When opposite angles are supplementary, could the quadrilateral be a
parallelogram? Explain.
A)No, because in a parallelograms, angles are supplementary if and only is they are consectutie
angles
B)No, because opposite angles are always congruent.
C)Yes, only when opposite angles are 90°
D)Yes, because only squares have that property and squares are both parallelograms and
quadrilaterals.

Fraction ⇒ Decimal

Divide the numerator by the denominator.

(Numerator is the number on the top, denominator is the number on the bottom: \(\frac{numerator}{denominator}\))

Decimal ⇒ Percent

Multiply by 100 (OR Move the decimal 2 places to the right.)

Decimal ⇒ Fraction

Say the decimal out-loud (correctly). Write the fraction as it sounds, then simplify if needed.

Percent ⇒ Decimal

Divide by 100 (OR Move the decimal 2 places to the left)

Answer:

b = 2

Step-by-step explanation:

Factor: 0 = (b - 2)(b - 2)

b = 2

Answer:

3x2 - x + 10 when x=-4 plug in -4 for x 3(-4)^2 -(-4)+10

3(16)+4+10

3x16= 48

48+4+10= 62

Suzie's work is incorrect because of her negative distributions

Step-by-step explanation:

Answer : The integral value is: ∫∫sin(r^2) dr dθ = -π(cos(49) - cos(9))

Given:  we have 3 ≤ r ≤ 7.

The integral becomes: ∫∫sin(r^2) dr dθ

with r ranging from 3 to 7 and θ ranging from 0 to 2π.

Now we can evaluate the integral:

∫(0 to 2π) dθ ∫(3 to 7) r sin(r^2) dr

To solve the inner integral, we use substitution. Let u = r^2, so du = 2r dr.

Then: (1/2)∫(9 to 49) sin(u) du = [-(1/2)cos(u)](9 to 49) = -(1/2)[cos(49) - cos(9)]

Now, evaluate the outer integral:

∫(0 to 2π) [-(1/2)(cos(49) - cos(9))] dθ = -[(1/2)(cos(49) - cos(9))][θ](0 to 2π)

= -(1/2)(cos(49) - cos(9))(2π - 0) = -π(cos(49) - cos(9))

So the integral value is:

∫∫sin(r^2) dr dθ = -π(cos(49) - cos(9))

Answer:

B) 5.8

Step-by-step explanation:

Answer:

x=38

Step-by-step explanation:

let we say the supplimentary y then

y+83=180

y=180-83

y=97x+y=125 b/c y and x are remote angles then

97+x=125

x= 125-97

x=28

No, X² and 2x are not like terms. Like terms are terms that have the same variables and exponents, but different coefficients. X² and 2x have the same variable (x) but different exponents (2 and 1, respectively).

Like terms are terms that have the same variables and exponents, but different coefficients. X² and 2x are not like terms because they have the same variable (x) but different exponents (2 and 1, respectively). When two terms have the same variable and exponent, they are considered like terms. For example, 3x² and 4x² are like terms because they have the same variable (x) and exponent (2). However, if the terms have different variables or different exponents, they are not like terms. For example, X² and 2y are not like terms because they have different variables (x and y). Similarly, 2x and x² are not like terms because they have different exponents (1 and 2, respectively). To sum up, like terms must have the same variables and exponents in order to be considered as such.

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The unit rate or miles per gallon for Blue car is 35.6 miles and for Red car is 36 miles. Red car could travel the greater distance on 1 gallon of gasoline.

44 1/2 = 44 + 1/2

         = (88+1)/2

         = 89/2

1 1/4 = 1 + 1/4

       = (4+1)/5

       = 5/4

28 4/5 = 28 + 4/5

           = (140+4)/5

           = 144/5

Distance blue car covers in 5/4 gallons of fuel = 89/2

     Distance blue car covers in 1 gallon of fuel = (89/2) / (5/4)

                                                                             = (89*4)/(2*5)

                                                                             = 178/5  

                                                                             = 35.6

Distance red car covers in 4/5 gallons of fuel = 144/5

     Distance red car covers in 1 gallon of fuel = (144/5) / (4/5)

                                                                             = (144*5)/(4*5)

                                                                             = 144/5  

                                                                             = 36

Hence, Red car could travel the greater distance on 1 gallon of gasoline by 0.4 miles.

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Answer:

the answer is 3

90÷3=30

so

10×3=300

10×3=3.00

Answer:

Step-by-step explanation:

To calculate the amount of ice needed for this sculpture, we would apply the formula for determining the volume of a rectangular base pyramid which is expressed as

Volume = 1/3 × base area × height

Since the base is rectangular, base area = length × width = 3 × 5 = 15 m

Given that height = 3.6m, therefore, the amount of ice needed for this sculpture is

1/3 × 15 × 3.6 = 18 m³

Answer: To calculate the amount of ice needed for this sculpture, we would apply the formula for determining the volume of a rectangular base pyramid which is expressed as Volume = 1/3 × base area × heightSince the base is rectangular, base area = length × width = 3 × 5 = 15 mGiven that height = 3.6m, therefore, the amount of ice needed for this sculpture is 1/3 × 15 × 3.6 = 18 m³

Step-by-step explanation:

The completed table with regards to terms of an expression are presented as follows;

     Condition      \({}\)              (6·x + 3) + (5·x - 4)            (-4·y - 16) - 8·y + 10 + 2·y

         Exactly 3 terms          N/A     \({}\)                              N/A

         Exactly 5 terms          N/A         \({}\)                          N/A

Includes a zero pair             No \({}\)                                   No

Uses distributive property   No                                    No

Includes a negative factor   No                                    

Has no like terms                False                                False

Condition       \(8 - \dfrac{1}{2} \cdot \left(4 \cdot x - \dfrac{1}{2} + 12\cdot x -\dfrac{1}{4} \right)\)  0.25·(8·m - 12) - 0.5·(-4·m + 2)

Exactly 3 terms       No                         \({}\)                    No

Exactly 5 terms       Yes                          \({}\)   \({}\)              No

Includes a zero pair No    \({}\)                         \({}\)              Yes

Uses the distributive property Yes            \({}\)             Yes

Includes a negative factor      Yes           \({}\)                Yes

Has no like terms                    No          \({}\)                  No

A mathematical expression is a collection of variables and numbers along with mathematical operators which are all properly arranged.

The details of the conditions in the question are as follows;

Terms of an expression

A term is a subunit of an algebraic expression which are joined together by operators such as addition or subtraction

Zero pair

A zero pair are two numbers that when added together have a zero result

Distributive property

The distributive property of multiplication states that the multiplication of a number or variable by an addend is equivalent to the sum of the multiplication of the number or variable and each member of the addend

Negative factor

A negative factor is a factor that has a negative sign prefix

Like terms

Like terms are terms consisting of identical variables with the same  powers of the variable

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Answer:

645,479

Step-by-step explanation:

874,248-228,759=645,479

Answer:

Parte 1

a) {0, 1, 2, 3, 4}

b) {11, 13, 17, 19}

c) {12, 24, 36, 48, 60}

d) Conjunto A = {4, 5, 6, 7, 8, 9, 10, 11}

Conjunto B = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}

Parte 2

a) {x ∈ Z / x > 10}

{11, 12, 13, 14, 15, ....}

b) {x ∈ Z / 2x}

{2, 4, 6, 8, 10, 12, ....}

c) {x ∈ Z / 1 ≤ x ≤ 5}

{1, 2, 3, 4, 5}

d) {x ∈ N / x < 10}

{0, 1, 2, 3, 4, 5, 6, 7, 8, 9} o {1, 2, 3, 4, 5, 6, 7, 8, 9}

e) {x ∈ Z / 2x < 15}

{2, 4, 6, 8, 10, 12, 14}

Step-by-step explanation:

Parte 1

a.) El conjunto de enteros no negativos menos de cinco.

Los enteros son números enteros. Los enteros no negativos son enteros positivos y 0 (0 es un entero neutral ya que es un número entero)

{0, 1, 2, 3, 4}

b.) El conjunto de números primos entre 10 y 20.

Los números primos son números divisibles por solo 1 y ellos mismos.

{11, 13, 17, 19}

c.) El conjunto de múltiplos de 12 que son menores de 65.

Los múltiplos de 12 son números que surgen cuando 12 se multiplica por números enteros.

{12, 24, 36, 48, 60}

d.) A = {x ∈ Z / 3 < x < 12}. y. B = {x / x es un número de un dígito}.

El conjunto A se refiere a todos los números enteros entre 3 y 12 (no incluido)

Conjunto A = {4, 5, 6, 7, 8, 9, 10, 11}

El conjunto B es un conjunto de todos los números de un dígito.

Conjunto B = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}

Parte 2

a. El conjunto de enteros mayores que diez.

El conjunto variará de 11 a infinito.

Se escribirá como {x ∈ Z / x> 10}

{11, 12, 13, 14, 15, ....}

b.) El conjunto de enteros pares.

El conjunto de enteros pares comenzará desde 2 e incluirá todos los números pares hasta el infinito.

Se escribirá como {x ∈ Z / 2x}

{2, 4, 6, 8, 10, 12, ....}

c.) El conjunto {1, 2, 3, 4, 5}.

Este es un conjunto de enteros entre 1 y 5 (inclusive)

Se escribirá como {x ∈ Z / 1 ≤ x ≤ 5}

{1, 2, 3, 4, 5}

d.) El conjunto de números naturales menores de 10.

Los números naturales son números de conteo.

Se escribirá como {x ∈ N / x < 10}

Algunos casos incluyen 0 como número natural y otros no.

{0, 1, 2, 3, 4, 5, 6, 7, 8, 9} o {1, 2, 3, 4, 5, 6, 7, 8, 9}

e.) El conjunto de múltiplos de 2 que son menores que 15.

Se puede escribir como {x ∈ Z / 2x <15}

{2, 4, 6, 8, 10, 12, 14}

¡¡¡Espero que esto ayude!!!

In English

Part 1

a. The set of nonnegative integers less than five.

Integers are whole numbers. Non-negative integers are positive integers and 0 (0 is a neutral integer as it is a whole number)

{0, 1, 2, 3, 4}

b. The set of prime numbers between 10 and 20.

Prime numbers are numbers divisible by only 1 and themselves.

{11, 13, 17, 19}

c. The set of multiples of 12 that are less than 65.

Multiples of 12 are numbers that arise when 12 is multiplied by whole numbers.

{12, 24, 36, 48, 60}

d. A = {x ∈ Z / 3 < x < 12}. and. B = {x / x is a one-digit number}.

Set A refers to all integer numbers between 3 and 12 (non-inclusive)

Set A = {4, 5, 6, 7, 8, 9, 10, 11}

Set B is a set of all the one digit numbers.

Set B = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}

Part 2

a. The set of integers greater than ten.

The set will range from 11 to infinity.

It'll be written as {x ∈ Z / x > 10}

{11, 12, 13, 14, 15, ....}

b. The set of even integers.

The set of even integers will start from 2, and include all the even numbers till infinity.

It'll be written as {x ∈ Z / 2x}

{2, 4, 6, 8, 10, 12, ....}

c. The set {1, 2, 3, 4, 5}.

This is a set of integers between 1 and 5 (inclusive)

It'll be written as {x ∈ Z / 1 ≤ x ≤ 5}

{1, 2, 3, 4, 5}

d. The set of natural numbers less than 10.

Natural numbers are counting numbers.

It'll be written as {x ∈ N / x < 10}

Some cases include 0 as a natural number and some others do not.

{0, 1, 2, 3, 4, 5, 6, 7, 8, 9} or {1, 2, 3, 4, 5, 6, 7, 8, 9}

e. The set of multiples of 2 that are less than 15.

It can written as {x ∈ Z / 2x < 15}

{2, 4, 6, 8, 10, 12, 14}

Hope this Helps!!!

Given equations are:3x - 2x² = 9  ...(i)1 - x₁ - x₂ = -2 ...(ii)-5x₁ - 3x₂ = 4   ...(iii)We can write the equations in matrix form: $\begin{bmatrix}-2 & 3\\-1 & -1\\-5 & -3\end{bmatrix}\begin{bmatrix}x^2 \\x^1\end{bmatrix}=\begin{bmatrix}9 \\-2 \\4\end{bmatrix}$

Then, $A=\begin{bmatrix}-2 & 3\\-1 & -1\\-5 & -3\

end{bmatrix}$ and $b=\begin{bmatrix}9 \\-2 \\4\end{bmatrix}$Let the determinant of the coefficient matrix A be D.

Since the augmented matrix of the syste

m is: $\begin{bmatrix}-2 & 3 & 9\\-1 & -1 & -2\\-5 & -3 & 4\end{bmatrix}$

Therefore, D = $\begin{vmatrix}-2 & 3 & 9\\-1 & -1 & -2\\-5 & -3 & 4\end{vmatrix}$Now,

we calculate D as follows:\[\begin{vmatrix}-2 & 3 & 9\\-1 & -1 & -2\\-5 & -3 & 4\end{vmatrix}\]\

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To summarize, the transformations from f(x) to g(x) and h(x) can be described as follows:

g(x) = f(x) - 1: a vertical shift downward by 1 unit

h(x) = f(x - 4): a horizontal shift to the right by 4 units

What did we do?

Starting with the function f(x) = -2x + 6, let's consider the transformations to the functions g(x) = f(x) - 1 and h(x) = f(x - 4).

For g(x) = f(x) - 1, we can see that the function is shifted downward by 1 unit from f(x). This means that every y-value of f(x) will be decreased by 1 to obtain the corresponding y-value of g(x). Therefore, the graph of g(x) will be the same as the graph of f(x), but shifted down by 1 unit.

For h(x) = f(x - 4), we can see that the function is shifted to the right by 4 units from f(x). This means that the input to f(x) will be increased by 4 to obtain the corresponding input to h(x). Therefore, the graph of h(x) will be the same as the graph of f(x), but shifted to the right by 4 units.

To summarize, the transformations from f(x) to g(x) and h(x) can be described as follows:

g(x) = f(x) - 1: a vertical shift downward by 1 unit

h(x) = f(x - 4): a horizontal shift to the right by 4 units

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Answer: the third one

Step-by-step explanation:Hope this is correct

a) The average estimated time for Bill to drive to work when he leaves on time at 6:30 with no red lights and no trains to wait for is approximately -19.9166 minutes. b) The estimated coefficients of REDS and TRAINS in the regression model are 1.3353 (REDS). c) The absolute value of the calculated t-value (-1.7733) is less than the critical t-value (1.9719), we fail to reject the null hypothesis.

a) To find the average estimated time in minutes for Bill to drive to work when he leaves on time at 6:30 and there are no red lights and no trains at the crossroad to wait, we substitute the values into the regression model:

TIME = -19.9166 + 0.3692(DEPART) + 1.3353(REDS) + 2.7548(TRAINS)

Given:

DEPART = 0 (as he leaves on time at 6:30)

REDS = 0 (no red lights)

TRAINS = 0 (no trains to wait for)

Substituting these values:

TIME = -19.9166 + 0.3692(0) + 1.3353(0) + 2.7548(0)

    = -19.9166

Therefore, the average estimated time for Bill to drive to work when he leaves on time at 6:30 with no red lights and no trains to wait for is approximately -19.9166 minutes. However, it's important to note that negative values in this context may not make practical sense, so we should interpret this as Bill arriving approximately 19.92 minutes early to work.

b) The estimated coefficients of REDS and TRAINS in the regression model are:

1.3353 (REDS)

2.7548 (TRAINS)

Interpreting the coefficients:

- The coefficient of REDS (1.3353) suggests that for each additional red light, the estimated time to drive to work increases by approximately 1.3353 minutes, holding all other factors constant.

- The coefficient of TRAINS (2.7548) suggests that for each additional train Bill has to wait for at the crossing, the estimated time to drive to work increases by approximately 2.7548 minutes, holding all other factors constant.

c) To test the hypothesis that each train delays Bill by 3 minutes, we can conduct a hypothesis test.

Null hypothesis (H0): The coefficient of TRAINS is equal to 3 minutes.

Alternative hypothesis (Ha): The coefficient of TRAINS is not equal to 3 minutes.

We can use the t-test to test this hypothesis. The t-value is calculated as:

t-value = (coefficient of TRAINS - hypothesized value) / standard error of coefficient of TRAINS

Given:

Coefficient of TRAINS = 2.7548

Hypothesized value = 3

Standard error of coefficient of TRAINS = 0.1390

t-value = (2.7548 - 3) / 0.1390

       = -0.2465 / 0.1390

       ≈ -1.7733

Using a significance level of 5% (or alpha = 0.05) and looking up the critical value for a two-tailed test, the critical t-value for 230 degrees of freedom is approximately ±1.9719.

Since the absolute value of the calculated t-value (-1.7733) is less than the critical t-value (1.9719), we fail to reject the null hypothesis. This means that there is not enough evidence to conclude that each train delays Bill by 3 minutes.

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Answer: A

Step-by-step explanation:

hope this helps

Answer:

Slope- intercept form is y = -1/2x - 3

1.) 10x-5=y

2.) $85

3.) 5.25 yards of fabric.

1.) To write an equation for this problem, we need to figure out what every value means:

the variable x, represents the amount of yards of fabric you can buy

the variable y, represents the total amount of money after you found out (x) or the amount of yards of fabric.

1 yard of fabric equals $10, and we have a $5 coupon (-5)

Our equation is:

10x-5=y

This equation represents the relationship between the total cost of fabric, and the amount that you can buy.

2. In order to solve this problem we need to use our equation, one value is given which is 9.

9 represents x, the amount of yards of fabric.

Substitute 9 into x:

10(9)-5=y

90-5=y

85=y

85 represents the cost of 9 yards of fabric, including the $5 coupon.

3.) This problem is similar to #2, although it's flipped.

The value given is $47.50, which represents (y), the total amount of money after (x) was found out.

But, we need to figure out (x).

Substitute $47.50 into (y):

10(x)-5=$47.50

Next, we need to figure out x by solving the opposite.

Adding the coupon to our answer can represent 10(x).

$47.50+5=52.50

10(x)=52.50

Lastly, divide both sides of the equation by 10.

10(x) / 10=52.50 /10

x=5.25

5.25 represents the amount of yards of fabric purchased if you paid $47.50.

Answer:

n² + 12 = p/4

or

n² + 12 = p ÷ 4

Step-by-step explanation:

The number n is squared, and you have to add 12 to complete the equality. So n squared plus 12 will equal p divided by 4.

a) Contribution (objective value) = $132.14

b) The firm earns $132.14 at the optimal solution.

c) An additional hour of time available for the third machine is worth $0.14 to the firm.

d) An additional 5 hours of time available for the second machine will increase the objective value by $3.69.

The best result obtained from using computer software to solve the LP problem is: X1 = 11.43, X2 = 12.86, X3 = 5.71

b) The number of unused hours at the ideal solution is:

Machine 1 still has 8.57 hours of time left.

There are no hours left on Machine 2 at the moment.

There are still 94.29 hours left on Machine 3.

c) The shadow price of the third limitation is worth an extra hour of time available for the third machine. With the exception of increasing the right-hand side of the third constraint by one unit, we can solve the LP problem using the same constraints to determine the shadow price. Using LP to solve this issue, we discover that the shadow price for the third constraint is

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Yes, the business did better on the second campaign. Since a higher ROI or ROAS value indicates better performance, the business did better on the second campaign.

ROI stands for Return on Investment, which is a measure of profitability. It is calculated by dividing the net profit by the total investment and expressing it as a percentage.

In this case, the ROI for the first campaign was 3, indicating that the business generated three times the profit compared to the investment.

ROAS stands for Return on Advertising Spend, which is a measure of the effectiveness of advertising campaigns. It is calculated by dividing the revenue generated by the advertising campaign by the cost of that campaign.

In this case, the ROAS for the second campaign was 4, indicating that the business generated four times the revenue compared to the advertising cost.

Since a higher ROI or ROAS value indicates better performance, the business did better on the second campaign. The ROAS value of 4 is higher than the ROI value of 3, indicating that the second campaign was more effective in terms of generating revenue compared to the investment made.

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The ROI (Return on Investment) measures the profitability of an investment and is calculated by dividing the net profit by the cost of the investment.

The ROAS (Return on Advertising Spend) measures the effectiveness of an advertising campaign and is calculated by dividing the revenue generated by the advertising campaign by the cost of the campaign. Based on the information provided, it can be concluded that the business did better on the second campaign.


In this case, the business had an ROI of 3 for the first campaign and a ROAS of 4 for the second campaign.

To compare the two campaigns, we can look at the ratios.

For the first campaign, the ROI was 3, meaning that for every dollar invested, the business received 3 dollars in profit.

For the second campaign, the ROAS was 4, meaning that for every dollar spent on advertising, the business generated 4 dollars in revenue.

Comparing these two ratios, we can see that the second campaign had a higher return on advertising spend (ROAS) compared to the first campaign's return on investment (ROI). This suggests that the second campaign performed better in terms of generating revenue in relation to the advertising cost.

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Answer:

x = 36

Step-by-step explanation:

180 - 126 = 54

90 + 54 + x = 180

144 + x = 180

180 - 144 = x

x = 36

Have a nice day!

Triangle DBG is isosceles and  BD = BG.

A triangle is a closed two-dimensional geometric shape with three straight sides and three angles. It is one of the basic shapes in geometry and has a wide range of applications in mathematics, science, and engineering.

To prove that triangle DBG is isosceles, we need to show that BD = BG.

First, we can use the given similarity to find the length of DF in terms of EB and EC. Since triangle ABC is similar to triangle EDF, we have:

AB:BC = ED:DF

Substituting the given values, we get:

2:3 = ED:DF

Multiplying both sides by DF, we get:

DF = (3÷2)ED

Next, we can use the fact that triangles EDF and EBG are similar (since they share angle E) to find the length of BG in terms of EB and DF:

ED/EB = BG/DF

Substituting the value we found for DF, we get:

ED/EB = BG/(3/2)ED

Multiplying both sides by (3/2)ED, we get:

BG = (3/2)ED²/ EB

Now we can use the Pythagorean theorem to find the lengths of BD and BG in terms of EB and EC:

BD² = BE² + ED²

BG² = BE² + EG²

Since EG = EC - BD, we can substitute BD = EC - EG in the first equation to get:

BD² = BE² + ED² = BE² + (3/2)ED²

Substituting the expression we found for BG in terms of ED and EB in the second equation, we get:

BG² = BE² + (3/2)ED²/EB² * BE²

Simplifying this expression, we get:

BG² = BE²(1 + 3ED²/2EB²)

Since we know that ED/EB = 2/3, we can substitute this value to get:

BG² = BE²(1 + (3/2)(4/9)) = BE²(25/18)

Therefore, we have:

BD² = BE² + (3/2)ED² = BE² + (3/2)(9/4)BE² = (15/8)BE²

BG² = BE²(25/18)

To show that BD = BG, we can compare the squares of these lengths:

BD² = (15/8)BE²

BG² = BE²(25/18)

Multiplying both sides of the first equation by 18/25, we get:

(18/25)BD² = (27/40)BE²

Substituting the expression for BG² in the second equation, we get:

(18/25)BD² = (27/40)BG²

Therefore, we have:

BD² = (27/40)BG²

Taking the square root of both sides, we get:

BD = (3/4)√(10) * BG

Substituting the expression we found for BG in terms of ED and EB, we get:

BD = (3/4)√(10) * (3/2)ED²/EB

Substituting the value of ED/EB = 2/3, we get:

BD = (3/4)√(10) * (3/2)(4/9)ED²

Simplifying this expression, we get:

BD = (2/3)√(10)ED²

Next, we can substitute the value we found for DF in terms of ED to get:

DF = (3/2)

Therefore, triangle DBG is isosceles and  BD = BG.

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The value of k is 1, the volume of the parallelepiped is 12 + 4λ, and the vector component of a + c orthogonal to c is (1,1,1.5).

a) Here the area of the parallelogram determined by d and e is given as 10. The area of the parallelogram is given as `|d×e|`.

We have,

d=(2,4)

and e=(4k,3k)

Then,

d×e= (2 * 3k) - (4 * 4k) = -10k

Area of parallelogram = |d×e|

= |-10k|

= 10

As we know, area of parallelogram can also be given as,

|d×e| = |d||e| sin θ

where, θ is the angle between the two vectors.

Then,10 = √(2^2 + 4^2) * √((4k)^2 + (3k)^2) sin θ

⇒ 10 = √20 √25k^2 sin θ

⇒ 10 = 10k sin θ

∴ k sin θ = 1

Therefore, sin θ = 1/k

Hence, the value of k is 1.

Part(b) The volume of the parallelepiped determined by vectors a, b and c is given as,

| a . (b × c)|

Here, a=(1,1,2),

b=(5,3,λ), and

c=(4,4,0)

Therefore,

b × c = [(3 × 0) - (λ × 4)]i + [(λ × 4) - (5 × 0)]j + [(5 × 4) - (3 × 4)]k

= -4i + 4λj + 8k

Now,| a . (b × c)|=| (1,1,2) .

(-4,4λ,8) |=| (-4 + 4λ + 16) |

=| 12 + 4λ |

Therefore, the volume of the parallelepiped is 12 + 4λ.

Part(c) The vector component of a + c orthogonal to c is given by [(a+c) - projc(a+c)].

Here, a=(1,1,2) and

c=(4,4,0).

Then, a + c = (1+4, 1+4, 2+0)

= (5, 5, 2)

Now, projecting (a+c) onto c, we get,

projc(a+c) = [(a+c).c / |c|^2] c

= [(5×4 + 5×4) / (4^2 + 4^2)] (4,4,0)

= (4,4,0.5)

Therefore, [(a+c) - projc(a+c)] = (5,5,2) - (4,4,0.5)

= (1,1,1.5)

Therefore, the vector component of a + c orthogonal to c is (1,1,1.5).

Conclusion: The value of k is 1, the volume of the parallelepiped is 12 + 4λ, and the vector component of a + c orthogonal to c is (1,1,1.5).

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Answer: B.pi/2

ABCD is a square

=> Δ ABD is a  isosceles right triangle at A

using pythago theorem, we have:

AB² + AD² = BD²

=> 2AD² = 4

⇔ AD² = 2

⇒ AD = √2

through O, draw EF parallel to AD and BC

=> EF is  diameter

because EF//AD => EF = AD = √2 (because circle O is inscribed the square ABCD)

=> the area of the circle is \(S=\frac{(\sqrt{2})^{2} }{4}.\pi =\frac{\pi }{2}\)

Step-by-step explanation:

Answer:

\(\sqrt{260\\\) or 16.12

Step-by-step explanation:

distance formula is \(} d = \sqrt{(x_2 - x_1)^2 + (y_2-y_1)^2}\)

Distance (d) = √(9 - 1)2 + (9 - -5)2

= √(8)2 + (14)2

= √260

= 2√65

= 16.124515496597