-3x+y=11,4x+3y=7 use stubstitution to solve

1. The Venn diagrams are attached

2. When the statistics students number = 10·x + 3, we have;

The number of students that study

When the statistics students number = 2·x + 3, we have;

The number of students that study

The parameters given are;

Total number of students = 30

Therefore;

The number of students that study either algebra or statistics = n(A∪B)

From set theory we have;

n(A∪B) = n(A) + n(B) - n(A∩B)

n(A∪B) = 30 - 3 = 27

Therefore, we have;

The number of students that study

a. Algebra

n(A) = 18/11 + 10 = 128/11

b. Statistic

n(B) = 213/11

Hence, we have;

n(A - B) = n(A) - n(A∩B) = 128/11 - 4 = 84/11

Similarly, we have;

n(B - A) = n(B) - n(A∩B) = 213/11 - 4 = 169/11

However, assuming n(B) = (2·x + 3), we have;

n(A∪B) = n(A) + n(B) - n(A∩B)

n(A∪B) = 30 - 3 = 27

Therefore, we have;

n(A∪B) = x + 10 + 2·x + 3 - 4 = 27

2·x+3 + x + 10= 27 + 4 = 31

3·x = 18

x = 6

Therefore, the number of students that study

a. Algebra

n(A) = 16

b. Statistics

n(B) = 15

Hence, we have;

n(A - B) = n(A) - n(A∩B) = 16 - 4 = 12

Similarly, we have;

n(B - A) = n(B) - n(A∩B) = 15 - 4 = 11

The Venn diagrams can be presented as follows;

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Answer:

{40.7,49.3}

Step-by-step explanation:

[tex]\displaystyle CI=\bar{x}\pm z\frac{s}{\sqrt{n}}\\\\CI_{80\%}=45\pm1.282\biggr(\frac{19}{\sqrt{32}}\biggr)\\ \\CI_{80\%}\approx45\pm 4.3\\\\CI_{80\%}=\{40.7,49.3\}[/tex]

Thus, we are 80% confident that the true sample mean amount spent is between $40.70 and $49.30

Answer:

A.

Step-by-step explanation:

Since [tex]5[/tex] can rewritten as [tex]\sqrt{25}[/tex] and [tex]6[/tex] rewritten as [tex]\sqrt{36},[/tex] we can see which options go between [tex]\sqrt{25}[/tex] and [tex]\sqrt{36}.[/tex] The only square root that goes between is [tex]\sqrt{29}.[/tex] Therefore the answer is [tex]\boxed{A.}[/tex] (or [tex]\sqrt{29}.[/tex])

Answer:

[tex]x > 7[/tex]

Step-by-step explanation:

Given the inequality

[tex]x+4 > 11[/tex]

Lets solve the inequality for [tex]x[/tex].

Subtract [tex]4[/tex] from both sides of the inequality.

[tex]x > 11-4[/tex]

Evaluate [tex]11-4[/tex].

[tex]x > 7[/tex]

[tex]x[/tex] is greater than 7

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Answer:

27,64,216,64,243

Step-by-step explanation:

Any number raise to power 3

Answer:

Step-by-step explanation:

Since Joe bought six bunches of seedless black grapes for $15, the cost per bunch of grapes is $15 / 6 = $2.50.

Since Anjali has $7.50, she can buy $7.50 / $2.50 = 3 bunches of seedless black grapes.

The scaled dimensions of a kitchen with dimensions 6.3m by 4.8m, if the kitchen plan is drawn to a scale of 1:500, are given as follows:

1.26 cm and 0.96 cm.

The scaled dimensions are obtained applying the proportions in the context of the problem.

The scale is of 1:500, meaning that each dimension is divided by 500, as follows:

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1. The amount of pizzas which the pizzeria must sell for employees to receive the bonus is 78.

2. The new number of pizzas sold is considered as percent increase of 20%.

The pizzeria sold 65 pizzas yesterday and the owner offers bonus if the number sold increases by 20% today. The number that must be sold, to receive a bonus is computed as follows:

= 65 pizza * (65 pizza * 20%)

= 65 pizza * 13 pizza

= 78 pizza.

It is a percent increase which is computed as follows. The formula for percentage increase is: ((Final value - Initial value)/Initial value * 100).

The Percent increase:

= (78 pizza - 65 pizza) / 65 pizza.

= 13 pizza / 65 pizza

= 0.2

= 20%

Full question "A local pizzeria sold 65 pizzas yesterday. (step 1) The owner offers the employees a bonus if the number of pizzas sold increases by 20% today. How many pizzas must the pizzeria sell for employees to receive the bonus? Explain your reasoning. (Part 2) Suppose the actual number of pizzas sold today was 60. Describe the number of pizzas sold as a percent increase or decrease, rounded to the nearest hundredth. Show your work. Answer:"

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Step-by-step explanation:

(a) To convert feet to fathoms, we can use the conversion factor 1 fathom = 6 feet. Hence, we can divide the depth in feet by 6 to find the depth in fathoms:

36,078 feet / 6 feet/fathom = 6,013 fathoms

So the depth of the seabed is 6,013 fathoms.

(b) To convert feet to leagues (marine), we can use the conversion factor 1 league (marine) = 5,556 feet. Hence, we can divide the depth in feet by 5,556 to find the depth in leagues (marine):

36,078 feet / 5,556 feet/league (marine) = 6.52 leagues (marine)

So the depth of the seabed is approximately 6.52 leagues (marine).

The probability that more than 2 students will attend is 0.9179.

Part (a) List the values that X may take on.

X may take on the values 0, 1, 2, 3, ..., 13.

Part (b) Give the distribution of X.

The distribution of X can be represented as follows: X-0:0, 1:0.23, 2:0.23, 3:0.23, 4:0.23, 5:0.23, 6:0.23, 7:0.23, 8:0.23, 9:0.23, 10:0.23, 11:0.23, 12:0.23, 13:0.23

Part (c) How many of the 13 students do we expect to attend the festivities? (Round your answer to the nearest whole number.)

We can expect 3 students to attend the festivities.

Part (d) Find the probability that at most 3 students will attend. (Round your answer to four decimal places.)

The probability that at most 3 students will attend is 0.6923.

Part (e) Find the probability that more than 2 students will attend. (Round your answer to four decimal places.)

The probability that more than 2 students will attend is 0.9179.

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Answer:

Step-by-step explanation:

m=3-1/2-7= -2/5

c⇒ -2

y=mx+c

y=-2/5x-2

a. The profit function is given by P(x) = R(x) - C(x). Substituting the expressions for R(x) and C(x), we have P(x) = 12x + 300√x - 2500.

b. The marginal profit function is the derivative of the profit function with respect to x. To find the marginal profit function, we will differentiate P(x) is dP/dx = 12 + 300/2√x.

c. The marginal average profit function is the derivative of the profit function divided by the number of units sold is dP/dx ÷ x = (12 + 300/2√x) ÷ x.

d. To estimate the profit from the sale of the 201st doll, we will evaluate the profit function at x = 201 is $16.62.

What do you mean by  profit function?

A profit function is a mathematical model that represents the profit made from selling a certain product or service as a function of the number of units sold. The profit function is calculated as the difference between the total revenue and the total cost of producing and selling the product.

Mathematically, if R(x) represents the total revenue from selling x units of a product and C(x) represents the total cost of producing and selling x units of the product, then the profit function can be represented as P(x) = R(x) - C(x).

a. The profit function is given by P(x) = R(x) - C(x). Substituting the expressions for R(x) and C(x), we have:

P(x) = 14x - (2x - 300√x + 2500)

P(x) = 12x + 300√x - 2500

b. The marginal profit function is the derivative of the profit function with respect to x. To find the marginal profit function, we will differentiate P(x):

dP/dx = 12 + 300/2√x

c. The marginal average profit function is the derivative of the profit function divided by the number of units sold.

dP/dx ÷ x = (12 + 300/2√x) ÷ x

d. To estimate the profit from the sale of the 201st doll, we will evaluate the profit function at x = 201:

P(201) = 12 * 201 + 300 * √201 - 2500 = $3347.09

Since we want the profit from the sale of one unit, we divide the profit by 201:

P(201) ÷ 201 = $16.62

So, the estimated profit from the sale of the 201st doll is $16.62.

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Answer:

each parking ticket costs $100 and each traffic violation costs $124.28.

Step-by-step explanation:

To find out the cost of each parking ticket and each traffic violation, we can set up two equations using the information given:

Let x be the cost of each parking ticket.

In the first month, 3 parking tickets cost 3x, and 14 traffic violations cost 14y.

The total cost for the first month is 2030, so we have:

3x + 14y = 2030

In the second month, 11 parking tickets cost 11x, and 11 traffic violations cost 11y.

The total cost for the second month is 2200, so we have:

11x + 11y = 2200

Now that we have two equations, we can use substitution to find the value of x and y.

From the first equation, we can solve for y:

y = (2030 - 3x) / 14

Substituting this expression for y into the second equation:

11x + 11((2030 - 3x) / 14) = 2200

Expanding the expression for y and simplifying the equation:

11x + 2030 - 33x / 14 = 2200

Multiplying both sides by 14:

154x + 28420 = 30800

Solving for x:

x = $100

So each parking ticket costs $100, and each traffic violation costs:

y = (2030 - 3x) / 14 = (2030 - 3 * 100) / 14 = (2030 - 300) / 14 = 1730 / 14 = $124.28

Therefore, each parking ticket costs $100 and each traffic violation costs $124.28.

Answer:

-727.41 dollars

Step-by-step explanation:

To find the cost of each parking ticket and each traffic violation, we can set up a system of two equations. Let's call the cost of a parking ticket "x" and the cost of a traffic violation "y".

For the first month:

3x + 14y = 2030

For the second month:

11x + 11y = 2200

We can solve for x and y using any method, such as substitution or elimination. For this problem, let's use substitution.

First, we'll solve for x in the first equation:

3x + 14y = 2030

3x = 2030 - 14y

x = (2030 - 14y) / 3

Next, we'll substitute this expression for x into the second equation:

11x + 11y = 2200

11((2030 - 14y) / 3) + 11y = 2200

2030 - 14y + 11y = 2200 * 3 / 11

2030 - 3y = 636

-3y = -1394

y = 464.67

Finally, we'll substitute this value of y back into the first equation to find x:

3x + 14y = 2030

3x + 14(464.67) = 2030

3x = 2030 - 6544.68

x = (2030 - 6544.68) / 3

x = -2180.23 / 3

x = -727.41

So each parking ticket costs approximately -727.41 dollars, and each traffic violation costs approximately 464.67 dollars. However, since these values are negative, it means the suspect has received a credit rather than being charged for these fees.

a) A linear equation for the total remaining oil reserves, R, in terms of t, the number of years since the current year, is 2,090 - 18t.

b) Based on the above linear equation, 15 years from now, the total oil reserves will be 1,820 billion barrels.

c) Using division operation, the world's oil reserves will be completely depleted approximately 116 years from now.

A linear equation is a mathematical or algebraic equation written in the form of y = mx + b.

A linear equation consists of a constant and a first-order term, where m is the slope, and variable b, the y-intercept.

The current total world oil reserves, R = 2,090 billion barrels

Average decrease per year = 18 billion barrels

a) Linear equation for R is 2,090 - 18t

b) In 15 years, the total oil reserves = 2,090 - 18(15)

= 2,090 - 270

= 1,820

c) The constant rate of decrease = 18 billion barrels per year

The world's oil reserves will be completely depleted approximately 116 years from now (2,090/18).

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This student would receive 10 bag of popcorns.

Generally speaking, a unit price can be defined as the price that is being charged by a seller for the sale of a single unit of product.

In order to determine the quantity of popcorn, we would make use of the information provided about the unit price for one bag of popcorn by using the following mathematical expression;

Number of bag of popcorns = Cost of one bag of popcorn/Unit price

Substituting the given parameters into the mathematical expression above, we have the following;

Number of bag of popcorns = 5/0.50

Number of bag of popcorns = 10 bag of popcorns.

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[tex]~~~~~~ \textit{Simple Interest Earned Amount} \\\\ A=P(1+rt)\qquad \begin{cases} A=\textit{accumulated amount}\dotfill & \pounds 17160\\ P=\textit{original amount deposited}\dotfill & \pounds 11000\\ r=rate\to r\%\to \frac{r}{100}\\ t=years\dotfill &16 \end{cases}[/tex]

[tex]17160 = 11000[1+(\frac{r}{100})(16)] \implies \cfrac{17160}{11000}=1+\cfrac{16r}{100}\implies \cfrac{39}{25}=1+\cfrac{4r}{25} \\\\\\ \cfrac{39}{25}=\cfrac{25+4r}{25}\implies 39=25+4r\implies 14=4r\implies \cfrac{14}{4}=r\implies \stackrel{\%\qquad }{\boxed{3.5=r}}[/tex]

Answer: The answer to your problem is choice b

The answer is not one of the given choices. The correct answer is 73.

To find the total number of individuals who took the quiz, we need to add up the frequency (f) for each score (x) and then take the sum:

5(6) + 4(5) + 3(5) + 2(3) + 1(2) = 30 + 20 + 15 + 6 + 2 = 73

Therefore, the answer is not one of the given choices. The correct answer is 73.

       Frequency distributions are portrayed as frequency tables or charts. Frequency distributions can show either the actual number of observations falling in each range or the percentage of observations. In the latter instance, the distribution is called a relative frequency distribution.

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Based on the proceeds and effective rate, Wilson should take out the simple discount note, which has a higher proceeds of $5,005 and a higher effective rate of 9.9% compared to the simple interest note's proceeds of $5,995 and effective rate of 9.0%.

To compare the simple discount note and the simple interest note, we can calculate the proceeds and effective rates for each.

For the simple discount note, the interest is deducted upfront and the borrower receives the difference, or the proceeds. The interest on a $5,500 note at 6% for 18 months can be calculated as:

Interest = Principal × Rate × Time

Interest = $5,500 × 6% × 1.5 years

Interest = $495

The proceeds for the simple discount note can be calculated as:

Proceeds = Principal - Interest

Proceeds = $5,500 - $495

Proceeds = $5,005

The effective rate for the simple discount note can be calculated using the formula:

Effective Rate = Interest / Proceeds × 100%

Effective Rate = $495 / $5,005 × 100%

Effective Rate = 9.9%

For the simple interest note, the interest is calculated on the principal amount and added to the principal at the end of the term. The interest on a $5,500 note at 6% for 18 months can be calculated as:

Interest = Principal × Rate × Time

Interest = $5,500 × 6% × 1.5 years

Interest = $495

The total amount due at the end of the term for the simple interest note can be calculated as:

Total Due = Principal + Interest

Total Due = $5,500 + $495

Total Due = $5,995

The effective rate for the simple interest note can be calculated using the formula:

Effective Rate = Interest / Principal × 100%

Effective Rate = $495 / $5,500 × 100%

Effective Rate = 9.0%

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The equation y² - y²x = 6  can be describe as a function of x as:

y = √(6/(x - x)).

A function is a relationship between inputs where each input is related to exactly one output.

Example:

f(x) = 2x + 1

f(1) = 2 + 1 = 3

f(2) = 2 x 2 + 1 = 4 + 1 = 5

The outputs of the functions are 3 and 5

The inputs of the function are 1 and 2.

We have,

y² - y²x = 6

Taking y² as common.

y²(1 - x) = 6

Divide both sides with (1 - x).

y² = 6/(1 - x)

Square root on both sides.

y = √(6/(x - x))

This is a function of x.

Thus,

y = √(6/(x - x)) is a function of x.

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The derivative formula for the division of two functions is equal to d [θ(x)] / dx = [g(x) · f'(x) - f(x) · g'(x)] / [g(x)]², where θ(x) = f(x) / g(x).

In this problem we need to derive the formula for the derivative of the division of two functions, that is, the expression θ(x) = f(x) / g(x). First, we write the definition of the derivative for the given expression:

[tex]\frac{d}{dx} [\theta (x)] = \lim_{h \to 0} \frac{\theta(x + h) - \theta (x)}{h}[/tex]

Second, substitute and expand the expression by algebra properties:

[tex]\frac{d}{dx} [\theta (x)] = \lim_{h \to 0} \frac{\frac{f(x + h)}{g(x + h)} - \frac{f(x)}{g(x)} }{h}[/tex]

[tex]\frac{d}{dx}[\theta (x)] = \lim_{h \to 0} \frac{f(x + h) \cdot g(x) - f(x) \cdot g(x + h)}{h\cdot g(x + h)\cdot g(x)}[/tex]

Third, expand and simplify the expression one more time by algebra properties and limit properties:

[tex]\frac{d}{dx} [\theta (x)] = \lim_{h \to 0} \frac{f(x + h) \cdot g(x) - f(x) \cdot g(x) + g(x) \cdot f(x) -f(x) \cdot g(x + h)}{h \cdot g(x + h) \cdot g(x)}[/tex]

[tex]\frac{d}{dx}[\theta (x)] = \lim_{h \to 0} g(x) \cdot \frac{f(x + h) - f(x)}{h\cdot g(x+h)\cdot g(x)} - \lim_{h \to 0} f(x) \cdot \frac{g(x + h) - g(x)}{h\cdot g(x+h)\cdot g(x)}[/tex]

[tex]\frac{d}{dx}[\theta (x)] = \lim_{h \to 0} \frac{g(x)}{g(x + h) \cdot g(x)} \cdot \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} - \lim_{h \to 0} \frac{f(x)}{g(x+ h) \cdot g(x)} \cdot \lim_{h \to 0} \cdot \frac{g(x + h) - g(x)}{h}[/tex]

Fourth, simplify the resulting expression by evaluating the limits:

d [θ(x)] / dx = [g(x) / [g(x)]²] · f'(x) - [f(x) / [g(x)]²] · g'(x)

d [θ(x)] / dx = [g(x) · f'(x) - f(x) · g'(x)] / [g(x)]²

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The volume of the parallelopiped is 4

We have to solve this with the use of matrix system in math

PQ, PR, PS where

P(-3, 3, 2), Q(-1, 6, 5), R(-4, 2, 1), S(3, 1, 4)

we have to find

Q - P

(-1, 6, 5) - (-3, 3, 2)

= (2, 3, 3)

R - P

R(-4, 2, 1) - (-3, 3, 2)

= ( -1 , -1, -1)

S - P

S(3, 1, 4) - (-3, 3, 2)

= (6 , -2, 2)

we would have [tex]\left[\begin{array}{ccc}2&3&3\\-1&-1&-1\\6&-2&2\end{array}\right][/tex]

the determinant of the matrix is the volume

2[(-1 * 2) - 3[(-1x2)-(6x-1)] + 3[(-1x-2) -(6 x -1)]

= -8 - 12 + 24

= 4

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To find the weight of 2.2 cubic feet of water in pounds, we can use the following formula:

Weight in pounds = Volume in cubic feet x Density of water in pounds per cubic foot

The density of water is 62.4 pounds per cubic foot, so we can substitute the values into the formula:

Weight in pounds = 2.2 x 62.4 = 138.08

So, 2.2 cubic feet of water weighs 138.08 pounds.

To convert this weight to tons, we can divide by 2000, as there are 2000 pounds in a ton:

Weight in tons = 138.08 / 2000 = 0.06904 tons

So, 2.2 cubic feet of water weighs 0.06904 tons.

Answer:

Step-by-step explanation:

First, we need to find how many gallons of water 2.2 cubic feet hold:

2.2 cubic feet * 7.48 gallons per cubic foot = 16.456 gallons

Next, we'll find the weight of the water in pounds:

16.456 gallons * 8.33 pounds per gallon = 136.08 pounds

Finally, we'll convert the weight to tons:

136.08 pounds / 2000 pounds per ton = 0.0680 tons

So 2.2 cubic feet of water weigh 136.08 pounds and 0.0680 tons.

The linear functions for this problem are classified as follows:

a. Parallel.

b. Neither parallel nor perpendicular.

c. Perpendicular.

The slope-intercept definition of a linear function is given as follows:

y = mx + b.

In which:

They are classified as parallel, perpendicular or neither according to their slopes as follows:

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Answer:

Step-by-step explanation:

C. There is a moderate correlation between the scores and grades.

Based on the scatter plot and the calculated correlation coefficient, we can conclude that there is a moderate positive correlation between the pretest scores and the course grades. This means that as the pretest scores increase, the course grades tend to increase as well. However, the correlation is not extremely strong, so other factors may be affecting the course grades besides the pretest scores.

The correlation between two variables describes the relationship between them. The strength of the correlation can be classified into four categories: no correlation, weak correlation, moderate correlation, and strong correlation.

In this case, since the values of X and Y appear to have a moderate association, it might be concluded that there is a moderate correlation between the scores on the pretest and the course grade.

However, correlation does not imply causality, meaning that even if there is a correlation between two variables, it does not mean that one variable causes the other. Further analysis, such as regression analysis, is needed to determine the strength and direction of the relationship and to determine if there is a causal relationship.

The number of pieces of candy that will be left on day 4 would be 2, 560 candies .

The number of pieces of candy left on day 8 would be 66 candies .

To find the number of candy left on the fourth day, given that Augustus only gives candies after the pile of 100, 000 candies is reached, the formula is:

= Number of starting candy x ( 1 - Percent given away ) ^ number of days

= 100, 000 x ( 1 - 60 % ) ⁴

= 100, 000 x 40 % ⁴

= 2, 560 candies

On Day 8, this would be:

= 100, 000 x ( 1 - 60 % ) ⁸

= 66 candies

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The question is:

How many pieces of candy will be left on day 4? On day 8?

Answer:

  EF = 7

Step-by-step explanation:

You want to know the length of midsegment EF in the triangle with it labeled 43-4x and the parallel base segment labeled 32-2x.

The midsegment is half the length of the base segment, so we have ...

  (43 -4x) = 1/2(32 -2x)

  43 -4x = 16 -x . . . . . simplify

  27 = 3x . . . . . . . . add 4x-16

  9 = x . . . . . . . . divide by 3

  EF = 43 -4x = 43 -4(9) = 43 -36

  EF = 7