrac{167}{21},\dfrac{71}{21},\dfrac{-131}{7}][/tex]

By the orthogonal decomposition theorem we have:

The orthogonal projection of a vector v onto the subspace W=span{w,w'} is given by:

[tex]proj_W(v)=(\dfrac{v\cdot w}{w\cdot w})w+(\dfrac{v\cdot w'}{w'\cdot w'})w'[/tex]

Here we have:

[tex]v=[19,12,14,-17]\\\\w=[-4,-1,-1,3]\\\\w'=[1,-4,4,3][/tex]

Now,

[tex]v\cdot w=[19,12,14,-17]\cdot [-4,-1,-1,3]\\\\i.e.\\\\v\cdot w=19\times -4+12\times -1+14\times -1+-17\times 3\\\\i.e.\\\\v\cdot w=-76-12-14-51=-153[/tex]

[tex]w\cdot w=[-4,-1,-1,3]\cdot [-4,-1,-1,3]\\\\i.e.\\\\w\cdot w=(-4)^2+(-1)^2+(-1)^2+3^2\\\\i.e.\\\\w\cdot w=16+1+1+9\\\\i.e.\\\\w\cdot w=27[/tex]

and

[tex]v\cdot w'=[19,12,14,-17]\cdot [1,-4,4,3]\\\\i.e.\\\\v\cdot w'=19\times 1+12\times (-4)+14\times 4+(-17)\times 3\\\\i.e.\\\\v\cdot w'=19-48+56-51\\\\i.e.\\\\v\cdot w'=-24[/tex]

[tex]w'\cdot w'=[1,-4,4,3]\cdot [1,-4,4,3]\\\\i.e.\\\\w'\cdot w'=(1)^2+(-4)^2+(4)^2+(3)^2\\\\i.e.\\\\w'\cdot w'=1+16+16+9\\\\i.e.\\\\w'\cdot w'=42[/tex]

Hence, we have:

[tex]proj_W(v)=(\dfrac{-153}{27})[-4,-1,-1,3]+(\dfrac{-24}{42})[1,-4,4,3]\\\\i.e.\\\\proj_W(v)=\dfrac{-17}{3}[-4,-1,-1,3]+(\dfrac{-4}{7})[1,-4,4,3]\\\\i.e.\\\\proj_W(v)=[\dfrac{68}{3},\dfrac{17}{3},\dfrac{17}{3},-17]+[\dfrac{-4}{7},\dfrac{16}{7},\dfrac{-16}{7},\dfrac{-12}{7}]\\\\i.e.\\\\proj_W(v)=[\dfrac{464}{21},\dfrac{167}{21},\dfrac{71}{21},\dfrac{-131}{7}][/tex]

Answer:

(x,y,z)=(5,0,3)

[tex]((x,y,z)-(2,3,0))*(-1,1,0)=0[/tex]

Step-by-step explanation:

a)

The problem requires to find the intersection point of the lines, at that point the position 'r' of the lines is the same:

[tex]r_{1} =r_{2} \\(2,3,0)+(3,-3,3)t=(5,0,3)+(-3,3,0)s\\[/tex]

First, built the parametric equation system; this is just a simplification coordinate to coordinate of the vector equation:

[tex]2+3t=5-3s\\3-3t=3s\\3t=3[/tex]

From the last equation,

[tex]t=1[/tex]

And for whatever of the other two,

[tex]s=0[/tex]

You can check that replacing t=1 and s=0 the point gotten is (5,0,3), which is the intersection point (the point that belongs to both lines).

b) The plane is defined by an orthogonal direction. The equation of the plane uses the fact that the dot product between two orthogonal vectors is always zero.

The general equation of a plane is:

[tex]((x,y,z)-(x_{0},y_{0},z_{0}))*(n)=0[/tex]

Where (x,y,z) are the variables that may be part of the plane or not, [tex](x_{0},y_{0},z_{0})[/tex] is a point that belongs to the plane and n is a vector which is orthogonal to the plane.

Due that both lines belong to the plane, the cross product between their direction vectors will give us the orthogonal vector.

[tex]n=(3,-3,3)X(-3,3,0)=(-9,-9,0)[/tex]

We can divide (-9,-9,0) by nine, because we only need the direction and the division does not affect it.

[tex]n=(-1,-1,0)[/tex]

Finally, we know that both lines are inside the plane, so any point that belong to a line, belong to the plane. For this reason, let's select any point, for example: (2,3,0) (It could be another). So, the equation of the plane is:

[tex]((x,y,z)-(2,3,0))*(-1,-1,0)=0[/tex]

The intersection point of the lines can be obtained by equating the parametric forms of the lines and finding the values of parameters. The equation of the plane containing these lines can be obtained using the directional vectors of these lines, which essentially define the plane.

First, we need to find the common point at which the given lines intersect. We can do this by setting r = 2, 3, 0 + t 3, −3, 3 and r = 5, 0, 3 + s −3, 3, 0 to be equal, and finding the values of t and s that make this true. This gives us the (x, y, z) coordinates of the intersection point.

To find the equation of the plane that contains these lines, we know that any point on this plane can be expressed as a linear combination of the directional vectors of these lines, which are (3, -3, 3) and (-3, 3, 0). Therefore, the equation of the plane can be written in the form of Ax + By + Cz = D, where (A, B, C) is a normal vector to the plane, and D is a constant that can be determined by substituting the coordinates of any point on the plane.

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Answer:

[tex](x-6)^2+(y-2)^2=16[/tex].

Step-by-step explanation:

[tex](x-h)^2+(y-k)^2=r^2[/tex] is the equation for a circle with center (h,k) and radius r.

You are given center (6,2) and radius 4.

So we will replace h with 6 and k with 2 and r with 4.

This gives us:

[tex](x-6)^2+(y-2)^2=4^2[/tex]

Simplify:

[tex](x-6)^2+(y-2)^2=16[/tex].

For this case we have that by definition, the equation of a circle is given by:

[tex](x-h) ^ 2 + (y-k) ^ 2 = r ^ 2[/tex]

Where:

[tex](h, k):[/tex]It is the center of the circle

r: It is the radius of the circle

According to the data we have to:

[tex](h, k) :( 6.2)\\r = 4[/tex]

Substituting:

[tex](x-6) ^ 2 + (y-2) ^ 2 = 4 ^ 2\\(x-6) ^ 2 + (y-2) ^ 2 = 16[/tex]

ANswer:

[tex](x-6) ^ 2 + (y-2) ^ 2 = 16[/tex]

Answer:

[tex] y=\frac{C}{x}[/tex].

Step-by-step explanation:

Given homogeneous equation

[tex] x^2ydy+xy^2dx=0[/tex]

[tex]\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{xy^2}{x^2y}[/tex]

Substitute y=ux , [tex]u=\frac{y}{x}[/tex]

[tex] \frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{y}{x}[/tex]

Now,

[tex]u+x\frac{\mathrm{d}u}{\mathrm{d}x}=\frac{\mathrm{d}y}{\mathrm{d}x}[/tex]

[tex]u+x\frac{\mathrm{d}u}{\mathrm{d}x}=-u[/tex]

[tex]\frac{\mathrm{d}u}{\mathrm{d}x}=-2u[/tex]

[tex]\frac{du}{u}=-\frac{dx}{x}[/tex]

Integrating both side we get

lnu=-2lnx+lnC

Where lnC= integration constant

[tex]lnu+ln{x}^2=lnC[/tex]

[tex]lnux^2=lnC[/tex]

Cancel ln on both side

[tex]ux^2=C[/tex]

Substitute [tex]u=\frac{y}{x}[/tex]

Then we get

xy=C

[tex]y=\frac{C}{x}[/tex].

Answer:[tex]y=\frac{C}{x}[/tex].

Answer: 462

Step-by-step explanation:

The general theorem of combination says that there are [tex]C(n+r-1, r)[/tex], with r-combinations from a set having n elements when repetition of elements is allowed.

Here the number of denomination: [tex]n = 7[/tex] , r =5

Also order doesn't matters.

Then the number of different possible ways can you choose the five bills is given by :-[tex]C(7+5-1, 5)= C(11,5)\\\\=\dfrac{11!}{5!(11-5)!}\\\\=462[/tex]

Hence, the number of different possible ways can you choose the five bills is  462.

Answer:

Step-by-step explanation:

Remember that in the geometric serie if | r | < 1 the serie converges and if | r | ≥1 the serie diverges.

I suppose that the serie starts at 0, so using the geometric serie with r = | [tex]\frac{-3}{2}[/tex] | > 1 the serie diverges.

Answer:

This example illustrates Homeostasis.

Step-by-step explanation:

Consider the provided information.

On a very hot summer day and a few months later on a very cold winter day, you go outside and take your temperature. Each time your body temperature is 37 degrees Celsius that means your body's tissues, and cells helps you to attain the stability and constancy required for proper functioning.

The medical definition of Homeostasis is:

Homeostasis is a property of tissues, and cells that helps the stability and constancy required for proper functioning to be maintained and regulated. it is a state maintained through the continuous adjustment of biochemical and physiological processes.

The provided example explain the process of Homeostasis.

Thus, this example illustrates Homeostasis.

Final answer:

The unchanging body temperature of 37 degrees Celsius in both hot and cold external conditions exemplifies thermoregulation, the body's ability to maintain a constant internal temperature through homeostasis.

Explanation:

The scenario of your body temperature remaining constant at 37 degrees Celsius, regardless of whether it is a hot summer day or a cold winter day, illustrates the concept of thermoregulation. Thermoregulation is the ability of an organism to keep its body temperature within certain boundaries, even when the surrounding temperature is very different.

Your body does this through negative feedback mechanisms similar to a thermostat in a house. For example, on a hot day, if your body temperature rises, your skin produces sweat and the blood vessels near your skin's surface dilate to release heat and cool you down. Conversely, in cold weather, the blood vessels constrict, and shivering generates heat to maintain your body temperature. This constant adjustment keeps your body's core temperature steady, enabling the efficient functioning of enzymes and bodily processes that are optimized for a temperature of around 37 degrees Celsius.

This biological thermostat works continuously to keep internal conditions stable, a state known as homeostasis. In physiological terms, about 60 percent of the energy generated from the production of ATP (adenosine triphosphate) by your cells is released in the form of heat, contributing to the maintenance of your body temperature.

Answer:

2.3775

Step-by-step explanation:

Answer:386

Step-by-step explanation:

We have given

Smoothing parameter [tex]\left ( \alpha \right )=0.56[/tex]

Forecasted demand[tex]\left ( F_t\right )=385[/tex]

Actual demand[tex]\left ( D_t\right )=386[/tex]

And Forecast is given by

[tex]F_{t+1}=\alpha D_t+\left ( 1-\alpha \right )F_t[/tex]

[tex]F_{t+1}=0.56\cdot 386+\left ( 1-0.56\right )385=385.56\approx 386[/tex]

[tex]F_{t+2}=0.56\cdot 386+\left ( 1-0.56\right )385.56=385.806\approx 386[/tex]

[tex]F_{t+3}=0.56\cdot 386+\left ( 1-0.56\right )385.806=385.914\approx 386[/tex]

[tex]F_{t+4}=0.56\cdot 386+\left ( 1-0.56\right )385.914=385.962\approx 386[/tex]

Answer:

[tex]N(x)=\frac{50000}{1+99e^{\ln(\frac{49}{99})x}}[/tex]

Step-by-step explanation:

The logistic equation is

[tex]N(x)=\frac{c}{1+ae^{-rx}}[/tex]

where:

c/(1+a) is the initial value.

c is the limiting value

r is constant determined by growth rate

So we are given that:

N(0)=500 or that c/(1+a)=500

If your not sure about his initial value of c/(1+a) then replace x with 0 in the function N:

[tex]N(0)=\frac{c}{1+ae^{-r \cdot 0}}[/tex]

Simplify:

[tex]N(0)=\frac{c}{1+ae^{0}}[/tex]

[tex]N(0)=\frac{c}{1+a(1)}[/tex]

[tex]N(0)=\frac{c}{1+a}[/tex]

Anyways we are given:

[tex]\frac{c}{1+a}=500[/tex].

Cross multiplying gives you [tex]c=500(1+a)[/tex].

We are also giving that N(1)=1000 so plug this in:

[tex]N(1)=\frac{c}{1+ae^{-r \cdot 1}}[/tex]

Simplify:

[tex]N(1)=\frac{c}{1+ae^{-r}}[/tex]

So this means

[tex]1000=\frac{c}{1+ae^{-r}}[/tex]

Cross multiplying gives you [tex]c=1000(1+ae^{-r})[/tex]

We are giving that c=50000 so we have these two equations to solve:

[tex]50000=500(1+a)[/tex]

and

[tex]50000=1000(1+ae^{-r})[/tex]

I'm going to solve [tex]50000=500(1+a)[/tex] first because there is only one constant variable here,[tex]a[/tex].

[tex]50000=500(1+a)[/tex]

Divide both sides by 500:

[tex]100=1+a[/tex]

Subtract 1 on both sides:

[tex]99=a[/tex]

Now since we have [tex]a[/tex] we can find [tex]r[/tex] in the second equation:

[tex]50000=1000(1+ae^{-r})[/tex] with [tex]a=99[/tex]

[tex]50000=1000(1+99e^{-r})[/tex]

Divide both sides by 1000

[tex]50=1+99e^{-r}[/tex]

Subtract 1 on both sides:

[tex]49=99e^{-r}[/tex]

Divide both sides by 99:

[tex]\frac{49}{99}=e^{-r}[/tex]

Take natural log of both sides:

[tex]\ln(\frac{49}{99})=-r[/tex]

Multiply both sides by -1:

[tex]-\ln(\frac{49}{99})=r[/tex]

So the function N with all the write values plugged into the constant variables is:

[tex]N(x)=\frac{50000}{1+99e^{\ln(\frac{49}{99})x}}[/tex]

Final answer:

The question involves applying the logistic growth equation to determine the number of people who will see an advertisement over time, given initial conditions and the carrying capacity. The process includes finding the growth rate from the provided data and using it to solve the logistic growth formula for any time t.

Explanation:

The number of people in a community who are exposed to a particular advertisement is described by the logistic growth equation. Given that initially N(0) = 500, and after one unit of time N(1) = 1000, and the carrying capacity is 50,000, we want to solve for N(t), the number of people who will see the advertisement at any time t.

The logistic growth model can be written as:

N(t) = K / (1 + (K - N_0) / N_0 ×[tex]e^{(-rt)}[/tex]

Where:

N(t) is the number of individuals at time t

K is the carrying capacity of the environment

N_0 is the initial number of individuals

r is the growth rate

e is the base of the natural logarithms

We are given that K = 50,000, N_0 = 500, and N(1) = 1000. From N(1), we can find the growth rate r. Re-arranging the logistic equation and substituting the values for N(1), t = 1, K, and N_0, we get an equation that we can solve for r. Once we have found r, we can substitute all known values back into the logistic equation to solve for N(t) for any given value of t.

To find the solution for this kind of problem it might require numerical methods or algebraic manipulation which is beyond this explanation, but once the value of r is found, the N(t) formula can be applied to predict the number of people who will see the advertisement at any given time.

Answer:

Each square should have 5 inches of side and area = 25 square inches.

Step-by-step explanation:

Candy box is made that measures 45 by 24 inches.

Let the squares of equal size x inches has been cut out of each corner.

The sides will then be folded up to form a rectangular box.

Now we have to find the size of square that should be cut from each corner to obtain maximum volume of the box.

Now the box is with length = (45 - 2x) inches

and width = (24 - 2x) inches

and height = x inches

Volume of the candy box = Length × width × height

V = (45 - 2x)(24 - 2x)(x)

V = x(1080 - 48x -90x + 4x²)

  = x(1080 - 138x + 4x²)

  = 4x³ - 138x² + 1080x

Now we will find the derivative of volume and equate it to zero.

[tex]\frac{dV}{dx}=12x^{2}-276x+1080=0[/tex]

12(x² - 23x + 90) = 0

x² - 23x + 90 = 0

x² - 18x - 5x + 90 = 0

x(x - 18) - 5(x - 18) = 0

(x - 5)(x - 18)=0

x = 5, 18

Now for x = 18 Width of the box will be = (24 - 2×18) = 24 - 36 = -12

Which is not possible.

Therefore, x = 5 will be the possible value.

Therefore, square having area 25 square inches should be cut out from each corner to get the maximum volume of candy box.

The size of the square that should be cut away from each corner to obtain the maximum volume for a box made from a cardboard measuring 45 by 24 inches is 3 inches.

To find the size of the square that should be cut from each corner to obtain the maximum volume, we should first make an equation for the volume of the box. If x is the length of the side of the square, then the dimensions of the box are (45-2x) by (24-2x) by x, thus the volume of the box V is (45-2x)(24-2x)x.

By using calculus, we can find the derivative of this function, set it to zero and solve, this will give the critical points where the maximum and minimum volumes will be.

The derivative is found to be -4x^2 + 138x - 1080. Setting this to zero and solving, we find that x = 3 and x = 90 are the critical points for the maximum and minimum volumes. Since we cannot cut corners more than 24 inches (this would make the width negative), x = 3 inches is the only feasible solution.

So, 3 inches should be cut away from each corner to obtain the maximum volume.

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The Wronskian determinant is

[tex]\begin{vmatrix}\sin x&x\sin x\\\cos x&x\cos x+\sin x\end{vmatrix}=\sin x(x\cos x+\sin x) - x\sin x\cos x=\sin^2x[/tex]

which is non-zero for all [tex]x\in(0,1)[/tex], so the solutions are linearly independent.

Answer:

(2,0)

Step-by-step explanation:

The solution of the inequality [tex]y>-3x+2[/tex] is shown in attached diagram.

The boundary line is dotted line, because the sign of inequality is without notion "or equal to". The dotted line means that points lying on this line are not solutions of the inequality. The solutions are those points lying in the shaded region.

From the points (0,2), (2,0), (1,-2), (-2,1) only point (2,0) lies on the shaded region, so only point (2,0) is a solution to the inequality

Answer:

on ed it says its B

Step-by-step expla

Give me robux on robloz

Step-by-step explanation:

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