the ratio of toddlers to infants at a day care center is 7 to 3. if twelve more infants join the day care to change the ratio to 7 to 5, how many toddlers are there at this day care center?

Let's analyze each series separately using the specified convergence tests:

For the series \(\(\sum_{n=1}^{\infty} \frac{4 + 3^n}{2^n}\),\) we can use the limit comparison test.

Taking the limit as \(\(n\)\) approaches infinity of the ratio of the nth term of this series to the nth term of the comparison series \((\(2^n\)),\) we get:

\(\[\lim_{n\to\infty} \frac{\frac{4 + 3^n}{2^n}}{2^n} = \lim_{n\to\infty} \frac{4 + 3^n}{2^n \cdot 2^n} = 0.\]\)

Since the limit is 0, and the comparison series converges, we can conclude that the original series also converges.

For the series \(\(\sum_{n=1}^{\infty} \frac{n^2 + 1}{2n^3 - 1}\),\) we can again use the limit comparison test.

Taking the limit as \(\(n\)\) approaches infinity of the ratio of the nth term of this series to the nth term of the comparison series \((\(\frac{1}{n^3}\)),\) we get:

\(\[\lim_{n\to\infty} \frac{\frac{n^2 + 1}{2n^3 - 1}}{\frac{1}{n^3}} = \lim_{n\to\infty} \frac{n^5 + n^3}{2n^3 - 1}.\]\)

Simplifying further, we divide each term by the highest power of \(\(n\),\) which is \(\(n^3\):\)

\(\[\lim_{n\to\infty} \frac{n^2 + \frac{1}{n^2}}{2 - \frac{1}{n^3}} = \infty.\]\)

Since the limit is infinity, the series diverges.

For the series \(\(\sum_{n=1}^{\infty} \frac{n}{\sqrt{n^5 + 5}}\),\) we can again apply the limit comparison test.

Taking the limit as \(\(n\)\) approaches infinity of the ratio of the nth term of this series to the nth term of the comparison series \((\(\frac{1}{n^{3/2}}\)),\) we get:

\(\[\lim_{n\to\infty} \frac{\frac{n}{\sqrt{n^5 + 5}}}{\frac{1}{n^{3/2}}} = \lim_{n\to\infty} (n^{5/2} + 5^{1/2}).\]\)

The limit is infinity, which means the series diverges.

For the series \(\(\sum_{n=2}^{\infty} (-1)^{n+1} \frac{2}{\ln(n)}\)\) , we can use the alternating series test.

The series satisfies the alternating series test if the terms decrease in absolute value and approach zero as \(\(n\)\) approaches infinity.

In this case, the terms \(\((-1)^{n+1} \frac{2}{\ln(n)}\)\) alternate in sign, and the absolute value of each term decreases as \(\(n\)\) increases. Additionally, \(\(\lim_{n\to\infty} \frac{2}{\ln(n)} = 0\).\)

Therefore, the series converges by the alternating series test.

To summarize:

The series \(\(\sum_{n=1}^{\infty} \frac{4 + 3^n}{2^n}\)\) converges.

The series \(\(\sum_{n=1}^{\infty}\)

\(\frac{n^2 + 1}{2n^3 - 1}\) diverges.\)

\(The series \(\sum_{n=1}^{\infty} \frac{n}{\sqrt{n^5 + 5}}\) diverges.\)

\(The series \(\sum_{n=2}^{\infty} (-1)^{n+1} \frac{2}{\ln(n)}\) converges.\)

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Answer:

5 degrees Fahrenheit.

Step-by-step explanation:

The temperature shows -5 degrees Fahrenheit. The opposite of a negative number is a positive number, so the opposite of the temperature is 5 degrees.

Answer:

Yes

Step-by-step explanation:

Their ratios are equivalent

Answer:

10 m/s

Step-by-step explanation:

KE = 0.5mv^2

400 = 0.5(8)(v^2)

100 = v^2

v = 10 m/s

Answer:

Step-by-step explanation:

600 secs=600/60 mins=10 mins

9 mins= 9 mins

Therefore, 9 mins : 600 secs

                =9 mins : 10 mins

                =9:10

Therefore, Andrea has 25 dimes in 37 coins, all nickels and dimes and the value of the 37 coins is $3.10.

Let's represent the number of nickels by "n" and the number of dimes by "d".

From the problem, we know that:

n + d = 37 (equation 1) ---(the total number of coins)

0.05n + 0.1d = 3.10 (equation 2) ---(the total value of the coins in dollars)

To solve for d, we can rearrange equation 1 to get:

n = 37 - d

Substitute this expression for n into equation 2 and simplify:

0.05(37-d) + 0.1d = 3.10

1.85 - 0.05d + 0.1d = 3.10

0.05d = 1.25

d = 25

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Answer:

5

Step-by-step explanation:

A(circle) = πr²
In this case, it's 78.5

r² = 78.5 : π = 24.9873260654 (round to 25.)
r = approx. sqrt(25) = 5

The researcher who has 80 participants for a study that involves four experimental conditions and who assigns participants to conditions using matched random assignment will form 20 matched clusters or blocks before assigning participants to conditions.

Matched Random Assignment is the process of grouping participants based on their similarities (matching variables) before randomly assigning them to different treatment groups (experimental conditions) to ensure that the groups are similar in all aspects except for the independent variable. This helps to control for extraneous variables that might affect the outcome of the study.

The number of matched clusters or blocks formed in matched random assignment is equal to the number of treatment groups (experimental conditions).

Since the researcher has four experimental conditions, she will form four matched clusters or blocks before assigning participants to conditions.

Therefore, the number of matched clusters or blocks that she will form before assigning participants to conditions is:

4 experimental conditions x 5 matched participants per block= 20 matched clusters or blocks.

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Answer:

see below

Step-by-step explanation:

Move R side to the L side to get

2y^2 -4y +3  = 0

Use quadratic formula to find y = 1 +- i sqrt (2) / 2

d is not a polynomial coz it has negative power

Answer:

There is a total of 200 dollars profit

Answer:

Step-by-step explanation:

cot x = adjacent side / opposite side

So cot 180 = x / 0 which is undefined.

Answer:

Step-by-step explanation:

Dont do it. Just take the detention

Answer:

Ok,first write the 2 points

(0,6)(9,3)

Second,find the slope with this formula

\(m=\frac{3-6}{9-0}\)

\(m=\frac{-3}{9}\)

\(m=\frac{-1}{3}\) this is the slope

The slope intercept form for 2 points is:

y-y1=m(x-x1)

\(y-6=\frac{-1}{3} (x-0)\) This is the final equation.

I hope this help :)

Answer:

so use the butterfly method which is multiplying the

Step-by-step explanation:

numerator by the denominator on the other side like I'll give you an example

2/3 4/5

you would multiply 2 and 5 together

Answer:

Hello!!! Princess Sakura here ^^

Step-by-step explanation:

The answer is the second one.

You put 15 quarts over one then you multiply it by the number of liters per 1 quart.

Tip: You should always keep the units that are diagonal the same.

Example:       15 quarts  x   .95 liter        See how they are diagonal.

                          1                1 quart

Answer:

32km

Step-by-step explanation:

speed=96km/h

time=20mins=1/3hour

➜96×1/3

➜96/3

➜32km

Answer: i think the answer is b

Step-by-step explanation:

Answer:

8

Step-by-step explanation:

You can think as vertices as the corners of the shape

Answer:

the shape have 8 vertices.

To determine which option would provide a higher rate of return, let's calculate the returns for both options:

Option 1: T-Bill

The T-Bill costs $23,250 and has a par value of $25,000. The return is the difference between the par value and the purchase price, which is:

Return = Par Value - Purchase Price

Return = $25,000 - $23,250

Return = $1,750

Option 2: 12-month CD

The CD has a 6.5% interest rate. To calculate the return, we need to determine the interest earned on the investment:

Interest = Principal * Interest Rate

Interest = $25,000 * 6.5% = $1,625

Therefore, for the 12-month CD, the return would be $1,625.

Comparing the returns for both options, we see that the T-Bill provides a higher return of $1,750 compared to the CD's return of $1,625. Thus, the T-Bill option would yield a higher rate of return.

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Given:

Total student = 50

each row has 6 students.

Find-:

Solve the question using the divisibility test.

Sol:

In row six student

then:

So a total of 8.33 rows in a class.

Answer:

Step-by-step explanation:

I'm just going to go out on a limb here and assume that those are a set of poorly made parenthesis. This, then, would be your equation:

\(x+\frac{x}{7}+\frac{1}{11}(x+\frac{x}{7})=60\)

The first thing to do is distribute through the parenthesis to get:

\(x+\frac{x}{7}+\frac{x}{11}+\frac{x}{77}=60\)

Next is to get rid of the denominators by multiplying by the LCM of 77:

\(77(x+\frac{x}{7}+\frac{x}{11}+\frac{x}{77}=60)\) which gives you:

77x + 11x + 7x + x = 4620 and

96x = 4620 so

\(x=\frac{385}{8}\)

Answer: middle

Step-by-step explanation:

Answer:

$12.65

Step-by-step explanation:

Given that the wall is 23 feet long and 8 feet high.

As the wall is rectangular, so the area of the wall,

A=23x8=184 square feet.

As a gallon of paint costs $11 and covers 320 square feet of area, so

Cost of paint for 1 square foot area = $ 11/320 per square foot.

Cost for 1 coat of paint = $ (11/320)x184

Cost for 2 coat of paint = $ (11/320)x184x2= $ 12.65

Hence, cost requires for 2 coats of paint is $ 12.65.

\(\frac{1}{2} x+6(x+15)=12\\\\\frac{1}{2} x+6x+90=12\\\\6\frac{1}{2} x+90=12\\6\frac{1}{2} x=-78\\x=-12\)

\(y=x+15\\y=-12+15\\y=3\)

Answer:

1) 35

2) 125

Step-by-step explanation:

1)

angle adc = x

x+ 90 + 20 = 180                (triangke sum property)\

x = 180 - 110 = 70

angle bdc = 70/ 2 = 35          ( the median cuts the angle into 2)

2) cbd = x = 125

x + c + bdc = 180

x + 35 + 20 = 180

x = 180 - 55 =  125

I hope im right!!!

Answer:

E. 8

Step-by-step explanation:

It's a special right angle triangle. Hypotenuse is

\( \sqrt{2} x\)

where x is the length of other legs. Also other two sides of triangle are always equal.

So to find the other leg divide by .

\( \sqrt{2} \)

The length will be 8.

Answer -: Solution

f(x)=6x-2

g(x)=2x-10

f(3)+g(2)=?

We know that

f(x)=6x-2

or,f(3)=6×3+2

=18+2

=20

Now,

g(x)=2x-10

or,g(2)=2×2+10

=4+10

=14

Again,

f(3)+g(2)=20+14

=34

Therefore the sum of f(3)+ f(2) is 34.

Answer:

20 miles < The distance between the cities < 200 miles

Step-by-step explanation:

The given information on the sign = "Wichita 90 mi, Topeka 10 mi"

Given that the two cites are not colinear, we have;

The number of points specified in the figure = 3 points

The specified points are;

1) The location of the family

2) The location of Wichita, relative to the family's location, = 90 mi

3) The location of Topeka, relative to the family's location, = 110 mi

Therefore, given that the three points are not colinear, they form the vertices of a triangle

The legs of the triangle are 90 mi and 110 mi

The maximum distance between Wichita and Topeka is therefore = 90 + 110 = 200 miles

The minimum distance distance from Wichita to Topeka = 110 - 90 = 20 miles

Therefore, the possible range of distances between the two cities is 20 miles to 200 miles.

Which gives;

20 miles < The distance between the cities < 200 miles.

I'll assume the ODE is

\(y'' - 3y' + 2y = e^x + e^{2x} + e^{-x}\)

Solve the homogeneous ODE,

\(y'' - 3y' + 2y = 0\)

The characteristic equation

\(r^2 - 3r + 2 = (r - 1) (r - 2) = 0\)

has roots at \(r=1\) and \(r=2\). Then the characteristic solution is

\(y = C_1 e^x + C_2 e^{2x}\)

For nonhomogeneous ODE (1),

\(y'' - 3y' + 2y = e^x\)

consider the ansatz particular solution

\(y = axe^x \implies y' = a(x+1) e^x \implies y'' = a(x+2) e^x\)

Substituting this into (1) gives

\(a(x+2) e^x - 3 a (x+1) e^x + 2ax e^x = e^x \implies a = -1\)

For the nonhomogeneous ODE (2),

\(y'' - 3y' + 2y = e^{2x}\)

take the ansatz

\(y = bxe^{2x} \implies y' = b(2x+1) e^{2x} \implies y'' = b(4x+4) e^{2x}\)

Substitute (2) into the ODE to get

\(b(4x+4) e^{2x} - 3b(2x+1)e^{2x} + 2bxe^{2x} = e^{2x} \implies b=1\)

Lastly, for the nonhomogeneous ODE (3)

\(y'' - 3y' + 2y = e^{-x}\)

take the ansatz

\(y = ce^{-x} \implies y' = -ce^{-x} \implies y'' = ce^{-x}\)

and solve for \(c\).

\(ce^{-x} + 3ce^{-x} + 2ce^{-x} = e^{-x} \implies c = \dfrac16\)

Then the general solution to the ODE is

\(\boxed{y = C_1 e^x + C_2 e^{2x} - xe^x + xe^{2x} + \dfrac16 e^{-x}}\)