Find the equation of the line going through the points (2,-1) and (5,2) 3x 2y

Answer:

The provided statement [tex]\exists x(x^2-4=1)[/tex] is true in the domain of real number.

Step-by-step explanation:

Consider the provided information.

Any equation or inequality with variables in it is a predicate in the domain of real numbers.

The provided statement is:

[tex]\exists x(x^2-4=1)[/tex]

Here, we need to find the value of x for which the above statement is true.

Since the value of x can be any real number so we can select the value of x as √5

(√5)²-4=1

5-4=1

Which is true.

Thus, the provided statement [tex]\exists x(x^2-4=1)[/tex] is true.

Answer:

The speed of the slower car is: 51 miles/hour and the speed of the faster car is 54 miles/hour.

Step-by-step explanation:

First, we can easily find the speed of the slower car, since we know that the speed is given by the formula v=d/t, where d stands for the traveled distance and t for the elapsed time.

[tex]v_{s}=\frac{d_{s}}{t}=\frac{153\ miles}{3\ hours} = 51\ miles/hour[/tex]

Next, for the faster car, we know that the distance traveled is 153+9 miles in the same time, therefore, its speed is given by:

[tex]v_{f}=\frac{d_{f}}{t}=\frac{153+9\ miles}{3\ hours} = 54\ miles/hour[/tex]

Final answer:

The statement is FALSE. Human population growth has experienced different rates and is not described accurately by an exponential growth model for all periods, with various factors affecting growth rates. Population is projected to stabilize in the future, not grow indefinitely.

Explanation:

The statement is FALSE. Human population growth has not been steady since prehistoric times. While it is true that the human population growth since 1000 AD has often exhibited exponential patterns, it has also undergone various phases and rates of growth due to many factors such as economics, wars, disease, technological advancement, and social changes. For instance, the population in Asia, which has many economically underdeveloped countries, is increasing exponentially. However, in Europe, where most countries are economically developed, population is growing much more slowly. Additionally, historic events like the Black Death and the World Wars caused noticeable dips in population growth. Hence, the exponential growth model does not accurately describe all periods in human history. Modern projections suggest that the world's population will stabilize between 10 and 12 billion, indicating that population growth will not continue exponentially indefinitely.

Some earlier descriptions of population growth suggested a more stable growth during early human history, with high birth and death rates. Over time and especially in the recent centuries, more data has allowed us to understand that the population growth has experienced different growth rates. The issue of ongoing exponential growth and its sustainability is a topic of significant debate, considering the finite resources of the planet and the impacts of overpopulation.

Answer:

[tex]\frac{1}{6}[/tex]

Step-by-step explanation:

y = x     .....(1)

[tex]y=\sqrt{x}[/tex]     .... (2)

By solving equation (1) and equation (2)

[tex]x = \sqrt{x}[/tex]

[tex]\sqrt{x}\left ( \sqrt{x}-1 \right )=0[/tex]

[tex]\sqrt{x}=0[/tex] or [tex]\left ( \sqrt{x}-1 \right )=0[/tex]

x = 0, x = 1

y = 0, y = 1

A = [tex]\int_{0}^{1}ydx(curve)-\int_{0}^{1}ydx(line)[/tex]

A = [tex]\int_{0}^{1}\sqrt{x}dx-\int_{0}^{1}xdx[/tex]

A = [tex]\frac{2}{3}[x^\frac{3}{2}]_{0}^{1}-\frac{1}{2}[x^2]_{0}^{1}[/tex]

A = [tex]\frac{2}{3}-\frac{1}{2}[/tex]

A = [tex]\frac{1}{6}[/tex]

Answer:

just saying

Step-by-step explanation:

subtract them lol use a mixed number calculator online

let's change the mixed fractions first to improper fractions and then get their sum.

we don't know the beginning point of the market, however from that point on, it spikes up by 30½and then goes down by 120¼, we're being asked on the total change or namely the displacement, and that'd be the sum of the spike line going up and the dive line going down.

[tex]\bf \stackrel{mixed}{30\frac{1}{2}}\implies \cfrac{30\cdot 2+1}{2}\implies \stackrel{improper}{\cfrac{61}{2}}~\hfill \stackrel{mixed}{120\frac{1}{4}}\implies \cfrac{120\cdot 4+1}{4}\implies \stackrel{improper}{\cfrac{481}{4}} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{481}{4}+\cfrac{61}{2}\implies \stackrel{\textit{using an LCD of 4}}{\cfrac{(1)481+(2)61}{4}}\implies \cfrac{481+122}{4}\implies \cfrac{603}{4}\implies 150\frac{3}{4}[/tex]

Answer: [tex]1.5\text{ square units}[/tex]

Step-by-step explanation:

We know that the area of triangle with coordinates [tex](x_1,y_1),(x_2,y_2)[/tex] and [tex](x_3,y_3)[/tex] is given by :-

[tex]\text{Area}=\dfrac{1}{2}|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|[/tex]

Given : The coordinates of ΔABC are  A = (-3,3), B=(-4,1), C = (-6,0).

Then, the area of ΔABC will be :-

[tex]\text{Area}=\dfrac{1}{2}|-3(1-0)+(-4)(0-3)+(-6)(3-1)|\\\\\Rightarrow\text{Area}=\dfrac{1}{2}|-3-4(-3)-6(2)|\\\\\Rightarrow\text{Area}=\dfrac{1}{2}|-3+12-12|\\\\\Rightarrow\text{Area}=\dfrac{1}{2}|-3|=\dfrac{1}{2}(3)=1.5 [/tex]

Hence, the area of ΔABC= [tex]1.5\text{ square units}[/tex]

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:  

(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:

Area is 321.6 square inches.

Step-by-step explanation:

A regular polygon with 16 sides is a Regular Hexadecagon. Regular means all sides are equal dimension. See image attached.

Side length is 4 inches.

side length=s=4 inches

Apothem=s*2.51367

Apothem=10.054

2.  You can use the formula to calculate de area of an hexadecagon:

[tex]A=\frac{Perimeter*Apothem}{2}[/tex]

Perimeter=16 sides*4inches

Perimeter= 64 inches

[tex]A=\frac{64*10.054}{2}[/tex]

[tex]A=321.6 inches^{2}[/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]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:

96

Step-by-step explanation:

Step 1

Divide your confidence interval by 2. In this case the confidence is 95% = 0.095, so 0.095/2 = 0.0475

Step 2

Use either a z-score table or a computer to find the closest z-score for 0.0475 and you will find this value is 1.96

Step 3

Divide the margin error by 2. In this case, the margin error is 2%. When dividing this figure by 2, we get 1% = 0.01

Step 4

Divide the number obtained in Step 2 by the number obtained in Step 3 and square it

1.96/0.01 = 196 and 196 squared is 38,416

Step 5

As we do not now a proportion of people that purchase on line, we must assume this value is 50% = 0.5. Square this number and you get 0.25

Step 6

Multiply the number obtained in Step 5  by the number obtained in Step 4, round it to the nearest integer and this is an appropriate size of the sample.

38,416*0.25 = 9,604

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:

Option B is right

Step-by-step explanation:

[tex]H_0: x bar =2\\H_a: x bar <2[/tex]

(One tailed test at 5% significance level)

n =16 and x bar =1.97

s =0.1

Std error of mean = [tex]\frac{s}{\sqrt{n} } \\=0.025[/tex]

Mean diff = [tex]1.97-2 =-0.03[/tex]

t statistic =Mean diff/se =-1.2

df =16-1=15

p value =0.124375

(B) do not reject the null because the test statistic (-1.2) is > the critical value (-1.7531).

The null hypothesis should not be rejected.

The question is asking whether to reject or not reject the null hypothesis based on the given information. The null hypothesis states that the house cleaning service can clean a four bedroom house in less than 2 hours. To test this hypothesis, a sample of 16 houses was taken, with a sample mean of 1.97 hours and a sample standard deviation of 0.1 hours. Using a significance level of 0.05, we compare the test statistic to the critical value to make the decision.

The correct conclusion in this case is to do not reject the null hypothesis, because the test statistic of -1.2 is greater than the critical value of -1.7531. Therefore, there is insufficient evidence to conclude that the cleaning service takes more than 2 hours to clean a four bedroom house.

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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:

See step-by-step explanation.

Step-by-step explanation:

a) If the old TV breaks down then we are buying a new TV.

b) If you have a good credit score in the United States, then you obtain a loan.

c) If you don't make me a better offer then I will keep my current job.

d) If we observe faster than light travel then we would question relativity theory.

Final answer:

Conditional statements, often structured in "if... then" form, express logical relationships between propositions. The provided statements have been reformulated as clear conditionals, preserving their logical meaning.

Explanation:

The statements provided by the student can be rephrased in standard "if... then" form as follows:

These rephrased statements maintain the same logical meaning but are now in the clear structure of conditionals that indicate a cause-and-effect relationship.

Answer:

A 95% confidence interval for the mean number of months elapsed since the last visit to a dentist, is [tex][5.79063, 28.20937][/tex]

Step-by-step explanation:

We will build a 95% confidence interval for the mean number of months elapsed since the last visit to a dentist. A (1 - [tex] \alpha [/tex]) x100% confidence interval for the mean number of months elapsed since the last visit to a dentist with unknown variance and is given by:

[tex][\bar x -T_{(n-1,\frac{\alpha}{2})} \frac{S}{\sqrt{n}}, \bar x +T_{(n-1,\frac{\alpha}{2})} \frac{S}{\sqrt{n}}][/tex]

[tex]\bar X = 17[/tex]

[tex]n = 5[/tex]

[tex]\alpha = 0.05[/tex]

[tex]T_{(n-1,\frac{\alpha}{2})}=2.7764[/tex]

[tex]S = 9.0277[/tex]

[tex][17 -2.7764 \frac{9.0277}{\sqrt{5}}, 17 +2.77644 \frac{9.0277}{\sqrt{5}}][/tex]

A 95% confidence interval for the mean number of months elapsed since the last visit to a dentist, is    [tex][5.79063, 28.20937][/tex]

Answer: [tex]|x-12.8|\leq3.93[/tex]

The interval in which the average is likely to fall : [tex]8.87\leq x\leq16.73[/tex]

Step-by-step explanation:

Given : A poll found that a particular group of people read an average of 12.8 books per year.

The pollsters are​ 99% confident that the result from this poll is off by fewer than 3.93 books from the actual average x.

The inequality to express this situation involving absolute​ value will be :-

[tex]|x-12.8|\leq3.93\\\\\Rightarrow\ -3.93\leq x-12.8\leq3.93 \\\\\text{Add 12.8 on both sides , we get}\\\\\Rightarrow\ -3.93+12.8\leq x\leq3.93+12.8\\\\\text{Simplify}\\\\\Rightarrow\ 8.87\leq x\leq16.73[/tex]

Hence, the interval in which the average is likely to fall : [tex]8.87\leq x\leq16.73[/tex]

The problem involves using confidence intervals and absolute value inequalities. The poll results suggest that the actual average number of books read by the group per year is between 8.87 books and 16.73 books, with a 99% confidence level.

In this problem, the pollsters are 99% confident which means that the actual average (x) of books read by the group is within 3.93 books of the given average (12.8 books). This situation can be expressed as an inequality involving absolute value as follows: |x - 12.8| < 3.93

To find the interval of values that x (the actual average) can take, we will solve the inequality for x. This inequality is saying that the distance between x (actual average) and 12.8 is less than 3.93.

This gives us two inequalities when broken down:  x - 12.8 < 3.93 and -(x - 12.8) < 3.93. Solving these two inequalities gives us an interval for x as 8.87 < x < 16.73.

So, the pollsters are 99% confident that the actual average number of books read by the group per year is between 8.87 books and 16.73 books.

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Answer:

a) There is an 8% probability that a randomly selected world class tennis player is over 6 feet tall and was a premature baby.

b) There is a 42% probability that a randomly selected world class tennis player is not over 6 feet tall and was not a premature baby.

Step-by-step explanation:

In this problem, we have these following probabilities:

-A 40% probability that a world class tennis player is over 6 feet tall.

-A 60% probability that a world class tennis player is not over 6 feet tall.

-A 20% probability that a world class tennis player over 6 feet tall was a premature baby. This also means that there is a 80% probability that a world class tennis player over 6 feet tall was not a premature baby

-A 30% probability that a world class tennis player who is not over 6 feet tall was a premature baby. This also means that there is a 70% probability that a world class tennis player who is not over 6 feet tall was not a premature baby.

a. Calculate the probability that a randomly selected world class tennis player is over 6 feet tall and was a premature baby.

[tex]P = P_{1}*P_{2}[/tex]

[tex]P_{1}[/tex] is the probability that a random selected world class tennis player is over 6 feet tall. So

[tex]P_{1} = 0.4[/tex]

[tex]P_{2}[/tex] is the probability that a world class tennis player over 6 feet tall was a premature baby. So

[tex]P_{2} = 0.2[/tex]

[tex]P = P_{1}*P_{2} = 0.4*0.2 = 0.08[/tex]

There is an 8% probability that a randomly selected world class tennis player is over 6 feet tall and was a premature baby.

b. Calculate the probability that a randomly selected world class tennis player is not over 6 feet tall and was not a premature baby

[tex]P = P_{1}*P_{2}[/tex]

[tex]P_{1}[/tex] is the probability that a random selected world class tennis player is not over 6 feet tall. So

[tex]P_{1} = 0.6[/tex]

[tex]P_{2}[/tex] is the probability that a world class tennis player under 6 feet tall was not a premature baby. So

[tex]P_{2} = 0.7[/tex]

[tex]P = P_{1}*P_{2} = 0.6*0.7 = 0.42[/tex]

There is a 42% probability that a randomly selected world class tennis player is not over 6 feet tall and was not a premature baby.

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 with Step-by-step explanation:

We are given that a statement ''If I have the disease , then I will test positive.''

Let p:I have the disease.

q:I will test positive.

a.Converse :[tex]q\implies p[/tex]

''If I will test positive, then I have the disease''.

b.Inverse :[tex]\neg p\implies \neg q[/tex]

''If I have not the disease, then I will not test positive.''

c. Contrapositive:[tex]\neg q\implies \neg p[/tex]

''If I will not test positive, then I have not the disease''.

d.Disjunction:p or q=[tex]p\vee q[/tex]

''I have the disease or I will test positive''.

e.Negation :If p is true then its negation is p is false.

Negation of conditional statement is equivalent to [tex]p\wedge \neg q[/tex]

I have disease and I will not test positive.

To calculate the annualized return of an investment with varying yearly returns, convert the percentage returns to decimal format, add 1, multiply them together, take the cube root (for a 3-year period), subtract 1 and convert back to a percentage. This provides a simplified approach to understanding how each year contributes to the overall growth, demonstrating the power of compound interest over time.

To calculate the annualized return for an investment over a multi-year period with varying yearly returns, you can use the formula for geometric mean. The returns provided are 4% in Year 1, 7% in Year 2, and 10.8% for the first half of Year 3. Since the return in Year 3 is only for half the year, we annualize this by considering the effective annual rate that would lead to this return over half a year, which could be simplified for calculation purposes here without altering the principle of geometric mean calculation for annualization. Assuming each return compounds, the computation involves converting the percentage returns to their decimal form, adding 1 to each (to account for the total value, not just the gain), multiplying these values together, then taking the cube root (since we're considering a period slightly over 2.5 years), and finally subtracting 1 and converting back to a percentage.

This simplifies to:
Annualized Return = ((1 + 0.04) * (1 + 0.07) * (1 + 0.108)) 1/3 - 1, which when solved gives the annualized return. However, correctly accounting for the exact duration (slightly over 2.5 years) could involve more precise financial formulas or software for exactitude.

It is crucial to recognize each year's return contributes to the calculation differently because of compound interest, which is why each year must be calculated based on its specific return, then all are combined to find the overall annualized performance of the investment.

Answer:

Angle 1 = 3 . 43.5+6 =136.5°

Step-by-step explanation:

To angles are supplementary when added up, the result is 180°.

So:

Angle 1 + Angle 2 = 180°

We know that angle 1 = 3x+6 and angle 2= x

So we get:

[tex]3x+6+x=180\\4x+6=180\\4x=180-6\\4x=174\\x=174 : 4\\x=43.5[/tex]

If we know now that x=43.5, we can calculate the value of angle 1.

Angle 1 = 3 . 43.5+6 =136.5°

Final answer:

There are 5 different ways to buy 42 cookies: three 21-cookie packages, two 21-cookie packages and one 7-cookie package, one 21-cookie package and three 7-cookie packages, one 21-cookie package, two 7-cookie packages, and one single, and three 7-cookie packages and seven singles.

Explanation:

To determine how many different ways you can buy 42 cookies, we can break it down into different combinations of packages. The packages are sold in singles, 7-cookie packages, and 21-cookie packages.

Let's start with the largest packages, the 21-cookie packages. We can buy zero, one, two, or three packages. For each number of packages, we can then calculate the number of remaining cookies we need to buy with singles and 7-cookie packages. We repeat this process for the 7-cookie packages, and then for the singles.

Using this method, we find that there are 5 different ways to buy 42 cookies:

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.

Answer:

You should make 250 quarts of Creamy Vanilla and 200 of Continental Mocha to use up all the eggs and cream.

Step-by-step explanation:

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 700 eggs in stock, so:

2x + y = 700.

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 1350 cups of cream in stock, so:

3x + 3y = 1350

Now we have to solve the following system of equations

1) 2x + y = 700

2) 3x + 3y = 1350

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

y = 700 - 2x

3x + 3(700 - 2x) = 1350

3x + 2100 - 6x = 1350

-3x = -750 *(-1)

3x = 750

x = 250

You should make 250 quarts of Creamy Vanilla

Now, replace it in 1)

y = 700 - 2x

y = 700 - 2(250)

y = 700 - 500

y = 200.

You should make 200 quarts of Continental Mocha

Final answer:

To use up all eggs and cream at the ice cream factory, the manager should produce 250 quarts of Creamy Vanilla and 200 quarts of Continental Mocha, by solving the system of linear equations derived from the given recipe requirements.

Explanation:

To solve the problem of how many quarts of Creamy Vanilla and Continental Mocha ice cream can be produced with 700 eggs and 1350 cups of cream, we need to set up a system of equations.

Let x be the number of quarts of Creamy Vanilla and y be the number of quarts of Continental Mocha.

From the information given:

We have the following equations:

To simplify the second equation we can divide by 3, resulting in:

Now we solve this system of linear equations. By subtracting the second equation from the first, we get:

Now, substitute x into the second equation:

To use up all the eggs and cream, the factory should make 250 quarts of Creamy Vanilla and 200 quarts of Continental Mocha.

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

A backward e means "there exists".

An upside down A means "for all".

The flipped E symbol (∃) is used to assert that there exists at least one element in a set that satisfies a given property in discrete math. The upside-down A symbol (∀) is used to assert that all elements in a set satisfy a given property.

The flipped E and upside-down A are symbols used in discrete math to represent logical operations. The flipped E symbol (∃) is called the Existential Quantifier and is used to assert that there exists at least one element in a set that satisfies a given property. The upside-down A symbol (∀) is called the Universal Quantifier and is used to assert that all elements in a set satisfy a given property.

For example, if we have a set of integers S = {1, 2, 3, 4, 5}, the statement ∃x(x > 3) would be true because there exists at least one element in set S (in this case, 4 or 5) that is greater than 3. On the other hand, the statement ∀x(x > 3) would be false because not all elements in set S are greater than 3, as 1, 2, and 3 are also included in the set.

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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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