The "absorption law" (theorem 2.1.1 in our book) states that p V (p Aq) is logically equivalent to p. Construct a truth table to show these statements are equivalent.

Answer:

Is needed 16 fluid oz of water to fill in 2 cups of water

Step-by-step explanation:

1 cup have 8 fluid oz

2 cups = 2 * 8 fluid oz = 16 fluid oz

Final answer:

The recipe requires 2 cups of water, which is equivalent to 16 fluid ounces since there are 8 fluid ounces in one cup.

Explanation:

To calculate how many fluid ounces of water is needed if a recipe requires 2 cups of water, you need to understand the unit equivalence between cups and fluid ounces. One cup is equivalent to 8 fluid ounces. Since the recipe requires 2 cups, you will multiply the number of cups by the unit equivalence to find the total number of fluid ounces.

Here is the calculation:

2 cups × 8 fluid ounces/cup = 16 fluid ounces

Therefore, the recipe requires 16 fluid ounces of water.

Answer:

you would multiply the 11 so 88-44y tjen divide which is 22

Step-by-step explanation:

Answer:

19-4y

Step-by-step explanation:

-(4y-8)+11

-4y+8+11

-4y+19

19-4y

Answer:

20 mg/ml.

Step-by-step explanation:

We have been given that a liquid oral concentrate of morphine sulfate contains 2.4 g of morphine sulfate in a 120-mL bottle.

[tex]\text{Concentration of morphine sulfate on g/mL basis}=\frac{\text{2.4 g}}{\text{120 ml}}[/tex]

To convert the concentration of morphine sulfate on a mg/mL basis, we need to convert 2.4 grams to milligrams.

1 gram equals 1000 milligrams.

[tex]\text{Concentration of morphine sulfate on mg/mL basis}=\frac{\text{2.4 g}}{\text{120 ml}}\times\frac{\text{1,000 mg}}{\text{1 g}}[/tex]

[tex]\text{Concentration of morphine sulfate on mg/mL basis}=\frac{2.4\times\text{1,000 mg}}{\text{120 ml}}[/tex]

[tex]\text{Concentration of morphine sulfate on mg/mL basis}=\frac{2400\text{ mg}}{\text{120 ml}}[/tex]

[tex]\text{Concentration of morphine sulfate on mg/mL basis}=\frac{20\text{ mg}}{\text{ml}}[/tex]

Therefore, the concentration of morphine sulfate would be 20 mg per ml.

Answer:

Step-by-step explanation:

Answer:

Vertex = (0,-1)

The x value of its largest x-intercept is [tex]x=\frac{1}{2}[/tex].

The y value of the y-intercept is y = -1.

Step-by-step explanation:

The given function is

[tex]f(x)=4x^2-1[/tex]           .... (1)

The vertex form of a parabola is

[tex]g(x)=a(x-h)^2+k[/tex]         ..... (2)

where, a is a constant, (h,k) is vertex of the parabola.

From (1) and (2) we get

[tex]a=4,h=0,k=-1[/tex]

So, the vertex of the parabola is (0,-1).

Substitute f(x)=0 in equation (1) to find x-intercepts.

[tex]0=4x^2-1[/tex]

Add 1 on both sides.

[tex]1=4x^2[/tex]

Divide both sides by 4.

[tex]\frac{1}{4}=x^2[/tex]

Taking square root both sides.

[tex]\pm \sqrt{\frac{1}{4}}=x[/tex]

[tex]\pm \frac{1}{2}=x[/tex]

The x-intercepts are [tex]-\frac{1}{2}[/tex] and [tex]\frac{1}{2}[/tex].

Therefore the x value of its largest x-intercept is [tex]x=\frac{1}{2}[/tex].

Substitute x=0 in equation (1) to find the y-intercept.

[tex]f(0)=4(0)^2-1=-1[/tex]

Therefore the y value of the y-intercept is y = -1.

The vertex of the quadratic function f(x) = 4x^2 - 1 is at (0, -1). The x-intercepts are x = 0.5 and x = -0.5. The y-intercept is at y = -1.

The quadratic function you provided is f(x) = 4x^2 - 1. This is indeed a second-order polynomial or more commonly referred to as a quadratic function.

The vertex of this quadratic function, which represents its maximum or minimum point, can be found using the formula -b/2a. Since this quadratic has no 'x' term, the vertex's x-value is 0. Plugging this into the function gives a y-value of -1. So, the vertex is at (0, -1).

The x-intercepts of the quadratic, i.e., the points where the function intersects the x-axis, can be found by setting f(x) = 0 and solving for x. Doing so gives two solutions: x = 0.5 and x = -0.5, with 0.5 being the larger of the two.

Finally, the y-intercept (the point where the function intersects the y-axis) is found by setting x = 0. Doing so gives a y-value of -1. So, the y-intercept is at y = -1.

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Answer:

Use the formula [tex]Area(S)=\iint_{S} 1 dS= \iint_{D} \lVert r_{u}\times r_{v} \rVert dudv[/tex]

Step-by-step explanation:

Let [tex]r(x,y)=(x,y,5-3x^2-2y^2)[/tex] be the explicit parametrization of the paraboid. The intersection of this paraboid with the xy plane is the ellipse given by

[tex]\dfrac{x^2}{\frac{5}{3}}+\dfrac{y^{2}}{\frac{5}{2}}=1[/tex]

The partial derivatives of the parametrization are:

[tex]\begin{array}{c}r_{x}=(1,0,-6x)\\r_{y}=(0,1,-4y)\end{array}[/tex]

and computing the cross product we have

[tex]r_{x}\times r_{y}=(6x,4y,1)[/tex]. Then

[tex]\lVert r_{x}\times r_{y}\rVert =\sqrt{1+36x^{2}+16y^{2}}[/tex]

Then,  if [tex]R[/tex] is the interior region of the ellipse the superficial area located above of the xy  is given by the double integral

[tex]\iint_{R}\sqrt{1+36x^2+16y^2}dxdy=\int_{-\sqrt{5/3}}^{\sqrt{5/3}}\int_{-\sqrt{5/2}\sqrt{1-\frac{x^2}{5/3}}}^{\sqrt{5/2}\sqrt{1-\frac{x^2}{5/3}}}\sqrt{1+36x^2+16y^2}dy dx=30.985[/tex]

The last integral is not easy to calculate because it is an elliptic integral, but with any software of mathematics you can obtain this value.

Answer:  [tex]224.9\ m^2[/tex]

Step-by-step explanation:

An isosceles right triangle is a right triangle having two legs (other than hypotenuse ) of same length  .

Given : The length of the hypotenuse of an isosceles right triangle  is 30 meters.

Let x be the side length of the other two legs, then by using the Pythagoras theorem for right triangle , we have

[tex](30)^2=x^2+x^2\\\\\Rightarrow\ 900=2x^2\\\\\Rightarrow\ x^2=\dfrac{900}{2}\\\\\Rightarrow\ x^2=450\\\\\Rightarrow\ x=\sqrt{450}=\sqrt{9\times25\times2}=\sqrt{3^2\times5^2\times2}\\\Rightarrow\ x=3\times5\sqrt{2}=15(1.414)=21.21[/tex]

Thus, the other two legs have side length of 21.21 m each.

Now, the area of a right triangle is given by :-

[tex]A=\dfrac{1}{2}\times base\times height\\\\\Rightarrow\ A=\dfrac{1}{2}(21.21)\times(21.21)=224.93205\approx224.9\ m^2[/tex]

Hence, the area of the given isosceles right triangle= [tex]224.9\ m^2[/tex]

Answer:

[tex](-\frac{85}{28},2)[/tex]

Step-by-step explanation:

We are asked to find the focus of the parabola that has a vertex [tex](-3,2)[/tex] opens horizontally and passes through the point [tex](-10,1)[/tex].

We know that equation of a horizontal parabola is [tex](y-k)^2=4p(x-h)[/tex], where p is not equal to zero.

[tex](h,k)[/tex] = Vertex of parabola.

Focus: [tex](h+p,k)[/tex]

Upon substituting the coordinates of vertex and given point in equation of parabola, we will get:

[tex](1-2)^2=4p(-10-(-3))[/tex]

[tex](-1)^2=4p(-10+3)[/tex]

[tex]1=4p(-7)[/tex]

[tex]1=-28p[/tex]

[tex]\frac{1}{-28}=\frac{-28p}{-28}[/tex]

[tex]-\frac{1}{28}=p[/tex]

Focus: [tex](h+p,k)[/tex]

[tex](-3+(-\frac{1}{28}),2)[/tex]

[tex](-3-\frac{1}{28},2)[/tex]

[tex](\frac{-3*28}{28}-\frac{1}{28},2)[/tex]

[tex](\frac{-84}{28}-\frac{1}{28},2)[/tex]

[tex](\frac{-84-1}{28},2)[/tex]

[tex](\frac{-85}{28},2)[/tex]

[tex](-\frac{85}{28},2)[/tex]

Therefore, the focus of the parabola is at point [tex](-\frac{85}{28},2)[/tex].

Answer:

>>syms t

>>f=sin(t)*cos(t);

>>int(f)

Step-by-step explanation:

It's actually pretty easy, just use symbolic variables.

First, create the symbolic variable t using this command:

syms t

Now define the function

f=sin(t)*cos(t)

Finally use the next command in order to calculate the indefinite integral:

int(f)

I attached a picture in which you can see the procedure and the result.

To evaluate the integral of sintcost tdt in MATLAB, use the symbolic math toolbox and the 'int' function.

To evaluate the integral sintcost tdt in MATLAB, you can use the symbolic math toolbox. First, define the extended variable t using 'syms t'. Then, determine the integrand using the symbolic expression 'f = sin(t) * cos(t)'. Finally, use the 'int' function to find the integral, 'int(f, t)'.

Here's the MATLAB code:

The resulting 'integral_value' will be the evaluated integral.

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Answer:

[tex]P(A'\cup B)=\frac{84}{100}[/tex]

Step-by-step explanation:

Let A and B represents the following events.

A denote the event that a disk has high shock resistance.

B denote the event that a disk has high scratch resistance.

Given probabilities:

[tex]P(A)=\frac{70+16}{100}=\frac{86}{100}[/tex]

[tex]P(B)=\frac{70+9}{100}=\frac{79}{100}[/tex]

[tex]P(A')=\frac{9+5}{100}=\frac{7}{50}[/tex]

[tex]P(A\cup B)=\frac{70+16+9}{100}=\frac{95}{100}[/tex]

The probability of intersection of A and B is,

[tex]P(A\cap B)=P(A)+P(B)-P(A\cup B)[/tex]

Substitute the above values.

[tex]P(A\cap B)=\frac{86}{100}+\frac{79}{100}-\frac{95}{100}=\frac{70}{100}[/tex]

The probability of union of A' and B is,

[tex]P(A'\cup B)=P(A')+P(A\cap B)[/tex]

Substitute the above values.

[tex]P(A'\cup B)=\frac{7}{50}+\frac{70}{100}[/tex]

[tex]P(A'\cup B)=\frac{14}{100}+\frac{70}{100}[/tex]

[tex]P(A'\cup B)=\frac{84}{100}[/tex]

Therefore, [tex]P(A'\cup B)=\frac{84}{100}[/tex].

Using Venn probabilities, it is found that the desired probability is given by:

[tex]P(A' \cup B) = \frac{21}{25}[/tex]

The or probability of Venn sets is given by:

[tex]P(A \cup B) = P(A) + P(B) - P(A \cap B)[/tex]

In this problem, we want:

[tex]P(A' \cup B) = P(A') + P(B) - P(A' \cap B)[/tex]

We have that:

[tex]P(A') = \frac{7}{50}[/tex]

[tex]P(B) = \frac{79}{100}[/tex]

[tex]P(A' \cap B)[/tex] is probability of A not happening and B happening, thus low shock resistance and high scratch resistance, thus [tex]P(A' \cap B) = \frac{9}{100}[/tex]

Then

[tex]P(A' \cup B) = P(A') + P(B) - P(A' \cap B)[/tex]

[tex]P(A' \cup B) = \frac{7}{50} + \frac{79}{100} - \frac{9}{100}[/tex]

[tex]P(A' \cup B) = \frac{14}{100} + \frac{70}{100}[/tex]

[tex]P(A' \cup B) = \frac{84}{100}[/tex]

[tex]P(A' \cup B) = \frac{21}{25}[/tex]

A similar problem is given at https://brainly.com/question/23508811

Answer:

$37,800.

Step-by-step explanation:

We have been given that the cost, in dollars, of making x items is given by the function [tex]C(x)=25x+300[/tex]. We are asked to find [tex]C(1500)[/tex].

To find [tex]C(1500)[/tex], we will substitute [tex]x=1500[/tex] in our given function as:

[tex]C(1500)=25(1500)+300[/tex]

[tex]C(1500)=37,500+300[/tex]

[tex]C(1500)=37,800[/tex]

Therefore, the cost of making 1500 items would be $37,800 and option D is the correct choice.

Answer:

1) The tournament can be played in 10 different ways

2) The probability of 5 games being played is 0.40

3) The probability of a team winning two games in a row is 0.80  

Step-by-step explanation:

a) From the tree diagram below we can observe that the tournament can be played in 10 different ways.

b)The probability of 5 games being played is

P= (number of possibilities where 5 games are being played) / (Total games)

P = 4 / 10

P= 0.40

c) The probability of a team winning two games in a row is

P = (number of possibilities where a team wins two games in a row) / Total games

P = 8 / 10

P = 0.80

Answer:

76075, 76076, 76077

Step-by-step explanation:

There are 3 teams; Team A, Team B and Team C

Team A has most wins

Team C has least wins

Team B is in between

All these will be consecutive numbers.

Team B: x

Team A: x + 1 (most wins)

Team C: x - 1 (least wins)

Team A + Team B + Team C = Total number of wins

x + x + 1 + x - 1 = 228228

3x = 228228

x = 76076

Wins of Team B : x = 76076

Wins of Team A : x + 1 = 76076 + 1 = 76077

Wins of Team C : x - 1 = 76076 - 1 = 76075

Therefore, in the 2010 season, Team A had 76077 wins, Team B had 76076 wins and Team C had 76075 wins.

!!

Answer: 0.1498 square units.

Step-by-step explanation:

Let x be any random variable that follows normal distribution.

Given : For a normal distribution with mean equal to - 30 and standard deviation equal to 9.

i.e.  [tex]\mu=-30[/tex] and [tex]\sigma=9[/tex]

Use formula [tex]z=\dfrac{x-\mu}{\sigma}[/tex] to find the z-value corresponds to -34.5 will be

[tex]z=\dfrac{-34.5-(-30)}{9}=\dfrac{-34.5+30}{9}=\dfrac{-4.5}{9}=-0.5[/tex]

Similarly,  the z-value corresponds to -38 will be

[tex]z=\dfrac{-39-(-30)}{9}=\dfrac{-39+30}{9}=\dfrac{-9}{9}=-1[/tex]

By using the standard normal table for z-values , we have

The  area under the curve that is between - 34.5 and - 39. will be :-

[tex]P(-1<z<-0.5)=P(z<-0.5)-P(z<-1)\\\\=(1-P(z<0.5))-(1-P(z<1))\\\\=1-P(z<0.5)-1+P(z<1)\\\\=P(z<1)-P(z<0.5)\\\\=0.8413-0.6915=0.1498[/tex]

Hence, the area under the curve that is between - 34.5 and - 39 = 0.1498 square units.

Answer:

Function:

c = f(w) = 0.49, 0 < w ≤ 1

            = 0.70,  1 < w ≤ 2

            = 0.91, 2 < w ≤ 3

Step-by-step explanation:

Yes, the relation described can be interpreted as a function.

Here, c is the cost of a mail letter. c depends upon w, which is the weights of the mail letter.

As described in the question, the relation can be expressed as a function.

c can be expressed as a function of w in the following manner:

c(cost of mail) = f(w), where w is the independent variable and c is the dependent variable

c = f(w) = 0.49, 0 < w ≤ 1

            = 0.70,  1 < w ≤ 2

            = 0.91, 2 < w ≤ 3

where, c is in dollars and w is in ounces.

Answer:

The deep body temperature of a healthy person is 310.15 K

Step-by-step explanation:

The formula we are going to use is [tex]T_{K}=T_{C}+273.15[/tex]

We know that the deep body temperature is 37°C, putting this value into the formula we have that [tex]T_{K}=37+273.15 = 310.15 K[/tex]

The correct answer is A, because the angles are proven same by the line of symmetry marked.

Answer:

A

explanation:

because the angles are proven same

Answer:

[tex]x=36.772[/tex]  

Step-by-step explanation:

Given : Expression [tex]x=25.36+0.45(25.36)[/tex]

To find : Solve the expression ?

Solution :

Step 1 - Write the expression,

[tex]x=1(25.36)+0.45(25.36)[/tex]

Step 2 - Apply distributive, [tex]ab+ac=a(b+c)[/tex]

Here, a=25.36, b=1, c=0.45

[tex]x=25.36(1+0.45)[/tex]

Step 3 - Solve the expression,

[tex]x=25.36\times 1.45[/tex]

[tex]x=36.772[/tex]

Therefore, [tex]x=36.772[/tex]

Answer:

A. Representative

Step-by-step explanation:

The sample is a subset of the population. We take samples because it is easy to analyze samples as compared to population and it is less time taken. Further, Sample is said to be good if the sample is the representative of the population.

Simple Random Sampling is the sampling where samples are chosen randomly, where each unit has an equal chance of being selected in a sample.

Since, observer is taken the weight of men over age 18. Thus, it is good sample which represent whole population.

Answer:

If he uses 1 hour to practice harmonica, one our on homework, and another hour doing chores. He will be spending twice as much time practicing harmonica and doing homework than he would spend on chores.

Answer:

2

Step-by-step explanation:

UP

Answer:

The cost for 17 ft length with diameter 5 in is $408.

Step-by-step explanation:

Consider the provided information.

It is given that the cost of stainless steel tubing varies jointly as the length and the diameter of the tubing. If a 5 foot length with a diameter 2 inches costs $48.00.

let [tex]C\propto L\cdot D[/tex]

[tex]C=k\cdot L\cdot D[/tex]

[tex]48=k\cdot 5\cdot 2[/tex]

[tex]\frac{48}{10}=k[/tex]

[tex]k=4.8[/tex]

For 17 ft length with diameter 5in the cost will be:

[tex]C=k\cdot L\cdot D[/tex]

Substitute the respective values in the above formula.

[tex]C=4.8\cdot 17\cdot 5[/tex]

[tex]C=408[/tex]

Cost = $408

Hence, the cost for 17 ft length with diameter 5in is $408.

Answer:  

(22.12, 27.48)

Step-by-step explanation:

Given : Significance level : [tex]\alpha: 1-0.95=0.05[/tex]

Sample size : n= 8 , which is a small sample (n<30), so we use t-test.

Critical values using t-distribution: [tex]t_{n-1,\alpha/2}=t_{7,0.025}=2.365[/tex]

Sample mean : [tex]\overline{x}=24.8\text{ hours}[/tex]

Standard deviation : [tex]\sigma=3.2\text{ hours}[/tex]

The confidence interval for population means is given by :-

[tex]\overline{x}\pm t_{n-1,\alpha/2}\dfrac{\sigma}{\sqrt{n}}[/tex]

i.e. [tex]24.8\pm(2.365)\dfrac{3.2}{\sqrt{8}}[/tex]

[tex]24.8\pm2.67569206001\\\\\approx24.8\pm2.68\\\\=(24.8-2.68, 24.8+2.68)=(22.12, 27.48)[/tex]

Hence, the 95% confidence interval, assuming the times are normally distributed.=  (22.12, 27.48)

Answer:

a) The percentage of adults who smoke are decreasing with time. b) the equation that best described this data is y=-0.3364x+22.809 (R^2=0.859) in which y is the percentage of adults who smoke and x the number of years. c) the percentage of adults who smoke will be 19.8% and it will not meet the expected 12%, it would take 32 years to reach that value.

Step-by-step explanation:

The data can be plotted to which years is the independent variable and percentage of adults who smoke is the dependent variable. The linear trendline that described this data has a negative slope which indicates that the percentage of adults is decreasing with time. In order to determine if the OSH target is being met, the x is replaced by 9 which is the goal period of nine years. The y is 19% which is higher than the 12% goal. In order to know the period it will take to the reach the goal of 12%, the y is replaced by 12 in the curve and the x is the answer in years = 32 years.  

Answer:

Your account is going to be worst $405,142.92

Step-by-step explanation:

This is a compound interest problem.

The compound interest formula is given by:

[tex]A = P(1 + \frac{r}{n})^{nt}[/tex]

Where A is the amount of money, P is the principal(the initial sum of money), r is the interest rate(as a decimal value), n is the number of times that interest is compounded per unit t and t is the time the money is invested or borrowed for.

So, for this problem:

We want to find A

[tex]P = 55,000[/tex]

[tex]n = 1[/tex]

[tex]r = 0.105[/tex]

[tex]t = 20[/tex]

[tex]A = P(1 + \frac{r}{n})^{nt}[/tex]

[tex]A = 55000(1 + \frac{0.105}{1})^{20}[/tex]

[tex]A = 405,142.92[/tex]

Your account is going to be worst $405,142.92

Answer:

Step-by-step explanation:

If we assume that [tex][(x \vee y) \wedge (x \rightarrow z) \wedge (\neg z)][/tex] is true, then:

[tex](x \vee y)[/tex] is true

[tex](x \rightarrow z)[/tex] is true

[tex](\neg z)[/tex] is true

If [tex](\neg z)[/tex] is true, then [tex]z[/tex] is false.

[tex](x \rightarrow z) \equiv (\neg x \vee z)[/tex], since [tex](x \rightarrow z)[/tex] is true, then [tex](\neg x \vee z)[/tex] is true

If [tex]z[/tex] is false and [tex](\neg x \vee z)[/tex] is true, then [tex]\neg x[/tex] is true.

If [tex]\neg x[/tex] is true, then [tex]x[/tex] is false, as [tex](x \vee y)[/tex] is true and [tex]x[/tex] is false, then [tex]y[/tex] is true.

Conclusion [tex]y[/tex] it's true.

Answer:

"[tex]e^x[/tex] is irrational for every nonzero integer x"

Step-by-step explanation:

The original statement is

"[tex]e^x[/tex] is rational for some nonzero integer x."

The negation is technically:

"It is NOT true that [tex]e^x[/tex] is rational for some nonzero integer x."

So it's expressing that it's false that [tex]e^x[/tex] can be rational for some nonzero integer x.

This just means that [tex]e^x[/tex] is always irrational when x is a nonzero integer.

Which can be worded as

"[tex]e^x[/tex] is irrational for every nonzero integer x"

The negation of the statement "e^x is rational for some nonzero integer x" is "For all nonzero integers x, e^x is not rational."

To write the negation of the statement "ex is rational for some nonzero integer x", we can express it as "For all nonzero integers x, ex is not rational." This implies that there does not exist any nonzero integer x such that ex is rational. The original statement is an existential statement, asserting the existence of an x that makes the statement true. Its negation is a universal statement, asserting that for every x the statement is false.

The approach to negating the statement involves changing the existential quantifier ("There is some" or "For some") to a universal quantifier ("For all") and negating the predicate of the original statement. This is similar to how we represent negations in symbolic logic, aligning with the principle of contradiction. To negate existential statements, the corresponding universal statement is negated and vice versa.

Answer:

To see the steps to the diagonal form see the step-by-step explanation. The solution to the system is [tex]x =  -\frac{1}{9}[/tex], [tex]y= -\frac{1}{9}[/tex], [tex]z= \frac{4}{9}[/tex] and [tex]w = \frac{7}{9}[/tex]

Step-by-step explanation:

Gauss elimination method consists in reducing the matrix to a upper triangular one by using three different types of row operations (this is why the method is also called row reduction method). The three elementary row operations are:

To solve the system using the Gauss elimination method we need to write the augmented matrix of the system. For the given system, this matrix is:

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

For this matrix we need to perform the following row operations:

After this step, the system has an upper triangular form

The triangular matrix looks like:

[tex]\left[\begin{array}{cccc|c}1 & 0 & 0 & -0.5 & -0.5  \\0 & 1 & 0 & -0.5 & -0.5\\0 & 0 & 1 & 2 &  2 \\0 & 0 & 0 & 1 &  \frac{7}{9}\end{array}\right][/tex]

If you later perform the following operations you can find the solution to the system.

After this operations, the matrix should look like:

[tex]\left[\begin{array}{cccc|c}1 & 0 & 0 & 0 & -\frac{1}{9}  \\0 & 1 & 0 & 0 &   -\frac{1}{9}\\0 & 0 & 1 & 0 &  \frac{4}{9} \\0 & 0 & 0 & 1 &  \frac{7}{9}\end{array}\right][/tex]

Thus, the solution is:

[tex]x =  -\frac{1}{9}[/tex], [tex]y= -\frac{1}{9}[/tex], [tex]z= \frac{4}{9}[/tex] and [tex]w = \frac{7}{9}[/tex]

Answer:

238.25 mg

Step-by-step explanation:

Given:

Molar mass of MgCl₂ = 95.3

atomic weight of Mg₂ = 243

Atomic weight of Cl = 35.5

Volume of solution required = 5.0 mEq of magnesium

Now,

mEq = [tex]\frac{\textup{Weight in mg\timesValency}}{\textup{Atomic mass}}[/tex]

on substituting the values, we get

5 = [tex]\frac{\textup{Weight in mg\times2}}{\textup{95.3}}[/tex]

or

weight of magnesium chloride = 238.25 mg

Therefore,

the required mass of MgCl₂ is 238.25 mg

Step-by-step explanation:

Let's assume that

P(n)=1.1! +2.2! +3.3! + ... +n.n! = (n + 1)! - 1.

For n = 1

L.H.S = 1.1!

         = 1

R.H.S = (n + 1)! - 1.

          =(1 + 1)! - 1.

          = 1

L.H.S = R.H.S

Hence the P(n) is true for n=1

Fort n = 2

L.H.S=1.1! +2.2!

        =1+4

        =5

R.H.S = (2 + 1)! - 1.

          =(2 + 1)! - 1.

          = 5

L.H.S = R.H.S

Hence the P(n) is true for n=2

Let's assume that P(n) is true for all n.

Then we have to prove that P(n) is true for (n+1) too.

So,

L.H.S = 1.1! +2.2! +3.3! + ... +n.n!+(n+1).(n+1)!

         = (n + 1)! - 1 +(n+1).(n+1)!

         = (n+1)![1+(n+1)]-1

         =(n+1)!(n+2)-1

         =(n+2)!-1

         =[(n+1)+1]!-1

So, P(n) is also true for (n+1).

So, P(n) is true for all integers n.