1.How much will $5,000 accumulate in 5 years if it earns 6% per annum for the first 3 years and then 8% per annum for the next 2 years?

Answer:

(1) [tex]y=(\frac{2}{a+bx})^3[/tex]

By differentiating w.r.t. x,

[tex]\frac{dy}{dx}=3(\frac{2}{a+bx})^2\times \frac{d}{dt}(\frac{2}{a+bx})[/tex]

[tex]=3(\frac{2}{a+bx})^2\times (-\frac{2}{(a+bx)^2})[/tex]

[tex]=-\frac{24}{(a+bx)^4}[/tex]

(2) [tex]y=(at^3-3bt)^3[/tex]

By differentiating w.r.t. t,

[tex]\frac{dy}{dt}=3(at^3-3bt)^2\times \frac{d}{dt}(at^3-3bt)[/tex]

[tex]=3(at^3-3bt)^2 (3at^2-3b)[/tex]

[tex]=9t^2(at^2-3b)^2(at^2-b)[/tex]

(3) [tex]y=(t^b)(e^\frac{b}{t})[/tex]

Differentiating w.r.t. t,

[tex]\frac{dy}{dt}=t^b\times \frac{d}{dt}(e^\frac{b}{t})+\frac{d}{dt}(t^b)\times e^\frac{b}{t}[/tex]

[tex]=t^b(e^\frac{b}{t})\times \frac{d}{dt}(\frac{b}{t}) + bt^{b-1}(e^\frac{b}{t})[/tex]

[tex]=t^be^\frac{b}{t}(-\frac{b}{t^2})+bt^{b-1}e^{\frac{b}{t}}[/tex]

(4) [tex]z = ax^2.sin (4x)[/tex]

Differentiating w.r.t. x,

[tex]\frac{dz}{dt}=ax^2\times \frac{d}{dx}(sin (4x))+sin (4x)\times \frac{d}{dx}(ax^2)[/tex]

[tex]=ax^2\times cos(4x).4+sin (4x)(2ax)[/tex]

[tex]=4ax^2cos (4x)+2ax sin (4x)[/tex]

Answer:

Functions with straight lines have this sort of equation: Y = mx + b where the m is called slope and b, the y-intercept. As you don't have the slope you have to find out. In this case, the slope if 10, so when u have the m, just replace with one of your points to get the b, (y-intercept). The final equation for this function is, Y= 10x+35

Step-by-step explanation:

Answer:

a) 0.0202

b)The seller's claim is not correct.

c) If the seller's claim is true we cannot have individual length of 72.15 inch.

Step-by-step explanation:

In the question it is given that

population mean, μ = 72 inch

Population standard deviation, σ = 0.5 inch

Sample size, n = 47

Sample mean, x =72.15 inch

a) z score = [tex]\frac{x-\mu}{\sigma/\sqrtr{n}}[/tex] = [tex]\frac{72.15-72}{0.5/\sqrt{47}}[/tex] = 2.0567

P(x ≥ 72.15) = P(z ≥ 2.0567) = 0.5 - 0.4798 = 0.0202

We calculated the probability with the help of standard normal table.

b) The sellers claim that the machine cuts the lumber with a mean length of 72 inch is not correct as we obtained a very low probability.

c)If we assume that the seller's claim is true that is the machine cuts the lumber into mean length of 72 inch then we cannot have an individual length 72.15.

Standard error = [tex]\frac{\sigma}{\sqrt{n} }[/tex] = 0.0729

Because it does not lie within the range of two standard errors that is (μ±2 standard error) = (71.8541,72.1458)

The distribution function for the length of the time that the first to arrive has to wait for the other is given by a triangular shape. This comes from the continuous uniform distribution nature of their arrival time. More probable waiting times are represented by the peak of the triangle.

This problem relates to the mathematical concept of a continuous uniform distribution. In the meeting scenario of Mr. Warren Buffet and Mr. Zhao Danyang, each person arrives between 12 pm and 1 pm at random with a uniform probability. This means there is equal likelihood for each time interval within the hour for them to arrive.

We are interested in the length of time the first person to arrive has to wait for the second. Let's denote the time of arrival of Mr. Buffet by X and that of Mr. Zhao by Y, both from 0 (12:00 pm) to 1 (1:00 pm). The time the first to arrive has to wait is |X - Y|. The distribution function of this waiting time (W = |X - Y|) is a triangle with its peak at W = 0 (no waiting time). The two ends of the base of the triangle fall at -1 and 1. This triangular shape comes from the fact that shorter waiting times (near the peak) are more probable because there is more overlap between X and Y.

Therefore, the cumulative distribution function for the waiting time, W, is given by:

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Answer:

[tex]x=-\frac{7}{26}, y=\frac{37}{26}, z=-\frac{21}{13}[/tex]

Step-by-step explanation:

The matrix representation of the system of linear equations is:

[tex]\left(\begin{array}{ccccc}15&13&4&\vdots&8\\11&13&9&\vdots&1\\3&5&7&\vdots&-5\\8&8&2&\vdots&6\\7&5&2&\vdots&2\end{array}\right)[/tex]

First apply the following row operations:

[tex]\begin{array}{c}R_{2}\to R_{2}+(-\frac{11}{15})R_{1}\\R_{3}\to R_{3}+(-\frac{1}{5})R_{1}\\R_{4}\to R_{4}+(-\frac{8}{15})R_{1}\\R_{5}\to R_{5}+(-\frac{7}{15})R_{1}\end{array}[/tex]

The resulting matrix is:

[tex]\left(\begin{array}{ccccc}15&13&4&\vdots&8\\0&\frac{52}{15}&\frac{91}{15}&\vdots&-\frac{73}{15}\\0&\frac{12}{5}&\frac{31}{5}&\vdots&-\frac{33}{5}\\0&\frac{16}{15}&-\frac{2}{15}&\vdots&\frac{26}{15}\\0&-\frac{16}{15}&\frac{2}{15}&\vdots&-\frac{26}{15}\end{array}\right)[/tex]

Then apply  the row operations:

[tex]\begin{array}{c}R_{3}\to R_{3}+(-\frac{12}{5}\cdot \frac{15}{52})R_{2}\\R_{4}\to R_{4}+(-\frac{-16}{15}\cdot \frac{15}{52})R_{2}\\R_{5}\to R_{5}+(\frac{16}{15}\cdot \frac{15}{52})R_{2}\end{array}[/tex]

The resulting matrix is:

[tex]\left(\begin{array}{ccccc}15&13&4&\vdots&8\\0&\frac{52}{15}&\frac{91}{15}&\vdots&-\frac{73}{15}\\0&0&2&\vdots&-\frac{42}{13}\\0&0&-2&\vdots&\frac{42}{13}\\0&0&2&\vdots&-\frac{42}{13}\end{array}\right)[/tex]

Now apply the row operations:

[tex]\begin{array}{c} R_{4}\to R_{4}+R_{3}\\R_{5}\to R_{5}+(-1)R_{3}\end{array}[/tex]

The resulting matrix is:

[tex]\left(\begin{array}{ccccc}15&13&4&\vdots&8\\0&\frac{52}{15}&\frac{91}{15}&\vdots&-\frac{73}{15}\\0&0&2&\vdots&-\frac{42}{13}\\0&0&0&\vdots&0\\0&0&0&\vdots&0\end{array} \right)[/tex]

The equivalent linear system associated to this matrix is

[tex]\begin{cases}15x+13y+4z=8\\\frac{52}{15}y+\frac{91}{15}z=-\frac{73}{15}\\2z=-\frac{42}{13}\end{cases}[/tex]

To Solve this last system is very simple by substitution. The solutions are:

[tex]x=-\frac{7}{26}, y=\frac{37}{26}, z=-\frac{21}{13}[/tex]

Answer:

c. a strawberry or a red fruit

Step-by-step explanation:

Since, Banana and apple, banana and kiwi are different fruits. Also, it is given that kiwi is brown, thus kiwi or a red fruit are different sets. So they are mutually exclusive.

Since strawberry is red fruit. Thus a strawberry or a red fruit is not mutually exclusive.

Further, If two sets A and B is Mutually Exclusive, then any units of Set A is not a member of Set B and vice-versa.

Answer:

The graph of given inequality is shown below.

Step-by-step explanation:

The given linear inequality is

[tex]x>7[/tex]

We need to graph the region of solution of the given linear inequality.

The related equation of given linear inequality is

[tex]x=7[/tex]

We know that x=a is a vertical line which passes through the point (a,0).

Here, a =7. So x=7 is a vertical line which passes through the point (7,0).

Relate line is a dotted line because the sign of inequality is >. It means the points on the line are not included in the solution set.

Shaded region is right side of the related line because the solution set contains all possible values of x which are greater than 7.

Answer:

Mean = 86.067

Median = 85

Step-by-step explanation:

The given data set is

{ 93, 94, 95, 89, 85, 82, 87, 85, 84, 80, 78, 78, 84, 87, 90 }

Number of observations = 15

Formula for mean:

[tex]Mean=\frac{\sum x}{n}[/tex]

where, n is number of observations.

Using the above formula, we get

[tex]Mean=\frac{93+94+95+89+85+82+87+85+84+80+78+78+84+87+90}{15}[/tex]

[tex]Mean=\frac{1291}{15}[/tex]

[tex]Mean\approx 86.067[/tex]

Therefore the mean of the given data set is 86.067.

Arrange the given data set in ascending order.

{ 78, 78, 80, 82, 84, 84, 85, 85, 87, 87, 89, 90, 93, 94, 95 }

Number of observation is 15 which is an odd term. So

[tex]Median=(\frac{n+1}{2})\text{th term}[/tex]

[tex]Median=(\frac{15+1}{2})\text{th term}[/tex]

[tex]Median=8\text{th term}[/tex]

[tex]Median=85[/tex]

Therefore the median of the data set is 85.

Answer:

$37685.56

Step-by-step explanation:

Given,

Total amount we want to accumulate,A = $500,000

Total time, we have,t = 30 years

Interest rate,r = 9%

We are asked to calculate how much money we should deposit to get the required amount after a certain time period.

So, according to compound interest formula,

[tex]A\ =\ P(1+r)^t[/tex]

Where, P = amount of money we need to deposit

[tex]=>\ 500,000\ =\ P(1+0.09)^{30}[/tex]

[tex]=>\ 500,000\ =\ P(1.09)^{30}[/tex]

[tex]=>\ 500,000\ =\ P\times 13.267[/tex]

[tex]=>\ \dfrac{500,000}{13.267}\ =\ P[/tex]

[tex]=>\ P\ =\ 37685.568[/tex]

So, we need to deposit total amount of $37,685.56.

Answer:

50%

Step-by-step explanation:

29 right and 29 wrong. 29+29=58

Divide 29 by 58 and you get .5

Move the decimal over two to the right, and there’s the 50%.

You could also multiply .5 by 100 and you’ll still get 50!

Answer:

Step-by-step explanation:

1. Find the total number of problems, incorrect and correct. 29 incorrect problems + 29 correct problems is 58 total problems.

2. Find the total number of correct problems divided by the total number of problems. 29 incorrect problems divided by 58 total problems. This is simplified to 1/2.

3. Finally, convert 1/2 into a fraction. We can do this by multiplying the fraction by 50/50 to get a fraction of 50/100, and we know that 50/100 50 percent.

If the intravenous fluid is adjusted to deliver 15 mg of medication per hour, the dosage rate would be 0.5 mg of medication delivered per half minute.

First, let's convert the 1 hour dosage rate of 15 mg to minutes. Since there are 60 minutes in an hour, divide 15 mg by 60 to find the dosage rate per minute.

15 mg / 60 min = 0.25 mg/min

Next, convert the dosage rate from minutes to half minutes. Since there are 2 half minutes in a minute, multiply the dosage rate per minute by 2 to find the dosage rate per half minute.

0.25 mg/min * 2 = 0.5 mg/half min

Therefore, if the intravenous fluid is adjusted to deliver 15 mg of medication per hour, the dosage rate would be 0.5 mg of medication delivered per half minute.

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If 15 mg of medication is delivered per hour, the rate per half-minute would be 0.125 mg. This is calculated by converting 15 mg/hour to 0.25 mg/minute, and then halving to find the dose per half-minute.

The student's question pertains to the calculation of dosage rates in medical treatment, specifically how much medication is delivered per half-minute given a rate of 15 mg of medication per hour. To figure this out, it's a simple conversion from hours to minutes and then to half-minutes.

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Answer: [tex]0.070<\sigma< 0.128[/tex]

Step-by-step explanation:

Confidence interval for population standard deviation :-

[tex]s\sqrt{\dfrac{n-1}{\chi^2_{\alpha/2, n-1}}}<\sigma<s\sqrt{\dfrac{n-1}{\chi^2_{1-\alpha/2, n-1}}}[/tex]

Given : Significance level : [tex]\alpha: 1-0.90=0.10[/tex]

Sample size : n= 17

Sample standard deviation: [tex]s= 0.09[/tex]

Then by using the chi-square distribution table, we have

[tex]\chi^2_{1-\alpha/2, n-1}}=\chi^2_{0.95, 16}=7.96[/tex]

[tex]\chi^2_{\alpha/2, n-1}}=\chi^2_{0.05, 16}=26.30[/tex]

Confidence interval for population standard deviation will be :-

[tex]( 0.09)\sqrt{\dfrac{16}{26.30}}<\sigma<( 0.09)\sqrt{\dfrac{16}{7.96}}\\\\0.070197981837<\sigma<0.12759861690\\\\\approx0.070<\sigma< 0.128[/tex]

Hence, 90% confidence interval of the standard deviation of the skating performance test.: [tex]0.070<\sigma< 0.128[/tex]

90% confidence interval of the standard deviation of the skating performance test is [tex]0.070 < \sigma < 0.127[/tex]

It is defined as the sampling distribution following an approximately normal distribution for known standard deviation.

The formula for finding the confidence interval for population standard deviation as follow:

[tex]\rm s \sqrt{\dfrac{n-1}{\chi ^2_{\alpha /2,n-1}}} < \sigma < s \sqrt{\dfrac{n-1}{\chi ^2_{1-\alpha /2,n-1}}}[/tex]

Where [tex]\rm s[/tex] is the standard deviation.

[tex]\rm n[/tex] is the sample size.

[tex]\rm{\chi ^2_{\alpha /2,n-1[/tex]  and  [tex]\rm\chi ^2_{1-\alpha /2,n-1[/tex] are the constant based on the Chi-Square distribution table.

[tex]\rm\alpha[/tex] is the significance level.

[tex]\rm\sigma[/tex] is the confidence interval for population standard deviation.

Calculating the confidence interval for population standard deviation:

We know significance level = 1 - confidence level

[tex]\rm \alpha = 1 - 0.90 = 0.10[/tex]

[tex]\rm n = 17[/tex]

[tex]\rm s = 0.09[/tex]

By the chi-square distribution table, the values of constants are below:

[tex]\rm \chi ^2_{1-\alpha/2,n-1} = \chi ^2_{0.95,16} = 7.96\\\rm \chi ^2_{ \alpha /2,n-1} = \chi ^2_{0.05,16} = 26.30[/tex]

putting all values in the above formula we will get the confidence interval for population standard deviation:

[tex]\begin{aligned} \rm (0.09) \sqrt{\dfrac{17-1}{26.30}}} & < \sigma < (0.09) \sqrt{\dfrac{17-1}{7.96}}}\\\\\rm s \sqrt{\dfrac{16}{26.30}}} & < \sigma < s \sqrt{\dfrac{16}{7.96}}\\\\\rm 0.0701197 & < \sigma < 0.12759 \\\\\end{aligned}\\[/tex]

or [tex]\approx 0.070 < \sigma < 0.127[/tex]

Thus, 90% confidence interval of the standard deviation of the skating performance test is [tex]0.070 < \sigma < 0.127[/tex]

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Answer:

The present value of K is, [tex]K=251.35[/tex]

Step-by-step explanation:

Hi

First of all, we need to construct an equation system, so

[tex](1)K=\frac{169}{(1+i)} +\frac{169}{(1+i)^{2}}[/tex]

[tex](2)K=\frac{225}{(1+i)^{2}} +\frac{225}{(1+i)^{4}}[/tex]

Then we equalize both of them so we can find [tex]i[/tex]

[tex](3)\frac{169}{(1+i)} +\frac{169}{(1+i)^{2}}=\frac{225}{(1+i)^{2}} +\frac{225}{(1+i)^{4}}[/tex]

To solve it we can multiply [tex](3)*(1+i)^{4}[/tex] to obtain [tex](1+i)^{4}*(\frac{169}{(1+i)} +\frac{169}{(1+i)^{2}}=\frac{225}{(1+i)^{2}} +\frac{225}{(1+i)^{4}})[/tex], then we have [tex]225(1+i)^{2}+225=169(1+i)^{3}+169(1+i)^{2}[/tex].

This leads to a third-grade polynomial [tex]169i^{3}+451i^{2}+395i-112=0[/tex], after computing this expression, we find only one real root [tex]i=0.2224[/tex].

Finally, we replace it in (1) or (2), let's do it in (1) [tex]K=\frac{169}{(1+0.2224)} +\frac{169}{(1+0.2224)^{2}}\\\\K=251.35[/tex]

Answer:  -8

Step-by-step explanation:

The slope of a line passing through two points (a,b) and (c,d) is given by:-

[tex]m=\dfrac{d-b}{c-a}[/tex]

Given : A line passes through the points ( 3 , -14 ) and ( x , 6 ) with a slope of -4, then we have

[tex]-4=\dfrac{6-(-14)}{x-3}\\\\\Rightarrow\ (x-3)(-4)=6+14\\\\\Rightarrow\ -4(x-3)=20\\\\\Rightarrow\ (x-3)=\dfrac{20}{-4}=-5\\\\\Rightarrow\ x-3=-5\\\\\Rightarrow\ x=-5-3=-8[/tex]

Hence, the value of x= -8

The expression 4d + 6 + 2d is equivalent to the expression (3d + 3) + (3d + 3). Then the correct option is C.

The equivalent is the functions that are in different forms but are equal to the same value.

The expression is given below.

→ 4d + 6 + 2d

Then the expression can be written as

→ 6d + 6  

→ 2(3d + 3)

→ (3d + 3) + (3d + 3)

Thus, the correct option is C.

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Final answer:

Choices A and C, which evaluate to 2(3d+3) and (3d+3)+(3d+3) respectively, are equivalent to the original expression 4d+6+2d.

Explanation:

The student is asking to identify which expressions are equivalent to 4d+6+2d. To solve this, we combine like terms in the original expression.

4d + 2d + 6 = 6d + 6.

Now let's analyze the provided choices:

(Choice A) 2(3d+3) simplifies to 6d + 6, which is the same as the original expression.

(Choice B) 6(d+6) simplifies to 6d + 36, which is not equivalent to 6d + 6.

(Choice C) (3d+3) + (3d+3) simplifies to 3d + 3 + 3d + 3, which simplifies further to 6d + 6, equivalent to the original expression.

Choices A and C are equivalent to 4d+6+2d.

Answer: If we have a set defined as the difference of two sets, this is

A = B - C, then A = B - B∩C

So this is defined as, if you subtract the set C to the set B, then you are subtracting the common elements between C and B, from the set B.

so if A = {a,b,c} - {a,b,d}, the common elements are a and b, so: A = {c}

1. Y-axis. 2. X-axis. 3. X-axis

Step-by-step explanation:

1. Look at the shapes, is it reflecting vertically (y axis) or horizontally (x axis)? Take the shape, and mentally flip it over over each axis. If it lines up with the reflection provided, the answer is the axis you flipped over.

Answer:

At x = 2 and 10.

Step-by-step explanation:

Given : The cost of producing x hundred items is given by the equation [tex]C(x) = x^2-3x + 7[/tex]

The revenue generated from sales of x hundred units is given by the equation [tex]R(x) = -x^2 + 21x-33[/tex]

To Find :What values of x will the company break even?

Solution:

Cost function : [tex]C(x) = x^2-3x + 7[/tex]

Revenue function : [tex]R(x) = -x^2 + 21x-33[/tex]

Now to find the company break even :

[tex]-x^2 + 21x-33= x^2-3x + 7[/tex]

[tex]24x= 2x^2+40[/tex]

[tex]12x= x^2+20[/tex]

[tex]x^2-12x+20=0[/tex]

[tex]x^2-10x-2x+20=0[/tex]

[tex]x(x-10)-2(x-10)=0[/tex]

[tex](x-2)(x-10)=0[/tex]

So, x = 2,10

Hence the company break even at x = 2 and 10.

Answer:

. . . False

Step-by-step explanation:

5 is not among the elements of the given set.

__

The symbol ∈ means "is an element of ...".

Answer:

8 ten-thousands is 80 thousands.

Step-by-step explanation:

To find : How do I write 8 ten-thousands?

Solution :

We have to write 8 ten thousand,

We know that, according to numeric system

[tex]\text{Thousand is 1000}[/tex]

[tex]\text{Ten-Thousand is 10,000}[/tex]

[tex]\text{8 ten thousand is}\ 8\times 10000[/tex]

[tex]\text{8 ten thousand is 80000}[/tex]

Therefore, 8 ten-thousands is 80 thousands.

Answer:

139,600.96

Step-by-step explanation:

We use the payment of a loan formula:

[tex] \displaystyle PMT = \frac{P \left(\displaystyle \frac{r}{n}\right)}{\left[ 1 - \left( 1 + \displaystyle \frac{r}{n}\right)^{-nt} \right]} [/tex]

P is the principal: $200,000. t is the number of years: 30, n is 12 since it is compounded monthly. And r is 0.039 which is 3.9% in decimal form (3.9/100)

So the formula becomes:  

[tex] \displaystyle PMT = \frac{200000 \left(\displaystyle \frac{0.039}{12}\right)}{\left[ 1 - \left( 1 + \displaystyle \frac{0.039}{12}\right)^{-12(30)} \right]} [/tex]

And using our calculator we get: PMT = $943.336

Then the total amount of money paid in the mortgage is:  

PMT*n*t = $943.336(12)(30) = $339,600.96

Therefore, the interest paid is:  

$339,600.96 - $200,000 = $139,600.96

You have to enter it without $ and rounded to the nearest cent so: 139,600.96

For part (a) and (b) suposse [tex]A\subset B[/tex].

To prove part (a) observe that we already have that [tex]B\subset A\cup B[/tex]. So we will prove that [tex]A\cup B \subset B[/tex]. Let [tex]x\in A\cup B[/tex], then [tex]x\in A[/tex] or [tex]x\in B[/tex]. If  [tex]x\in B[/tex] we finish the proof, and if [tex]x\in A[/tex] implies  [tex]x\in B[/tex] because we assume [tex]A\subset B[/tex], and the proof is complete.

For part (b) we always have [tex]A\cap B\subset A[/tex]. We finish the proof showing [tex]A\subset A\cap B[/tex]. Let [tex]x\in A[/tex], then [tex]x\in B[/tex] by the asumption that [tex]A\subset B[/tex]. So, we have both [tex]x\in A[/tex] and  [tex]x\in B[/tex], that implies [tex]x\in A\cap B[/tex]. Therfore [tex]A\subset A\cap B[/tex], which completes the proof.

Step-by-step explanation:

If X is a finite Hausdorff space then every two points of X can be separated by open neighborhoods. Say the points of X are [tex]x_1, x_2, ..., x_n[/tex]. So there are disjoint open neighborhoods [tex]U_{12}[/tex] and [tex]U_2[/tex], of [tex]x_1[/tex] and [tex]x_2[/tex] respectively (that's the definition of Hausdorff space). There are also open disjoint neighborhoods [tex]U_{13}[/tex] and [tex]U_3[/tex] of [tex]x_1[/tex] and [tex]x_3[/tex] respectively, and disjoint open neighborhoods [tex]U_{14}[/tex] and [tex]U_4[/tex] of [tex]x_1[/tex] and [tex]x_4[/tex], and so on, all the way to disjoint open neighborhoods [tex]U_{1n}[/tex], and [tex]U_n[/tex] of [tex]x_1[/tex] and [tex]x_n[/tex] respectively. So [tex]U=U_2 \cup U_3 \cup ... \cup U_n[/tex] has every element of [tex]X[/tex] in it, except for [tex]x_1[/tex]. Since [tex]U[/tex] is union of open sets, it is open, and so [tex]U^c[/tex], which is the singleton [tex]\{ x_1\}[/tex], is closed. Therefore every singleton is closed.

Now, remember finite union of closed sets is closed, so [tex]\{ x_2\} \cup \{ x_3\} \cup ... \cup \{ x_n\}[/tex] is closed, and so its complemented, which is [tex]\{ x_1\}[/tex] is open. Therefore every singleton is also open.

That means any two points of [tex]X[/tex] belong to different connected components (since we can express X as the union of the open sets [tex] \{ x_1\} \cup \{ x_2,...,x_n\}[/tex], so that [tex]x_1[/tex] is in a different connected component than [tex]x_2,...,x_n [/tex], and same could be done with any [tex]x_i[/tex]), and so each point is in its own connected component. And so the space is totally disconnected.

Answer:

a) MAX--> PC (R,P) = 0,3R+ 0,5P

b) Optimal solution: 40.000 units of R and 10.000 of PC = $17.000

c) Slack variables: S3=1000, is the unattended demand of P, the others are 0, that means the restrictions are at the limit.

d) Binding Constaints:

1. 0.3 R+0.6 P ≤ 18.000

2. R+P ≤ 50.000

3. P ≤ 20.000

4. R ≥ 0

5. P ≥ 0

Step-by-step explanation:

I will solve it using the graphic method:

First, we have to define the variables:

R : Regular Gasoline

P: Premium Gasoline

We also call:

PC: Profit contributions

A: Grade A crude oil

• R--> PC: $0,3 --> 0,3 A

• P--> PC: $0,5 --> 0,6 A

So the ecuation to maximize is:

MAX--> PC (R,P) = 0,3R+ 0,5P

The restrictions would be:

1. 18.000 A availabe (R=0,3 A ; P 0,6 A)

2. 50.000 capacity

3. Demand of P: No more than 20.000

4. Both P and R 0 or more.

Translated to formulas:

Answer d)

1. 0.3 R+0.6 P ≤ 18.000

2. R+P ≤ 50.000

3. P ≤ 20.000

4. R ≥ 0

5. P ≥ 0

To know the optimal solution it is better to graph all the restrictions, once you have the graphic, the theory says that the solution is on one of the vertices.

So we define the vertices: (you can see on the graphic, or calculate them with the intersection of the ecuations)

V:(R;P)

• V1: (0;0)

• V2: (0; 20.000)

• V3: (20.000;20.000)

• V4: (40.000; 10.000)

• V5:(50.000;0)

We check each one in the profit ecuation:

MAX--> PC (R,P) = 0,3R+ 0,5P

• V1: 0

• V2: 10.000

• V3: 16.000

• V4: 17.000

• V5: 15.000

As we can see, the optimal solution is  

V4: 40.000 units of regular and 10.000 of premium.

To have the slack variables you have to check in each restriction how much you have to add (or substract) to get to de exact (=) result.  

To maximize the total profit contribution, a linear programming model is formulated with decision variables, objective function, and constraints. The model is: Maximize 0.30x + 0.50y, Subject to: 0.3x + 0.6y ≤ 18,000, x + y ≤ 50,000, y ≤ 20,000, x, y ≥ 0.

To formulate a linear programming model, we need to define the decision variables, objective function, and constraints. Let's assume that x represents the number of gallons of regular gasoline produced and y represents the number of gallons of premium gasoline produced.

The objective function is to maximize the total profit contribution, which can be expressed as: 0.30x + 0.50y.

The constraints are:

So, the linear programming model can be formulated as:

Maximize 0.30x + 0.50y

Subject to:

0.3x + 0.6y ≤ 18,000

x + y ≤ 50,000

y ≤ 20,000

x, y ≥ 0

Answer:

the dimension called by the specification is larger

Step-by-step explanation:

Data provided in the question:

micrometer reading = 7.5 in

Dimension called by the specifications = 7.59 in

Now,

Subtracting the Micrometer reading from the dimension called by specification , we get

 7.59

- 7.5

--------

0.09

since, the result is positive thus,

the dimension called by the specification is larger

The specification of 7.59 inches is larger than the micrometer reading of 7.57 inches. Therefore, the part does not meet the required specification.

To determine whether the micrometer reading or the specification is larger, we need to compare the two values. The micrometer reading is 7.57 inches, and the specification calls for a dimension of 7.59 inches.

When comparing these two measurements:

Clearly, 7.59 inches is larger than 7.57 inches.

This comparison shows that the specification is larger than the micrometer reading.

Answer:

You should make 200 quarts of Creamy Vanilla and 50 quarts of Continental Mocha to earn the largest profit.

Step-by-step explanation:

You are going to earn the largest profit when you manage to use all the eggs and cups of cream that you have in stock.

This problem can be solved by a first order equation

I am going to call x the number of quarts of Creamy Vanilla and y the number of quarts of Continental Mocha.

The problem states that each quart of Creamy Vanilla uses 2 eggs and each quart of Continental Mocha uses 1 egg. There are 450 eggs in stock, so:

2x + y = 450.

The problem also states that each quart of Creamy Vanilla uses 3 cups of cream and that each quart of Continental Mocha uses 3 cups of cream. There are 750 cups of cream in stock, so:

3x + 3y = 750

Now we have to solve the following system of equations

1) 2x + y = 450

2) 3x + 3y = 750

The first thing i am going to do is simplify the equation 2) by 3

2) 3(x+y)/3 = 750/3

2) x+y = 250

So now, we have the following system

1) 2x + y = 450

2) x + y = 250

I am going to write y as a function of x in 2), and replace in 1)

y = 250 - x

Replacing in 1)

2x + 250 - x = 450

2x - x = 450 - 250

x = 200

You should make 200 quarts of Creamy Vanilla

From 2), we have

y = 250 - x = 250 - 200 = 50

You should make 50 quarts of Continental Mocha

Answer:

To multiply two numbers in scientific notation, multiply their coefficients and add their exponents. To divide two numbers in scientific notation, divide their coefficients and subtract their exponents. In either case, the answer must be converted to scientific notation.

Step-by-step explanation:

Answer:

See picture below

Step-by-step explanation:

The total area of the sheet of paper is 80, so we can divide the sheet of paper into 4 different sections, each one of them with area 20 (you can verify this by counting the squares in each section, there are 20 squares per section).

In the picture below we can see that the perimeter of each of the sections are from left to right:

The first one is a rectangle, with sides 5 and 4. Therefore the perimeter is 2(5) + 2(4) = 18.

The next section is irregular so we sum up the sides: 2 + 6 + 5 + 1 + 2 + 1 + 5 + 4 = 26.

For the next section we also sum up the sides: 3 + 7 + 2 + 1 + 1 + 6 = 20.

For the bottom section, we will sum up the sides too: 2 + 1 + 6 + 1 + 2 + 1 + 10 + 3 = 26.

So the perimeters are 8, 26, 20 and 26. This satisfies the condition of no more than two sections having the same perimeter.

Answer:

10 growth per 1000.

Step-by-step explanation:

Given,

Rate of birth = 30 births per 1000

Rate of death = 20 deaths per 1000

As the growth in population is the difference in the number of the child take birth and the person die.

As we are calculating the rate of birth and rate of growth in per thousands of members, so the growth rate will be also in per thousands.

As we can see on every one thousand people,

total birth = 30

total death = 20

so, total growth = total birth - total growth

                           = 30 - 20

                           = 10

As at every 1000 persons, there are 10 persons survive, so the rate of growth will be 10 growth per 1000.

Answer:

Part 1) The distance is [tex]d=7.3\ units[/tex]

Part 2) The measure of angle 2 is 121°

Part 3) The coordinates of endpoint V are (7,-27)

Part 4) The value of x is 10

Step-by-step explanation:

Part 1) Find the distance between M(6,16) and Z(-1,14)

we know that

the formula to calculate the distance between two points is equal to

[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]

substitute the given values in the formula

[tex]d=\sqrt{(14-16)^{2}+(-1-6)^{2}}[/tex]

[tex]d=\sqrt{(-2)^{2}+(-7)^{2}}[/tex]

[tex]d=\sqrt{53}\ units[/tex]

[tex]d=7.3\ units[/tex]

Part 2) Find the measure m∠2

we know that

If two angles are supplementary, then their sum is equal to 180 degrees

In this problem we have

m∠1+m∠2=180°

substitute the given values

[tex](4y+7)\°+(9y+4)\°= 180\°[/tex]

Solve for y

[tex](13y+11)\°= 180\°[/tex]

[tex]13y= 180-11[/tex]

[tex]13y=169[/tex]

[tex]y=13[/tex]

Find the measure of  m∠2

[tex](9y+4)\°[/tex]

substitute the value of y

[tex](9(13)+4)=121\°[/tex]

Part 3) The midpoint of UV is (5,-11). The coordinates of one endpoint are U(3,5) Find the coordinates of endpoint V

we know that

The formula to calculate the midpoint between two points is

[tex]M=(\frac{x1+x2}{2},\frac{y1+y2}{2})[/tex]

we have

[tex]M(5,-11)[/tex]

[tex](x1,y1)=(3,5)[/tex]

substitute and solve for (x2,y2)

[tex](5,-11)=(\frac{3+x2}{2},\frac{5+y2}{2})[/tex]

so

Equation 1

[tex]5=(3+x2)/2[/tex]

[tex]10=3+x2[/tex]

[tex]x2=7[/tex]

Equation 2

[tex]-11=(5+y2)/2[/tex]

[tex]-22=(5+y2)[/tex]

[tex]y2=-27[/tex]

therefore

The coordinates of endpoint V are (7,-27)

Part 4) GI bisects ∠DGH so that ∠DGI is (x-3) and ∠IGH is (2x-13) Find the value of x

we know that

If GI bisects ∠DGH

then

∠DGI=∠IGH

Remember that bisects means, divide into two equal parts

substitute the given values

[tex]x-3=2x-13[/tex]

solve for x

[tex]2x-x=-3+13[/tex]

[tex]x=10[/tex]