a) Choose two vectors vw in R2 and a third vector b also in R2, and express b as a linear combination of v,w by finding scalars c₁,c2 such that c₁v + c₂w = b. Sketch the situation in R2 with an illustration that uses the parallelogram rule. (b) Repeat part (a) with vectors in R3 such that the augmented matrix [v w | b] gives a consistent system, again illustrating by graphing but this time in R3. (c) Why is it harder to find a consistent system for part (b) compared to part (a)? Explain your idea clearly using complete sentences.
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Answer:

x = -2,  x = 9

Step-by-step explanation:

Given quadratic equation:

[tex]x^2-7x-18=0[/tex]

To factor a quadratic in the form ax²+bx+c, first find two numbers that multiply to ac and sum to b.

[tex]\implies ac=1 \cdot -18=-18[/tex]

[tex]\implies b=-7[/tex]

Therefore, the two numbers are: -9 and 2.

Rewrite b as the sum of these two numbers:

[tex]\implies x^2-9x+2x-18=0[/tex]

Factor the first two terms and the last two terms separately:

[tex]\implies x(x-9)+2(x-9)=0[/tex]

Factor out the common term (x - 9):

[tex]\implies (x+2)(x-9)=0[/tex]

Apply the zero-product property:

[tex]\implies x+2=0 \implies x=-2[/tex]

[tex]\implies x-9=0 \implies x=9[/tex]

Therefore, the solutions to the given quadratic equation are:

[tex]35 cm ^{3}[/tex] is the volume of the prism displayed. The triangular cross section is 1.25 cm tall.

length , l = 7 cm , width w = 4 cm , height , h= 1.25 ,diagonal ,   d=8.15858444 cm , total surface area, [tex]$S_{\text {tot }}=83.5 \mathrm{~m}^2$[/tex] , lateral surface area , [tex]$S_{\text {lat }}=27.5 \mathrm{~m}^2$[/tex] , top surface area , [tex]$S_{\text {top }}=28 \mathrm{~m}^2$[/tex] , bottom surface area , [tex]$S_{\text {bot }}=28 m^2$[/tex] , volume [tex]$V=35 \mathrm{~m}^3$[/tex] .

To calculate the height of a prism, divide its volume by its area at the base. Let's say that the prism has a volume of 600 cubic inches to wrap up this scenario. The height is determined by multiplying 600 cubic inches by 100 square inches.

A prism's volume is calculated using the formula V=Bh, where B denotes the base area and h the height. A rectangular area makes up the prism's base. The rectangle's dimensions are 9 cm long by 7 cm wide. The formula V=Bh, where V stands for volume, B for the prism's area at its base, and h for its height, can be used to determine a prism's volume. B should be replaced with the area of a rectangle formula because this is a rectangular prism.

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The choice that shows 13 · (7 · 4) correctly rewritten using the associative property and then correctly simplified is presented as follows;

The associative property of multiplication stated that the multiplication of three numbers produces the same result in any way the numbers are grouped.

Mathematically, the associative property of multiplication can be presented as follows;

(a × b) × c = a × (b × c)

The specified operation involving the multiplication of three numbers are presented as follows;

13 · (7 · 4)

The associative property of multiplication indicates that we get;

13 · (7 · 4) = (13 · 7) · 4

13 × 7 = 91

Therefore, (13 · 7) · 4 = (13 × 7) × 4

(13 × 7) × 4 = 91 × 4 = 364

The option that shows the associative property correctly written and then correctly simplified is therefore;

The possible options in the question, obtained from a similar online question are;

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Answer:

In an absolute value graph, the slope of one side of the graph is the negative of the other side.

For example, if the slope of one side of an absolute value function is 2, then the other side of the equation has a slope of -2.

Anazlysing the conditions given in the question we can  conclude that there is a statistically significant increase in the consensus estimate of economists.

Financial analysts often use the whisper number, which is a late-breaking estimate of a key economic indicator, to make predictions about the stock market and inform their investment decisions. They may use it to identify potential opportunities for buying or selling stocks based on whether the whisper number is higher or lower than expected.

While the whisper number can provide insight into the current economic conditions, it is not a foolproof way to predict the stock market. It is just an estimate based on a sample of economists' predictions and can be affected by various unpredictable factors. Therefore, it's important to take other economic indicators and market conditions into account when making investment decisions.

To conduct a hypothesis test, we need to first define the null hypothesis and alternative hypothesis.

Null Hypothesis: The whisper number (sample mean) is equal to the consensus estimate (population mean)

H0: μ = 250,000

Alternative Hypothesis: The whisper number (sample mean) is greater than the consensus estimate (population mean)

H1: μ > 250,000

The next step is to calculate the test statistic and p-value. We can use a one-tailed t-test since the sample size is less than 30 and we don't know the population standard deviation.

Test statistic:

t = (266,000 - 250,000) / (24,000 / sqrt(20))

P-value:

Using the t-distribution table with 19 degrees of freedom, the p-value is less than 0.01, which is the significance level we set.

Since the p-value is less than the significance level (0.01), we can reject the null hypothesis and conclude that there is a statistically significant increase in the consensus estimate of economists.

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The conversion from cents to Dollar we get :

4 cent  = 4/100 = $ 0.04

70 cents = 70/100 = $ 0.7

58 cents = 58/100 = $.58

204 cents = 204/100 = $ 2.04 = $2 and 4 cents

.75 cents = .75 / 100 = $ 0.0075

3009 cents = 3009/100 = $ 30.09 = $30 and 9 cents

A conversion factor is a number that is used to multiply or divide one set of units into another. If a conversion is required, it must be done using the correct conversion factor to get an identical value. For instance, 12 inches equals one foot when converting between inches and feet.

Cents are equivalent to one dollar. Dollars are represented by paper banknotes, which come in denominations of $100, $50, $20, $10, $5, and $1. Coins include cents.

The act of converting one value into another, such as from inches to millimeters or from liters to gallons, is known as conversion. The following measurements: length, weight, capacity, temperature, and speed are all measured in units.

The values in cents

4 cent  = 4/100 = $ 0.04

70 cents = 70/100 = $ 0.7

58 cents = 58/100 = $.58

204 cents = 204/100 = $ 2.04 = $2 and 4 cents

.75 cents = .75 / 100 = $ 0.0075

3009 cents = 3009/100 = $ 30.09 = $30 and 9 cents.

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The equation that represents the relationship is:

x + (x + 1) + (x + 2) = 42

And the 3 consecutive numbers are 13, 14, and 15.

Here we know that the sum of three consecutive integers is 42.

If the first number is x, then the two consecutive numbers are:

x + 1

x + 1 + 1 = x + 2

Then the sum of these 3 consecutive numbers can be written as:

x + (x + 1) + (x + 2)

And that is equal to 42, then the equation is:

x + (x + 1) + (x + 2) = 42

So the first option is the correct one.

And the solution is:

x + (x + 1) + (x + 2) = 42

3x + 3 = 42

3x = 42 - 3

3x = 39

x = 39/3

x = 13

The 3 numbers are:

13, 14, and 15.

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Answer:

a) General Equation for this scenario is

[tex]\boxed{P_n = P(.98)^n\\\\}[/tex]

where P is the starting population and Pₙ is the population at the end of n years

For this specific case,
[tex]\boxed{P_n = 20000(0.98)^n}[/tex]

b) Population in 2025 will be the population after 25 years:
[tex]P_{25} = 20000(0.98)^{25} = \boxed{12,069}[/tex]

Step-by-step explanation:

In general, if the population at year 0 is P and it decreases by 2% each year then the population after one year will be given by

P - decrease in one year

Decrease in population for 1 year = 2% of P [tex]= 0.02P[/tex]

So P(after 1 year) :  
[tex]P - 0.02P\\\\ = P(1-0.02) \\\\= 0.98P[/tex]

After 2 years the population will be population after 1 year (0.98P) - decrease in population for 1 year (0.02 x 0.98P)

So population at end of 2 years would be
 [tex]0.98P - 0.98P(0.02)\\\\= 0.98P(1-0.02)\\\\=0.98P(0.98)\\\\= P(0.98)^2[/tex]

In general after n years the population would be
[tex]P(0.98)^n\\\\[/tex]

If the population in year 2000 is 20,000 then population at the end of 25 years is:

[tex]20000(0.98)^{25}[/tex]
[tex]\rm\:{=\:12,069\:\;\:rounded\:down\:to\:the\:nearest\:integer}[/tex]  

Answer:

first answer is

1.31

Step-by-step explanation:

and second is

7.47

hope it help you

hi friend

Using the mathematical operations of addition and subtraction, the quantity of almond milk that Lucas should add to his recipe to make three 8-fluid-ounce servings of the smoothie is 1¹/₈ cups.

Mathematical operations include addition, division, subtraction, and multiplication.

In this situation, we can determine the needed quantity of almond milk using multiplication, subtraction, and addition.

The total quantity of smoothie is obtained by multiplying three cups by 8 fluid ounces.  The quantity of almond milk is determined by subtracting the sum of other ingredients from the total quantity.

Lucas' usual recipe for two 8 fluid ounces:

Berries 3/8 cups

Mango 1/2 cups

Yogurt = 3/4 cups

Almond Milk = 3/8 cups

Total cups made = 2 cups (16 fluid ounces = 2 x 8)

New recipe for three 8 fluid ounces:

Berries 5/8 cups

Mango 1/2 cups

Yogurt = 3/4 cups

Almond milk = x

1 cup = 8 fluid ounces

Total cups to be made = 3 cups (24 fluid ounces = 3 x 8)

5/8 + 1/2 + 3/4 + x = 24/8

= 15/8 + x = 24/8

x = 9/8 (24/8 - 15/8)

= 1¹/₈ cups

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Lucas makes a smoothie using the recipe (berries 3/8, mango 1/2, yogurt 3/4 cups) and adds almond milk for two 8 fluid-ounce servings. (There are 8 fluid ounces in 1 cup.​)

The firm needs to sample 138.297 products to achieve a ±5% accuracy at a 95% confidence level, given that it believes that the true proportion of defects is no greater than 10%.

Generally, To determine the sample size needed to achieve a ±5% accuracy at a 95% confidence level, with a maximum expected proportion of defects of 10%, you can use the formula for sample size for a proportion. The formula is:

n = (z^2 * p * (1-p)) / E^2

Where:

Plugging in the given values, we get:

n = (1.96^2 * 0.10 * (1-0.10)) / 0.05^2

n =138.297

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Answer:

  170,483

Step-by-step explanation:

You have N varying jointly with P1 and P2, such that N=109,000 for P1=74000 and P2=189000. You want to know N for P1=125000 and P2=175000.

The relation "N varies jointly with P1 and P2" can be described by the equation ...

  N = k·P1·P2

The value of k can be found as ...

  k = N/(P1·P2)

  k = 109000/(74000·189000) ≈ 7.793508×10^-6

Then the number of calls given the populations P1=125000 and P2=175000 is predicted to be ...

  N = (7.793508×10^-6)(125000)(175000) ≈ 170482.982...

  N ≈ 170483

About 170,483 calls are expected between cities with populations of 125 and 175 thousand.

If the number of phone calls between two nearby cities during a given time period. The number of calls  that are made between two cities with population 125,000 and 175,000 is: 170,482,987,500.

Calls  made between two cities with populations 74,000 and 189,000,

Call = 109,000/(74000 × 189000)

Call = 109,000 / 13,986,000,000

Call = 7.793508

Calls made between two cities with populations 125,000 and 175,000?

Calls =  7.793508 × 125000 ×175000

Calls =170,482,987,500

Therefore the calls made is 170,482,987,500.

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The area of the triangle shown is 51.96 unit²

An equation is a expression that shows the relationship between numbers and variables.

The area of a triangle can be found given two adjacent sides and an angle between the two sides. It is given as:

Area = (1/2) * product of adjacent sides * sin(angle)

Hence:

Area of triangle shown = (1/2) * 10 * 12 * sin(60) = 51.96

The area is 51.96

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The constants A and B for the differential equation are given as follows:

The solutions to the differential equation are given as follows:

This means that the roots of the characteristic equation are given as follows:

r = -8 and r = 8.

Considering that the leading coefficient is of 1, as 1 multiplies y'', the characteristic equation is given as follows:

(r + 8)(r - 8) = r² - 64.

Hence the differential equation is given as follows:

y'' - 64y = 0.

Hence the constants are given as follows:

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Answer:

[tex]\boxed{-2 < x\le1}[/tex]

Representation on number line is in attached image.

Notice that -2 has an unfilled circle and 1 has a filled circle. Important

Step-by-step explanation:

The given inequality is

[tex]2 < 4x+10\le14[/tex]

Subtract 10 throughout:

[tex]2-10 < 4x+10-10\le14-10\\\[/tex]

[tex]-8 < 4x\le4[/tex]

Divide throughout by 4:

[tex]-2 < x\le1[/tex]

Answer:    [tex]\boxed{-2 < x\le1}[/tex]

This is represented on the number line as follows: Notice the use of the

Answer:

B. point; vertices; point

Step-by-step explanation:

A pyramid is a 3-dimensional geometric shape that has a base, which is typically a polygon, and a number of triangular faces that all meet at a single point called the vertex. The pyramid has vertices at the corners of the base and all other points where the triangular faces meet.

Answer:

(14x+13)(x-1)

Step-by-step explanation:

14x²-x-13

14 × 13 = 182

Search for factors of 182 that contain a difference of 1 when subtracted because -x is the same as -1x.

Since 14 × 13 = 182

and

13 - 14 = -1

They are our 2 chosen numbers which will be displayed as :

14x² - 14× + 13x - 13

where -x is replaced by -14x + 13x

Proceed to factorise normally

14x(x-1) + 13(x-1)

Grouping

(14x+13)(x-1)

We know that Nevaeh initially deposited $2,100, and now she has $2,131.50 in her account. The interest earned is $2,131.50 - $2,100 = $31.50. To find out how many months has she been investing her money, we can use the following formula:

Time (in months) = (Interest / Principal) x (12 / Interest Rate)

Where Principal is the initial deposit amount, Interest Rate is the annual interest rate, and Interest is the amount of interest earned.

Plugging in the given values, we get:

Time (in months) = ($31.50 / $2,100) x (12 / 0.06)

Time (in months) = 0.0149 x 200

Time (in months) = 2.98

Since time cannot be in fraction of months, it has to be rounded off. So, Nevaeh has been investing her money for approximately 3 months.

According to the given expression, the account's initial deposit is 750, the balance after one year is 845, and the balance after two years is 952.

Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation.

Given, The amount of money in a savings account is modeled by A(x) = 750(1.01)^12x, where x represents number of years.

Expression for money is given

Thus, balance at x = 0

Balance A(0) = 750

Balance at x = 1

Balance A(1) = 750(1.01)^12*1

A(1) = 845.11

Balance at x = 2

Balance A(2) = 750(1.01)^12*2

A(2) = 952.43

Therefore, In the account initial deposit is 750, balance after 1 year  is 845, and balance after 2 years will be 952 as per the given expression.

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3/17

Both 3 and 17 are prime numbers, therefore they cannot be simplified by any number expect 1 and itself.

Answer:

94

Step-by-step explanation: PQS = (7x - 6)°

SQR = (4x + 15)° since QS bisect PQR these two expressions must be equal

so

7x - 6 = 4x + 15 transfer like terms to the same side of the equation

7x - 4x = 15 + 6

3x = 21 divide both sides by 3

x = 7

also the sum of these two would give us the measure of PQR

7x + 4x + 15 - 6 = PQR

11x + 9 = PQR replace x with 7

11*7 + 9 = 86 this is the measure of angle PQR and also supplementary to PQT so the measure of PQT = 180 - 86

Answer:

don't

Step-by-step explanation:

get that one wrong don't spend to much time on it

Answer:

infinitely many solutions and x=y

The z score for a student who received a score of 310 is 0.75

An equation is an expression composed of variables and numbers linked together by mathematical operations.

The z score shows by how many standard deviation the raw score is above or below the mean. It is given by:

z = (raw score - mean)/standard deviation

Given that the eighth-grade math test is Normal, (282, 40), hence mran = 282, stand. deviation = 40

For a score of 310:

z = (310 - 282) / 40 = 0.75

The z score is 0.75

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Answer:

43%

Step-by-step explanation:

28 questions - 16 wrongs = 12 correct

(12/28) × 100= 42.857.....%

To the nearest percent=43%

Answer:no

Step-by-step explanation:

Answer: The interest rates charged in both formulas are different.

Step-by-step explanation:

To begin with, the simple interest = PRT/100

In this, the interest is the same as the number of years.

For example, P=10,000 R=10% T=2yrs

SI = PRT/100

    = 10,000*10*2/100

    = 2000

This rupee 2000 is the same interest for two years.

compound interest = A= P(1+r/100)^n

In this, the interest changes every year up to the period of time.

For example, P= 10,000 R=10% N=2yrs

A= 10,000(1+10/100)^2

  = 10,000*11/10*11/10

   = 12100

I= A-P

 =12100-10000

 = 2100

This interest varies for 2 years.

Answer:

C{-3,-2.5}

Step-by-step explanation:

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