A nurse infuses a dose in 50 minutes. She needs to infuse the dose at a 20% faster rate. How much time does she need ?

Answer with Step-by-step explanation:

1.We are given

2783 and 7283

We have to compare the values of the underline digits

Place value of 2 in 2783 =2000 because 2 is at thousand place

2 in 7283 is at hundred place

Therefore, the place value of 2 in 7283=200

Therefore, the value of 2 in 2783 is 10 times the value of 2 in 7283.

2.We have to compare the values of 7  in both numbers

Place of 7 in 503497=one's

Therefore, place value of 7 =7

7 in 26475 is at tens place

Therefore, the place value of 7 in 26475=70

Hence, the value of 7 in 26475 is 10 times the value of 7 in 503497.

3.We have to compare the values of 4 in both given numbers

Place of 4 in 34258=Thousand

Place value of 4 in 34258=4000

Place of 4 in 47163=Ten thousand

Place value of 4 in 47163=40000

Hence, the value of 4 in 47163 is 10 times the value of 4 in 34258.

Answer:

The 105th term of given sequence is [tex]\frac{105}{2}[/tex].

Step-by-step explanation:

The given sequence is

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

It can be rewritten as

[tex]0.5,1,1.5,2[/tex]

Here the first term is 0.5.

It is an arithmetic​ sequence because it has common difference.

[tex]d=a_2-a_1=1-0.5=0.5[/tex]

[tex]d=a_3-a_2=1.5-1=0.5[/tex]

[tex]d=a_4-a_3=2-1.5=0.5[/tex]

The nth term of an AP is

[tex]a_n=a_1+(n-1)d[/tex]

where, [tex]a_1[/tex] is first term and d is common difference.

Substitute [tex]a_1=0.5[/tex] and [tex]d=0.5[/tex] in the above formula.

[tex]a_n=0.5+(n-1)0.5[/tex]

[tex]a_n=0.5+0.5n-0.5[/tex]

[tex]a_n=0.5n[/tex]

We need to find the 105th term of given sequence.

Substitute n=105 in the above equation.

[tex]a_n=0.5(105)[/tex]

[tex]a_n=52.5[/tex]

[tex]a_n=\frac{105}{2}[/tex]

Therefore the 105th term of given sequence is [tex]\frac{105}{2}[/tex].

[tex]\Rightarrow[/tex]

Suppose first that [tex]H\subset G[/tex] is a normal subgroup. Then by definition we must have for all [tex]a\in H[/tex], [tex]xax^{-1} \in H[/tex] for every [tex]x\in G[/tex]. Let [tex]a\in G[/tex] and choose [tex](ab)\in aH[/tex] ([tex]b\in H[/tex]). By hypothesis we have [tex]aba^{-1} =abbb^{-1}a^{-1}=(ab)b(ab)^{-1} \in H[/tex], i.e. [tex]aba^{-1}=c[/tex] for some [tex]c\in H[/tex], thus [tex]ab=ca \in Ha[/tex]. So we have [tex]aH\subset Ha[/tex]. You can prove [tex]Ha\subset aH[/tex] in the same way.

[tex]\Leftarrow[/tex]

Suppose [tex]aH=Ha[/tex] for all [tex]a\in G[/tex]. Let [tex]h\in H[/tex], we have to prove  [tex]aha^{-1} \in H[/tex] for every [tex]a\in G[/tex]. So, let [tex]a\in G[/tex]. We have that [tex]ha^{-1} =a^{-1}h'[/tex] for some [tex]h'\in H[/tex] (by the hypothesis). hence we have [tex]aha^{-1}=h' \in H[/tex]. Because [tex]a[/tex] was chosen arbitrarily  we have the desired .

By considering the given information we have ,

[tex]H_0: \mu = 10\\\\ H_a: \mu\neq10[/tex]

Since, the alternative hypothesis is two tailed so the test is a two-tailed test.

Given : Population mean : [tex]\mu=10[/tex]

Standard deviation: [tex]\sigma= 3[/tex]

Sample size : n=25 , whihc is less than 30 so the sample is small and we use t-test.

Sample mean : [tex]\overline{x}=11.3[/tex]

Significance level : [tex]\alpha= 0.5[/tex]

Formula to find t-test statistic is given by :-

[tex]t=\dfrac{\overline{x}-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]

i.e. [tex]t=\dfrac{11.3-10}{\dfrac{3}{\sqrt{25}}}\approx2.17[/tex]

By using the standard normal distribution table,

The p-value corresponds 2.17 (two-tailed)=0.0300068

Since , the p-value is less than the significance level, so we reject the null hypothesis.

Hence, we conclude that there are enough evidence to to support the claim that movie rental rates in Yuma, Arizona, were different from those of the country as a whole .

A hypothesis test is conducted to see if the average movie rental rate in Yuma, Arizona, is statistically different from the national average. This problem is resolved in several steps including stating hypotheses, formulating an analysis plan, analyzing sample data, and interpreting the results. The rental rate is then compared to a critical value determined by the significance level (α = .05).

The subject here is a hypothesis testing problem related to the mean rental rate of DVDs in Yuma, Arizona. Blockbuster found that the average nation-wide movie rental rate was 10 movies per year with a standard deviation of 3. In Yuma, a sample of 25 people resulted in a mean rental rate of 11.3 movies per year. The company wanted to check whether this difference was significant or not. So, they used an alpha level of .05 to conduct this hypothesis test.

Here are the steps of the hypothesis test:

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Answer:

Step-by-step explanation:

Given are the observations are on stabilized viscosity (cP) for specimens of a certain grade of asphalt with 18% rubber added:

2767 2924 3042 2844 2895

No of items = 5

If written in ascending order the order would be

2767   2844   2895   2924   3042

Hence median is the middle value in the ordered row = 2895

Mean = sum/5

=[tex]\frac{14472}{5} =2894.4[/tex]

Answer:

The value of the 3 is 30,000,000.

Step-by-step explanation:

From the digit at the right, you go multiplying each element by 10 powered to a counter that starts at zero and increases at every digit. So:

Our counter is i

i = 0;

v(7) is the value of the 7

[tex]v(7) = 7*10^{0} = 7[/tex]

i = 1;

v(5) is the value of the 5

[tex]v(5) = 5*10^{1} = 50[/tex]

i = 2;

v(1) is the value of the 1

[tex]v(1) = 1*10^{2} = 100[/tex]

i = 3;

v(5) is the value of the 5

[tex]v(5) = 5*10^{3} = 5,000[/tex]

i = 4;

v(9) is the value of the 9

[tex]v(9) = 9*10^{4} = 90,000[/tex]

i = 5;

v(6) is the value of the 6

[tex]v(6) = 6*10^{5} = 600,000[/tex]

i = 6;

v(8) is the value of the 8

[tex]v(8) = 8*10^{6} = 8,000,000[/tex]

i = 7;

v(3) is the value of the 3

[tex]v(3) = 3*10^{7} = 30,000,000[/tex]

The value of the 3 is 30,000,000.

The digit 3 in the number 138,695,157 represents a value of 300 million, indicating its substantial contribution to the overall magnitude of the figure in the context of place value and powers of 10.

The value of the digit 3 in the number 138,695,157 is 3 hundred million. In this number, each place value represents a power of 10, with the rightmost digit being ones, the next one being tens, the next hundreds, and so on. The digit 3 in the hundred million's place means that it represents 3 multiplied by 100,000,000.

In other words, the digit 3 in this context signifies 300 million. This is because when you see a digit in a number, its place value determines its weight in terms of powers of 10. So, the digit 3 in the hundred million's place is equivalent to 3 x 100,000,000, which is indeed 300 million.

So, in the number 138,695,157, the digit 3 holds the value of 300 million, contributing significantly to the overall magnitude of the figure.

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Answer:757

Step-by-step explanation:

Answer:

Maximize C =[tex]9x + 7y[/tex]

[tex]8x + 10y \leq 17[/tex]

[tex]11x + 12y\leq 25[/tex]

and x ≥ 0, y ≥ 0

Plot the lines on graph

[tex]8x + 10y \leq 17[/tex]

[tex]11x + 12y\leq 25[/tex]

[tex]x\geq 0[/tex]

[tex]y\geq 0[/tex]

So, boundary points of feasible region are (0,1.7) , (2.125,0) and (0,0)

Substitute the points in  Maximize C

At  (0,1.7)

Maximize C =[tex]9(0) + 7(1.7)[/tex]

Maximize C =[tex]11.9[/tex]

At  (2.125,0)

Maximize C =[tex]9(2.125) + 7(0)[/tex]

Maximize C =[tex]19.125[/tex]

At   (0,0)

Maximize C =[tex]9(0) + 7(0)[/tex]

Maximize C =[tex]0[/tex]

So, Maximum value is attained at   (2.125,0)

So, the optimal value of x is 2.125

The optimal value of y is 0

The maximum value of the objective function is 19.125

There are several ways to calculate optimal solution; one of them, is by using graphs.

The given parameters are:

[tex]\mathbf{Max\ C = 9x + 7y}[/tex]

[tex]\mathbf{8x + 10y \le 17}[/tex]

[tex]\mathbf{11x + 12y \le 25}[/tex]

[tex]\mathbf{x,y\ge 0}[/tex]

See attachment for the graphs of [tex]\mathbf{8x + 10y \le 17}[/tex] and [tex]\mathbf{11x + 12y \le 25}[/tex]

From the graph, the feasible regions are:

[tex]\mathbf{(x,y) = (0,1.7), (2.125,0)}[/tex]

Test these values in the objective function

[tex]\mathbf{Max\ C = 9x + 7y}[/tex]

[tex]\mathbf{C = 9 \times 0 + 7 \times 1.7 = 11.9}[/tex]

[tex]\mathbf{C = 9 \times 2.125 + 7 \times 0 = 19.125}[/tex]

So, the value of C is maximum at: [tex]\mathbf{(x,y) = (2.125,0)}[/tex]

So, we have:

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Answer:

(A) [tex]y=ke^{2t}[/tex] with [tex]k\in\mathbb{R}[/tex].

(B) [tex]y=ke^{2t}/2-1/2[/tex] with [tex]k\in\mathbb{R}[/tex]

(C) [tex]y=k_1e^{2t}+k_2e^{-2t}[/tex] with [tex]k_1,k_2\in\mathbb{R}[/tex]

(D) [tex]y=k_1e^{2t}+k_2e^{-2t}+e^{3t}/5[/tex] with [tex]k_1,k_2\in\mathbb{R}[/tex],

Step-by-step explanation

(A) We can see this as separation of variables or just a linear ODE of first grade, then [tex]0=y'-2y=\frac{dy}{dt}-2y\Rightarrow \frac{dy}{dt}=2y \Rightarrow  \frac{1}{2y}dy=dt \ \Rightarrow \int \frac{1}{2y}dy=\int dt \Rightarrow \ln |y|^{1/2}=t+C \Rightarrow |y|^{1/2}=e^{\ln |y|^{1/2}}=e^{t+C}=e^{C}e^t} \Rightarrow y=ke^{2t}[/tex]. With this answer we see that the set of solutions of the ODE form a vector space over, where vectors are of the form [tex]e^{2t}[/tex] with [tex]t[/tex] real.

(B) Proceeding and the previous item, we obtain [tex]1=y'-2y=\frac{dy}{dt}-2y\Rightarrow \frac{dy}{dt}=2y+1 \Rightarrow  \frac{1}{2y+1}dy=dt \ \Rightarrow \int \frac{1}{2y+1}dy=\int dt \Rightarrow 1/2\ln |2y+1|=t+C \Rightarrow |2y+1|^{1/2}=e^{\ln |2y+1|^{1/2}}=e^{t+C}=e^{C}e^t \Rightarrow y=ke^{2t}/2-1/2[/tex]. Which is not a vector space with the usual operations (this is because [tex]-1/2[/tex]), in other words, if you sum two solutions you don't obtain a solution.

(C) This is a linear ODE of second grade, then if we set [tex]y=e^{mt} \Rightarrow y''=m^2e^{mt}[/tex] and we obtain the characteristic equation [tex]0=y''-4y=m^2e^{mt}-4e^{mt}=(m^2-4)e^{mt}\Rightarrow m^{2}-4=0\Rightarrow m=\pm 2[/tex] and then the general solution is [tex]y=k_1e^{2t}+k_2e^{-2t}[/tex] with [tex]k_1,k_2\in\mathbb{R}[/tex], and as in the first items the set of solutions form a vector space.

(D) Using C, let be [tex]y=me^{3t} [/tex] we obtain that it must satisfies [tex]3^2m-4m=1\Rightarrow m=1/5[/tex] and then the general solution is [tex]y=k_1e^{2t}+k_2e^{-2t}+e^{3t}/5[/tex] with [tex]k_1,k_2\in\mathbb{R}[/tex], and as in (B) the set of solutions does not form a vector space (same reason! as in (B)).  

Answer:

the patient receiving is 15 mg/hr

Step-by-step explanation:

given data

infusing = 30 mL/hr

drug mixed = 250 mg

D5W = 500ml

to find out

How many mg/hr is the patient receiving

solution

we know infusing rate is = 30 mL/hr   ...................1

and we know mixed is = [tex]\frac{250mg}{500ml}[/tex]   ..............2

so

now we multiply both equation 1 and 2

patient receiving = [tex]\frac{250mg}{500ml}[/tex]  × [tex]\frac{30ml}{hr}[/tex]

solve we get here

patient receiving = [tex]\frac{15mg}{hr}[/tex]

so the patient receiving is 15 mg/hr

The patient is receiving 15 mg/hr of aminophylline.

To calculate the mg/hr that the patient is receiving, we need to find the concentration of aminophylline in the infusion and multiply it by the infusion rate. The concentration of aminophylline is 250 mg in 500 ml of D5W. This means that there are 250 mg of aminophylline in 500 ml, or 0.5 mg/ml. The infusion rate is 30 ml/hr. So, to find the mg/hr, we multiply the concentration by the infusion rate: 0.5 mg/ml * 30 ml/hr = 15 mg/hr. Therefore, the patient is receiving 15 mg/hr of aminophylline.

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To calculate the time Izzy takes to run to school on rainy days, we use the distance and speed to find that she runs 1.7 miles in approximately 13.6 minutes. We are unable to calculate the distance through the woods or the time saved without further information.

Calculation of Time and Distance

To calculate the time Izzy takes to run to school on rainy days, we use the following formula:

Time (in hours) = Distance (in miles) / Speed (in miles per hour)

Izzy runs a total distance of 1.2 miles on West St and then 0.5 miles on North Ave, summing up to 1.7 miles. Given Izzy's speed is 7.5 miles per hour, the time taken to run to school on rainy days can be calculated as:

Time = (1.2 + 0.5) miles / 7.5 mph = 1.7 / 7.5

To find the time in minutes, multiply the time in hours by 60:

Time in minutes = (1.7 / 7.5)  imes 60
= 13.6 minutes (approximately)

As the dashed path through the woods on dry days is not described in the question, we cannot calculate the exact distance Izzy is traveling through the woods. Without this information, we also cannot calculate the time saved by cutting through the woods.

Answer:

No. of juniors = 14

No. of seniors = 16

Total students = 30

A) From the 30 members, how many ways are there to arrange 5 members of the club in a line?

Since we are asked about arrangement so we will use permutation

Formula : [tex]^nP_r=\frac{n!}{(n-r)!}[/tex]

n = 30

r = 5

[tex]^{30}P_5=\frac{30!}{(30-5)!}[/tex]

[tex]^{30}P_5=17100720[/tex]

So, From the 30 members, there are 17100720 ways to arrange 5 members of the club in a line?

B) How many ways are there to arrange 5 members of the club in a line if there must be a senior at the beginning of the line and at the end of the line?

Out of 16 seniors 2 will be selected

So, 3 places are vacant

Remaining students = 30-2 = 28

So, out of 28 students 3 students will be selected

No. of ways = [tex]^{16}P_2 \times ^{28}P_3[/tex]

No. of ways = [tex]\frac{16!}{(16-2)!}\times\frac{28!}{(28-3)!}[/tex]

                   = [tex]4717440[/tex]

There are 4717440 ways to arrange 5 members of the club in a line if there must be a senior at the beginning of the line and at the end of the line.

C)If the club sends 2 juniors and 2 seniors to the tournament, how many possible groupings are there?

Since we are not asked about arrangement so we will use combination

Out of 16 seniors 2 will be selected

Out of 14 juniors 2 will be selected

Formula : [tex]^nC_r=\frac{n!}{r!(n-r)!}[/tex]

So, No. of possible groupings = [tex]^{16}C_2 \times ^{14}C_2[/tex]

                                                  = [tex]\frac{16!}{2!(16-2)!} \times \frac{14!}{2!(14-2)!}[/tex]

                                                  = [tex]10920[/tex]

If the club sends 2 juniors and 2 seniors to the tournament, there are 10920 possible groupings

D) If the club sends either 4 juniors or 4 seniors, how many possible groupings are there?

Out of 16 seniors 4 will be selected

or

Out of 14 juniors 4 will be selected

So, No. of possible groupings = [tex]^{16}C_4 + ^{14}C_4[/tex]

                                                  = [tex]\frac{16!}{4!(16-4)!} + \frac{14!}{4!(14-4)!}[/tex]

                                                  = [tex]2821[/tex]

So,If the club sends either 4 juniors or 4 seniors, there are 2821 possible groupings .

Final answer:

To find the percent of the original price you end up paying after a 25% discount and an additional 40% discount, first calculate the discounted prices and then determine the final price. In this case, you end up paying 45% of the original price.

Explanation:

To find the percent of the original price you end up paying, you need to calculate the final price after both discounts. Let's say the original price of the item is $100. First, apply the 25% discount by multiplying the original price by 0.75 (1 - 0.25 = 0.75). This gives you a price of $75. Next, apply the additional 40% discount by multiplying the discounted price by 0.60 (1 - 0.40 = 0.60). This gives you a final price of $45. Therefore, you end up paying 45% of the original price.

Answer:

(x,y,z)=(2,0,-4)

Step-by-step explanation:

[tex]\left[\begin{array}{ccc|c}2&2&-3&16\\2&-3&2&-4\\4&-1&3&-4\end{array}\right][/tex]

Using the elementary operations

[tex]\left[\begin{array}{ccc|c}2&2&-3&16\\0&-5&5&-20\\0&-5&9&-36\end{array}\right][/tex]

[tex]\left[\begin{array}{ccc|c}1&1&-3/2&8\\0&1&-1&4\\0&0&4&-16\\\end{array}\right][/tex]

[tex]\left[\begin{array}{ccc|c}1&1&-3/2&8\\0&1&-1&4\\0&0&1&-4\\\end{array}\right][/tex]

[tex]\left[\begin{array}{ccc|c}1&1&-3/2&8\\0&1&0&0\\0&0&1&-4\\\end{array}\right][/tex]

[tex]\left[\begin{array}{ccc|c}1&0&-3/2&8\\0&1&0&0\\0&0&1&-4\\\end{array}\right][/tex]

[tex]\left[\begin{array}{ccc|c}1&0&0&2\\0&1&0&0\\0&0&1&-4\\\end{array}\right][/tex]

The answer is determined:

x=2

y=0

z=-4

You can check they are correct, by entering in the original formulas.

Answer:

0.2 kW-hr

Step-by-step explanation:

First, we are going to transform 100W in kW. We can use a rule of three in which we know that 1000W is equivalent to 1 kW, then how many kW are equivalent to 100W. This is:

1000W ------------- 1 kW

100W -------------- X

Where X is the the number of kW that are equivalent to 100W. Solving for X, we get:

[tex]X=\frac{100W * 1kW}{1000W} =0.1kW[/tex]

Then, for calculate the number of kW-hr we need to multiplicate the number of kW with the number of hours, This is:

0.1 kW * 2 hours = 0.2 kW-hr

Finally, when one 100W light bulbs is used for 2 hours, it used 0.2 kW-hr

Answer:

The two readings that are cutoff values are T=1.84 deg C and T=-1.84 deg C.

Step-by-step explanation:

Thermometers rejected by measurements above normal represent 3.3% of the total, which indicates, by normal probability distribution data, that accepted thermometers are 96.7% likely to measure less than the maximum allowable temperature.  

This value (P(X>x)=0.967) corresponds to z = 1.8388. Since the mean and standard deviation values are the same as the standard normal probability distribution (mean = 0, sd = 0), the z value is equivalent to the measured value (temperature).

Given the symmetry of the probability distribution, we can affirm that the thermometers rejected by measurements below the permissible measured a temperature lower than -1.8388.

Final answer:

The cutoff values separating the rejected thermometers are -1.88°C for the low end and 1.88°C for the high end, based on a normal distribution with a mean of 0°C and a standard deviation of 1.00°C.

Explanation:

To find the two readings that are cutoff values separating the rejected thermometers, we look at the normal distribution curve with a mean of 0°C and a standard deviation of 1.00°C. Since 3% of thermometers are rejected for high readings and another 3% for being too low, these correspond to the tail ends of the distribution.

Using the Z-score table, find the Z-score that has 3% in the tail. For the lower end, we seek the Z-score where the left tail (the area to the left of the Z-score) is 0.03, and for the higher end, we look for a Z-score where the right tail is 0.03. These Z-scores are approximately -1.88 for the low end and +1.88 for the high end (using the 97th percentile since we want the upper 3%).

To find the actual thermometer readings:

For the low cutoff: cutoff low = mean + (Z-score * standard deviation) = 0 + (-1.88 * 1) = -1.88°C.

For the high cutoff: cutoff high = mean + (Z-score * standard deviation) = 0 + (1.88 * 1) = 1.88°C.

Therefore, thermometers reading lower than -1.88°C or higher than 1.88°C will be rejected.

Answer:

Chris lives farther from school than Alex lives, and Chris walks to school at a faster rate than Alex

Step-by-step explanation:

Chris walked 4 miles.

Alex walked 2.

Alex takes 25 minutes to walk one mile. (50÷2) and Chris takes 15 minutes to walk one mile (60÷4), meaning Chris walks at a faster rate although he lives farther away.

Answer:

Chris lives farther from the school than Alex lives, and Chris walked to school at a faster rate than Alex walked.

Answer:

The estimated GDP in 2007 is 12782 billion dollars.

The estimated GDP in 2011 is 15010 billion dollars.

The estimated GDP in 2012 is 15567 billion dollars.

Step-by-step explanation:

Consider the provided function.

[tex]g(y) = 557y +8883[/tex]

Where y = 10 corresponds to the year 2010.

Part (b) Estimate the GDP (to the nearest billion dollars) in 2007.

If y = 10 corresponds to the year 2010 then y = 7 corresponds to the year 2007.

Substitute the value of y = 7 in the provided function.

[tex]g(y) = 557(7) +8883[/tex]

[tex]g(y) = 3899+8883[/tex]

[tex]g(y) = 12782[/tex]

Hence, the estimated GDP in 2007 is 12782 billion dollars.

Part (b) Estimate the GDP (to the nearest billion dollars) in 2011.

If y = 10 corresponds to the year 2010 then y = 11 corresponds to the year 2011.

Substitute the value of y = 11 in the provided function.

[tex]g(y) = 557(11) +8883[/tex]

[tex]g(y) = 6127+8883[/tex]

[tex]g(y) = 15010[/tex]

Hence, the estimated GDP in 2011 is 15010 billion dollars.

Part (C) Estimate the GDP (to the nearest billion dollars) in 2012.

If y = 10 corresponds to the year 2010 then y = 12 corresponds to the year 2012.

Substitute the value of y = 12 in the provided function.

[tex]g(y) = 557(12) +8883[/tex]

[tex]g(y) = 6684+8883[/tex]

[tex]g(y) = 15567[/tex]

Hence, the estimated GDP in 2012 is 15567 billion dollars.

Final answer:

To estimate the GDP of a certain country for different years using the function g(y) = 557y + 8883, the value of y is adjusted according to the respective year, and the GDP is calculated by substituting this value into the formula. For 2007, 2011, and 2012, respective GDPs estimated are 12019, 14311, and 14868 billion dollars.

Explanation:

The question asks us to estimate the GDP of a certain country for different years using the function g(y) = 557y + 8883, where y=10 corresponds to the year 2010. To find the GDP for the given years, we first need to calculate the correct value of y for each year and then substitute these values into the function.

This approach allows us to quantify economic growth over different years, comparing these figures to gain insight into the economic health and trends of the country.

Step-by-step explanation:

Proof for i)

We will prove by mathematical induction that, for every natural [tex]n\geq 4[/tex], the number of diagonals of a convex polygon with n vertices is [tex]\frac{n(n-3)}{2}[/tex].

In this proof we will use the expression d(n) to denote the number of diagonals of a convex polygon with n vertices

Base case:

First, observe that:, for n=4, the number of diagonals is

[tex]2=\frac{n(n-3)}{2}[/tex]

Inductive hypothesis:  

Given a natural [tex]n \geq 4[/tex],

[tex]d(n)=\frac{n(n-3)}{2}[/tex]

Now, we will assume the induction hypothesis and then use this assumption, involving n, to prove the statement for n + 1.

Inductive step:

Observe that, given a convex polygon with n vertices, wich we will denote by P(n), if we add a new vertix (transforming P(n) into a convex polygon with n+1 vertices, wich we will denote by P(n+1)) we have that:

Because of these observation we know that, for every [tex]n\geq 4[/tex],

[tex]d(n+1)=d(n)+1+(n-2)=d(n)+n-1[/tex]

Therefore:

[tex]d(n+1)=d(n)+n-1=\frac{n(n-3)}{2}+n-1=\frac{n^2-3n+2n-2}{2}=\frac{n^2-n-2}{2}=\frac{(n+1)(n-2)}{2}[/tex]

With this we have proved our statement to be true for n+1.    

In conlusion, for every natural [tex]n \geq 4[/tex],

[tex]d(n)=\frac{n(n-3)}{2}[/tex]

Proof for ii)

Observe that:

Then, the statement is not true for n=1,2,3.

We will prove by mathematical induction that, for every natural [tex]n \geq 4[/tex],

[tex]2n<n![/tex].

Base case:

For n=4, [tex]2n=8<24=n![/tex]

Inductive hypothesis:  

Given a natural [tex]n \geq 4[/tex], [tex]2n<n![/tex]

Now, we will assume the induction hypothesis and then use this assumption, involving n, to prove the statement for n + 1.

Inductive step:

Observe that,

[tex]n!+2\leq (n+1)! \iff n!+2\leq n!(n+1) \iff 1+\frac{2}{n!}\leq n+1 \iff 2\leq n*n![/tex]

wich is true as we are assuming [tex]n\geq 4[/tex]. Therefore:

[tex]2(n+1)=2n+2<n!+2\leq (n+1)![/tex]

With this we have proved our statement to be true for n+1.    

In conlusion, for every natural [tex]n \geq 4[/tex],

[tex]2n<n![/tex]

Answer:   99 minutes

Step-by-step explanation:

Given: While completing a race, Edward spent 54 minutes walking.

The ratio of time walking to jogging was 6:5 i.e. [tex]\dfrac{6}{5}[/tex]     (1)

Let x be the time taken ( in minutes ) by him for jogging.

then, the ratio of time walking to jogging will be [tex]\dfrac{54}{x}[/tex]  (2)

From (1) and (2), we have

[tex]\dfrac{6}{5}=\dfrac{54}{x}\\\\\Rightarrow\ 6x=54\times5\\\\\Rightarrow\ x=\dfrac{54\times5}{6}=45[/tex]  

So, the number of minutes he took for jogging = 45 minutes

Now, the total time he spent on completing the race= 54+45=99 minutes

Answer:

n = 60

Step-by-step explanation:

GIVEN DATA:

Total sum of consecutive number is 1800

sum of n number is given as

[tex] sum = \frac{ n(n+1)}{2}[/tex]

where n is positive number and belong to natural number i.e 1,2,3,4,...

from the data given we have[tex]1800 = \frac{n(n+1)}{2}[/tex]

solving for n we get

[/tex]n^2 + n -3600 = 0[/tex]

n = 59.5, -60.5

therefore n = 60

Answer:

Mean = 3640

Mode = 4100

Median = 3830.

Step-by-step explanation:

We are given the following data in the question:

Strength of casts (in psi):

3970,4100,3100,3200,2950,3830,4100,4050,3460

Formula:

[tex]Mean = \displaystyle\frac{\text{Sum of all observation}}{\text{Total number of observations}}[/tex]

[tex]\displaystyle\frac{3970+4100+3100+ 3200+ 2950+ 3830+4100+ 4050+ 3460}{9} = \displaystyle\frac{32760} {9} = 3640[/tex]

Mode is the entry with most frequency. Thus, for the given sample mode = 4100.

Median

Since n = 9 is odd,

Formula:

[tex]Median = \displaystyle\frac{n+1}{2}th~term[/tex]

Data in ascending order:

2950,3100,3200,3460,3830,3970,4050,4100,4100

Median = 5th term = 3830.

The mean strength of the concrete is 3640.

The mode strength of the concrete is 4100.

The median strength of the concrete is 3830.

Given

A concrete mix is designed to withstand 3000 pounds per square inch​ (psi) of pressure.

The following data represent the strength of nine randomly selected casts​ (in psi).

3970​, 4100​, 3100​, 3200​, 2950​, 3830​, 4100​, 4050​, 3460

The mean value of the data is given by;

[tex]\rm Mean = \dfrac{Sum \ of \ all \ observation}{Total \ number \ of \ observation}[/tex]

The mean of the strength of the concrete is;

[tex]\rm Mean = \dfrac{Sum \ of \ all \ observation}{Total \ number \ of \ observation}\\\\Mean =\dfrac{3970+4100+3100+3200+2950+3830+ 4100+4050+3460 }{9}\\\\Mean =\dfrac{32760}{9}\\\\Mean=3640[/tex]

The mean strength of the concrete is 3640.

Mode is the entry with the most frequency.

Thus, for the given sample mode = 4100.

The mode strength of the concrete is 4100.

The mid-value of the data is called the median.

The median strength of the concrete is 3830.

To know more about Mean click the link given below.

https://brainly.com/question/12513463

Answer:

[tex]1\ \frac{\text{cal}}{m^2\times sec\times ^\circ C}=0.03926\frac{\text{BTU}}{ft^2\times hr\times ^\circ F}[/tex]

Step-by-step explanation:

To find : Convert [tex]1\ \frac{\text{cal}}{m^2\times sec\times ^\circ C}[/tex] into [tex]\frac{\text{BTU}}{ft^2\times hr\times ^\circ F}[/tex]

Solution :

We convert units one by one,

[tex]1\text{ m}^2=10.7639\text{ ft}^2[/tex]

[tex]1\text{ sec}=\frac{1}{3600}\text{ hour}[/tex]

[tex]1\text{ cal}=0.003968\text{ BTU}[/tex]

Converting temperature unit,

[tex]^\circ C\times \frac{9}{5}+32=^\circ F[/tex]

[tex]1^\circ C\times \frac{9}{5}+32=33.8^\circ F[/tex]

So, [tex]1^\circ C=33.8^\circ F[/tex]

Substitute all the values in the unit conversion,

[tex]1\ \frac{\text{cal}}{m^2\times sec\times ^\circ C}=\frac{0.003968}{10.7639\times \frac{1}{3600}\times 33.8}\frac{\text{BTU}}{ft^2\times hr\times ^\circ F}[/tex]

[tex]1\ \frac{\text{cal}}{m^2\times sec\times ^\circ C}=\frac{0.003968}{0.101061}\frac{\text{BTU}}{ft^2\times hr\times ^\circ F}[/tex]

[tex]1\ \frac{\text{cal}}{m^2\times sec\times ^\circ C}=0.03926\frac{\text{BTU}}{ft^2\times hr\times ^\circ F}[/tex]

Therefore, The conversion of unit is [tex]1\ \frac{\text{cal}}{m^2\times sec\times ^\circ C}=0.03926\frac{\text{BTU}}{ft^2\times hr\times ^\circ F}[/tex]

Answer:

96%

Step-by-step explanation:

Conditional probability is defined as:  

P(A|B) = P(A∩B) / P(B)  

Or, in English:  

Probability that A occurs, given that B has occurred = Probability that both A and B occur / Probability that B occurs

We want to find the probability that a woman uses an ATM to get cash, given that she uses an ATM to check her balance.

P(withdraws cash | checks account)

Using the definition of condition probability, this equals:

P = P(withdraws cash AND checks account) / P(checks account)

We know that P(checks account) = 0.28.

But we don't know what P(withdraws cash AND checks account) is.  To find that, we need to use the definition of P(A∪B):

P(A∪B) = P(A) + P(B) − P(A∩B)

This says that the probability of A or B occurring (or both) is the probability of A occurring plus the probability of B occurring minus the probability of both A and B occurring.

P(withdraws cash OR checks account) = P(withdraws cash) + P(checks account) − P(withdraws cash AND checks account)

0.95 = 0.94 + 0.28 − P(withdraws cash AND checks account)

P(withdraws cash AND checks account) = 0.27

Therefore:

P = 0.27 / 0.28

P ≈ 0.96

Final answer:

The probability that a woman who checks her account balance at an ATM also withdraws cash is approximately 96.43%.

Explanation:

To solve the problem, we can apply the probability rule for conditional probability. We are provided with the following probabilities:

Using this information, we're interested in finding the probability that a user who checks their account balance also withdraws cash, represented as P(Cash|Balance).

The formula for conditional probability is:

P(A|B) = P(A ∩ B) / P(B)

Where A and B are two events, and P(A|B) is the conditional probability of A given B.

Using the inclusion-exclusion principle, we can express P(Cash ∩ Balance) as:

P(Cash ∩ Balance) = P(Cash) + P(Balance) - P(Cash ∪ Balance)

Substitute the given probabilities:

P(Cash ∩ Balance) = 0.94 + 0.28 - 0.95 = 0.27

The probability that a woman who checks her balance also gets cash (P(Cash|Balance)) is:

P(Cash|Balance) = P(Cash ∩ Balance) / P(Balance)

P(Cash|Balance) = 0.27 / 0.28 ≈ 0.9643

Therefore, the probability is approximately 96.43%.

Final answer:

To calculate the sustainable growth rate of Mercoxit Energy Inc., the retention ratio is multiplied by the calculated Return on Equity (ROE), which is derived from the ROA and EM figures given. The result yields a sustainable growth rate of 5.21% for Mercoxit Energy Inc.

Explanation:

The sustainable growth rate (SGR) is a company's maximum growth rate in sales using internal financial resources while maintaining a constant debt to equity ratio. To calculate the SGR for Mercoxit Energy Inc., one needs to use the following formula:

SGR = Retention Ratio x Return on Equity (ROE)

While we are given the retention ratio and the Return on Assets (ROA), we are not directly given the ROE. However, we can calculate ROE using the provided Equity Multiplier (EM), since ROE can be written as:

ROE = ROA x EM

Considering the given figures:

Retention Ratio = 0.79

ROA = 5.94%

EM = 1.11

Now calculate ROE:

ROE = 0.0594 x 1.11 = 0.065934 (or 6.5934%)

Finally, we calculate the SGR:

SGR = 0.79 x 0.065934 = 0.05208786 (or 5.208786%)

If we round to two decimal places, Mercoxit Energy Inc. has a sustainable growth rate of 5.21%.

Answer:

The statement is true

Step-by-step explanation:

We will prove by mathematical induction that, for every natural n,

[tex]\sum^{n}_{i=1}\frac{1}{(2i-1)(2i+1)} =\frac{n}{2n+1}[/tex]

We will prove our base case, when n=1, to be true.

base case:

[tex]\sum^{1}_{i=1}\frac{1}{(2-1)(2+1)} =\frac{1}{3}=\frac{n}{2n+1}[/tex]

Inductive hypothesis:

[tex]\sum^{n}_{i=1}\frac{1}{(2i-1)(2i+1)} =\frac{n}{2n+1}[/tex]

Now, we will assume the induction hypothesis and then uses this assumption, involving n, to prove the statement for n + 1.

Inductive step:

[tex]\sum^{n+1}_{i=1}\frac{1}{(2i-1)(2i+1)} =\sum^{n}_{i=1}\frac{1}{(2i-1)(2i+1)}+\frac{1}{(2(n+1)-1)(2(n+1)+1)}=\frac{n}{2n+1}+\frac{1}{(2n+1)(2n+3)}=\frac{n(2n+3)+1}{(2n+1)(2n+3)}=\frac{2n^2+3n+1}{(2n+1)(2n+3)}=\frac{(2n+1)(n+1)}{(2n+1)(2n+3)}=\frac{n+1}{2n+3}=\frac{n+1}{2(n+1)+1}[/tex]

With this we have proved our statement to be true for n+1.

In conlusion, for every natural [tex]n[/tex].

[tex]\sum^{n}_{i=1}\frac{1}{(2i-1)(2i+1)} =\frac{n}{2n+1}[/tex]

Answer:

2h-2=16

Step-by-step explanation:

Let

h -----> the number of horses

c ----> the number of a cows

we know that

h+c=16 -----> equation A

h=c+2

c=(h-2) ----> equation B

Substitute equation B in equation A and solve for h

h+(h-2)=16

2h-2=16

2h=16+2

2h=18

h=9

therefore

The equation that represent the situation is

2h-2=16

Answer:

a) The probability that an order requests at least one optional feature is 34%+26% = 60%.

b) The probability that an  order does not request more than one optional feature is 40% + 34% = 74%.

Step-by-step explanation:

Probability:

What you want to happen is the desired outcome.

Everything that can happen iis the total outcomes.

The probability is the division of the number of possible outcomes by the number of total outcomes.

In our problem, the probabilities are:

-40% that no optional features are requested.

-34% that one optional feature is requested

-26% that more than one optional feature is requested.

(a) What is the probability that an order requests at least one optional feature?

There is a 34% probability that one optional feature is requested and a 26% probability that more than one optional feature is requested.

So the probability that an order requests at least one optional feature is 34%+26% = 60%.

(b) What is the probability that an order does not request more than one optional feature?

There is a 40% probability that no optional features are requested and a 34% probability that one optional feature is requested.

So the probability that an  order does not request more than one optional feature is 40% + 34% = 74%.

Final answer:

The probability an order requests at least one optional feature is 0.60, and the probability an order does not request more than one optional feature is 0.74.

Explanation:

The question involves calculating probabilities based on provided proportions of orders with optional features.

Part (a): Probability of at least one optional feature

The proportion of orders with no optional features is 0.40. Therefore, the probability that an order requests at least one optional feature is 1 - 0.40 = 0.60. So, the probability is 0.60 when rounded to two decimal places.

Part (b): Probability of not more than one optional feature

We are given that orders with one optional feature make up 0.34 and those with no optional features constitute 0.40. Adding these together gives us a probability of 0.34 + 0.40 = 0.74 for orders not requesting more than one optional feature. Thus, this probability is 0.74, rounded to two decimal places.

Answer:

The price of calculator before 30 years = $30+$122.36 = $145.36

Step-by-step explanation:

We have given price current price of calculator = $23

It is given that current price of calculator is after decrease of 532 %

We have to find the price of calculator before 30 years

The price of calculator will be more than 532 % from the current price

So 532% of 23 [tex]=\frac{23\times 532}{100}=$122.36[/tex]

So the price of calculator before 30 years = $30+$122.36 = $145.36

Answer:

Neither

Step-by-step explanation:

We are given that a sequence

5,8,13,21,34,55,....

We have to determine the sequence is an arithmetic , geometric or neither.

[tex]a_1=5,a_2=8,a_3=13,...a_6=55[/tex]

We know that if a sequence is an arithmetic then the difference between consecutive terms are equal.

If a sequence is geometric then the ratio of consecutive terms is constant.

[tex]d_1=a_2-a_1=8-5=3[/tex]

[tex]d_2=a_3-a_2=13-8=5[/tex]

[tex]d_1\neq d_2[/tex]

Hence, the difference between the consecutive terms is not equal .Therefore, sequence is not an arithmetic sequence.

[tex]r_1=\frac{a_2}{a_1}=\frac{8}{5}[/tex]

[tex]r_2=\frac{a_3}{a_2}=\frac{13}{8}[/tex]

[tex]r_1\neq r_2[/tex]

Hence, the ratio of consecutive terms is not equal .Therefore, given sequence is not geometric sequence.

Therefore, given sequence is not an arithmetic nor geometric sequence.