Solve the following triangles:

A = 60 degree, a = 4, b = 5

the third term of the expansion of (11x+3y)⁶ is 798,720x³y³.we can use the binomial theorem to solve this .

what is  binomial theorem ?

The binomial theorem is a theorem in algebra that describes the algebraic expansion of powers of a binomial. A binomial is a polynomial with two terms, such as (a + b) or (x - y).

In the given question,

To find the third term of the expansion of (11x+3y)⁶, we can use the binomial theorem, which states that:

(a + b)ⁿ = C(n,0)aⁿb⁰ + C(n,1)*aⁿ⁻¹*b¹+ C(n,2)*aⁿ⁻¹*b² + ... + C(n,n)a⁰bⁿ

where C(n,k) denotes the binomial coefficient, which is the number of ways to choose k items from a set of n items.

In this case, we have a = 11x and b = 3y, and n = 6. We want to find the third term, which corresponds to k = 3 in the expansion. Therefore, we need to compute the binomial coefficient C(6,3), as well as the powers of 11x and 3y that appear in the third term.

The binomial coefficient C(6,3) can be computed as follows:

C(6,3) = 6! / (3! * (6-3)!) = 20

To find the powers of 11x and 3y that appear in the third term, we need to look at the general pattern in the expansion. Each term in the expansion consists of a product of powers of a and b, with the powers of a decreasing by 1 from n to 0, and the powers of b increasing by 1 from 0 to n. Therefore, the kth term in the expansion will have powers of a and b given by aⁿ⁻ᵇand bᵃ, respectively.

In the third term, we have k = 3, so we need to find the coefficient of a³ = a³ = (11x)³ and b³ = (3y)³. The third term is given by:

C(6,3)(11x)³(3y)³

Plugging in the values for C(6,3), (11x)³, and (3y)³, we get:

20*(11x)³*(3y)³ = 2011³x³3³y³ = 798,720x³y³

Therefore, the third term of the expansion of (11x+3y)⁶ is 798,720x³y³.

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a) Jill receives £45

b) Jack receives £20 the following Christmas

c) Jack will be 5/4 years old, which is equivalent to 1 year and 3 months when he receives £25 from Uncle Fred.

Explain the term years

Years are a unit of time measurement used to quantify the duration of one complete orbit of the Earth around the sun. A year consists of 365 days, except in a leap year when an extra day is added. The concept of a year is used in various fields, including astronomy, geology, and history.

According to the given information

(a) Let's first find the total ratio of their ages:

1 + 3 = 4

So Jack's share will be:

1/4 x £60 = £15

And Jill's share will be:

3/4 x £60 = £45

Therefore, Jill receives £45.

(b) The next Christmas, Jack will be 2 and Jill will be 4. We can use the same ratio to find Jack's share:

2/6 x £60 = £20

So Jack receives £20.

(c) Let x be the age of Jack when he receives £25 from Uncle Fred. We know that:

1/4 x £60 = £25

Multiplying both sides by 4/60, we get:x = 5/4

So Jack will be 5/4 years old, which is equivalent to 1 year and 3 months, when he receives £25 from Uncle Fred.

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Answer:

According to the Pythagorean Theorem, in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. So we have:

c^2 = a^2 + b^2

where c is the length of the hypotenuse and a and b are the lengths of the other two sides.

Substituting the given values, we get:

c^2 = 14^2 + 48^2

c^2 = 196 + 2304

c^2 = 2500

Taking the square root of both sides, we get:

c = 50

Therefore, the length of the hypotenuse is 50 units.

Answer:

[tex]\frac{sin(A)}{a}[/tex]

The mean of the data set is 25, the median is 25, the range is 20, the variance is 20, and the standard deviation is 4.47.

Given the dataset:

x = 30, 20, 35, 25, 15

N = 5

Mean of the data set is the average of the dataset. To find the average, we add up all the numbers and divide by the total number of values:

Mean (Xbar) = (30 + 20 + 35 + 25 + 15) / 5 = 25

Median is the number that fall at the middle of the dataset. To find the median of the data set, we need to arrange the numbers in order from smallest to largest:

15, 20, 25, 30, 35

The median is the middle number, which is 25 in this case.

Range of the data set can be calculated by subtract the smallest value from the largest value:

Range = 35 - 15 = 20

Standard Deviation and Variance

First find the deviations of each value from the mean:

(x - Xbar) = d

30 - 25 = 5

20 - 25 = -5

35 - 25 = 10

25 - 25 = 0

15 - 25 = -10

To find the variance, we square each deviation, add them up, and divide by the number of values:

Variance = d²/N

               = (5² + (-5)² + 10² + 0² + (-10)²) / 5

               = 100/5 = 20

To find the standard deviation, we take the square root of the variance:

Standard Deviation = sqrt(variance)

                                 = sqrt(20)

                                 = 4.47 (rounded to two decimal places)

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Answer: A. 251.3 in.

Step-by-step explanation:

        We can use this formula to solve for the circumference of a circle. r is equal to the radius, or half of the circle's width.

                 C = 2πr

C = 2πr

C = 2π(40)

C ≈ 251.3

                 A. 251.3 in.

 The third player hit 4 more home runs than the first player. The algebraic expression that shows the relationship between a and b is:  a = 3b. So option A is correct

A linear equation in one variable is an equation that has only one variable. It is of the form Ax B = 0, where A and B are  two real numbers and x is an unknown variable with only one solution. For example, 9x + 78 = 18 is a linear equation in one variable.

 2) Each player's home runs are labeled A, B, C, and D, where A is the first baseman, B is the second baseman, C is the third baseman, and D is the shortstop. We know this:

C = 2B (the third player hit twice as many home runs as the second player)

C/B = D/B = A/C (numbers for first baseman and shortstop are proportional to numbers for third baseman and second baseman)

B = D 4 (second baseman hit 4 more home runs than  shortstop)

To solve for the values ​​of A, B, C, and D, we can use the second equation to write:

C/B = D/B

C = D (multiply both sides by B)

C = B - 4 (using the fourth equation to substitute  D)

2B = B - 4 (using the first equation to substitute  C)

B = 4 (interface to both sides 4)

C = 8 (using the first equation to find C)

D = 4 (using the fourth equation to find D)

A = C/2 = 4 (use another equation to find A)

Therefore the third player (C) hit 8 home runs and the first player (A) hit 4 home runs, so the third player hit 4 more home runs than the first player.

3) The relative relationship between a and b can be expressed as follows:

a/b = 6/2 = 3/1

To write this relationship as an algebraic equation, we can cross multiply to get:

a = 3b

Therefore, the algebraic expression that shows the relationship between a and b is:

a = 3b. So option A is correct

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Answer: 6 units

Step-by-step explanation: To find the mass of the lamina, we need to integrate the density function fi(x,y,z) over the volume of the lamina. The lamina is a portion of the cone z=sqrt(x^2+y^2) between the planes z=1 and z=3. We can express this volume using cylindrical coordinates, where rho is the distance from the origin to the point (x,y,z), phi is the angle between the positive x-axis and the line connecting the origin to the point (x,y,z), and z is the height of the point (x,y,z) above the xy-plane.

We have:

1 <= z <= 3

0 <= rho <= 3sin(phi)

0 <= phi <= 2pi

The density function is given by fi(x,y,z) = x^2z. In cylindrical coordinates, we have x = rhocos(phi), y = rhosin(phi), and z = z. Therefore, we can express the density function as:

fi(rho, phi, z) = (rhocos(phi))^2 * z = rho^2cos^2(phi) * z

The mass of the lamina is given by the triple integral of the density function over the volume of the lamina:

M = ∭ fi(rho, phi, z) dV

= ∫∫∫ rho^2cos^2(phi) * z dz dA

= ∫∫[1,3] ∫[0,2pi] ∫[0,3sin(phi)] rho^2cos^2(phi) * z dz d(phi) drho

= ∫[0,2pi] ∫[0,3] ∫[0,rhosin(phi)] rho^2cos^2(phi) * z dz drho d(phi)

= ∫[0,2pi] ∫[0,3] (1/3)rho^3sin(phi)cos^2(phi) d(phi) drho

= ∫[0,2pi] (1/3)[9sin(phi)cos^2(phi)] d(phi)

= (1/3)[9(2/3)]

= 6

Therefore, the mass of the lamina is 6 units.

Programme 2 in Los Angeles will be well received by 22 out of 25 people.

Given that the chances of a positive response from Los Angeles residents to two government programmes are 78.4% for Programme 1 and 88.1% for Programme 2, the likelihood of a positive response for Programme 2 given that the person is from Los Angeles must be calculated as follows:

88.1 / 100 = X 44.05 / 50 = X 22,025 / 25 = X

Therefore, Programme 2 will be well-received by 22 out of 25 people in Los Angeles.

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The given exponents 9⁻³/9¹² in the form 9ⁿ is 9⁻¹⁵ respectively.

A number's power or exponent, in other words, tells how many times it must be multiplied by itself.

Any integer, fraction, or decimal can serve as the basis in this situation.

Additionally, the exponent might have either a positive or negative value.

Subtract the exponents to divide exponents (or powers) with the same base.

Given that division is the inverse operation of multiplication, it stands to reason that, just as exponents are added when multiplying numbers with the same base, exponents are subtracted when dividing numbers with the same base.

So, we have the exponents as;
9⁻³/9¹²

Then, solve as follows:

9⁻³/9¹²

9⁻³⁻¹²

9⁻¹⁵

Therefore, the given exponents 9⁻³/9¹² in the form 9ⁿ is 9⁻¹⁵ respectively.

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The percentage of Kira's free throws is 68%.

We have,

From the grid,

We can see that,

There are 25 boxes.

Now,

The number of shaded boxes = 17

Now,

The percentage of the shaded boxes.

= 17/25 x 100

= 17 x 4

= 68%

Thus,

The percentage of Kira's free throws is 68%.

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The complete question.

The shaded area on the grid represents the part of Kira's free throws that she made in the basketball practice today.

Find the percentage of Kira's free throws.

Answer:

Vertically opposite angles are always same.

Straight angle=180°

Angle MQN=70°

The expressions that are equivalent to 6x²y + 2xy⁵; are 2(3x²y + xy⁵), 2xy(3x + y⁴), xy(6x + 2y⁴) and 4xy(3/2x + 1/2y⁴). So, the correct options are a, b, e and g.

Algebra is known to be a branch of mathematics in which problems are represented as mathematical expressions using letters or variables (x, y, z, etc.) to represent unknown values ​​that may change. The purpose of algebra is to understand what the unknown values ​​are in order to find solutions to problems. The degree of an algebraic equation is the largest power that exists for the variables in the equation. Algebraic equations can be classified by degree as follows:

Equation A: 2(3x²y + xy⁵) = 6x²y + 2xy⁵

Equation B: 2xy(3x + y⁴) = 6x²y + 2xy⁵

Equation C: 2x²y(6 + 5y) = 12x²y + 10x²y²

Equation D: 6x²y.2xy⁵

Equation E: xy(6x + 2y⁴) = 6x²y + 2xy⁵

Equation F: 2y(3x + xy⁴)⁰ = 2y

Equation G: 4xy(3/2x + 1/2y⁴) = 6xy + 2xy⁵

Hence, Equation A, B, E and G are the equations found equivalent to the given equation, 6x²y + 2xy⁵.

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Answer: 228[tex]cm^2[/tex]

Step-by-step explanation:

The lateral area is 2lh + 2wh

= (2 * 14 * 6) + (2 * 5 * 6)

= 228[tex]cm^2[/tex]

Answer:

(-8.89, -26.69)

Step-by-step explanation:

To find the point with coordinates of the form (a, 3a) that is in the third quadrant and is a distance 5 from P(2,1), we can use the following steps:

[tex]\sqrt{(2 - (-|a|))^2 + (1 - (-3|a|))^2} = 5\\\sqrt{(2 + |a|)^2 + (1 + 3|a|)^2} = 5[/tex]

[tex]a^2 + 8a - 8 = 0[/tex]

[tex]a = \frac{-8 ±\sqrt{8^2 - 4(1)(-8)}} {(2(1))}\\a = \frac{-8 ± \sqrt{96}}{ 2}\\a = -4 ± 2\sqrt{6}[/tex]

[tex]a = -4 - 2\sqrt(6)[/tex]

(x, y) =[tex](-|-4 - 2\sqrt(6)|, -3|-4 - 2\sqrt(6)|)[/tex]

(x, y) ≈ (-8.89, -26.69)

Therefore, the point with coordinates of the form (a, 3a) that is in the third quadrant and is a distance 5 from P(2,1) is approximately (-8.89, -26.69).

The statement that is false, given the table of Ms Stewart's class would be A. 15 % of her students are in biology.

The table shows that out of all he students, Ms. Stewart has 35 % that are doing Biology and not 15 %. But, she does have 15 % of Freshmen doing Biology and not of all her students.

30 % of the students are indeed engaged in doing physical science, and 35 % of her students are in chemistry from the table. 45 % of the total cohort that Ms. Stewart teaches are freshmen.

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a) The break-even point for the division is: 4000 units

b) The level of sales for 10% profit is: 4681 units

The break-even point is defined as the point at which total cost and total revenue are equal, meaning there is no loss or gain for your small business. In other words, you've reached the level of production at which the costs of production equals the revenues for a product.

a) We are told that:

Selling price for income tax app = $8

Monthly fixed cost = $20000

Variable cost producing each app = $3

Thus:

8x = 3x + 20000

5x = 20000

x = 4000 units

b) 8x = 1.1(3x + 20000)

8x = 3.3x + 22000

4.7x = 22000

47x =220000

x = 4681 units

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Missing questions are:

(a) Find the break-even point for the division.

(x,y)=

(b) What should be the level of sales in order for the division to realize a 10% profit over the cost of making the income tax apps? (Round your answer up to the nearest whole number.)

To determine if (2, 6) is a solution to the system of equations:

y = 3/2x + 3 ... (equation 1)

y = -9x + 1 ... (equation 2)

We need to substitute x = 2 and y = 6 into both equations and check if they are true.

Substituting x = 2 and y = 6 into equation 1, we get:

6 = 3/2(2) + 3

6 = 3 + 3

6 = 6

This equation is true.

Substituting x = 2 and y = 6 into equation 2, we get:

6 = -9(2) + 1

6 = -18 + 1

6 = -17

This equation is false.

Since (2, 6) does not satisfy equation 2, it is not a solution to the system of equations. Therefore, the answer is: no.

Answer:

To translate triangle ABC 3 units up, we need to add 3 to the y-coordinate of each vertex:

A = (-3, 3 + 3) = (-3, 6)

B = (0, 7 + 3) = (0, 10)

C = (-3, 0 + 3) = (-3, 3)

Therefore, the coordinates of the vertices for the image triangle A'B'C' are A(-3, 6), B'(0, 10), and C(-3, 3).

So the correct answer is: A(−3, 6), B(0, 10), C(−3, 3).

Since p and q are real numbers, the value of q is 21.

In Mathematics and Geometry, a quadratic equation can be defined as a mathematical expression that can be used to define and represent the relationship that exists between two or more variable on a graph.

In Mathematics, the standard form of a quadratic equation is represented by the following equation;

ax² + bx + c = 0

Next, we would solve the given quadratic equation by using the completing the square method;

x² - 8x - 5 = 0

x² - 8x = 5

In order to complete the square, we would have to add (half the coefficient of the x-term)² to both sides of the quadratic equation as follows:

x² - 8x + (-8/2)² = 5 + (-8/2)²

x² - 8x + 16 = 5 + 16

x² - 8x + 16 = 21

By simplifying, we have;

(x - 4)² = 21

p = 4 and q = 21

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The solution to the system of equations is:

x = -19/5, y = 2, z = 1/10

4x + 6y - 2z = 2 (multiply first equation by 2)

x + 2y - 3z = -5 (third equation)

Add the two equations, we get:

5x + 8y = -3 (eliminated z)

-7x + 7y - 6z = -33 (multiplying second equation by 3 and subtracting from the first)

3x + y + 2z = 12 (second equation)

Add these two equations, we have:

-4x + 8y = -21 (eliminated z)

5x + 8y = -3

5x = -8y - 3

x = (-8y - 3)/5

Substituting this expression for x into the second equation, we get:

3((-8y - 3)/5) + y + 2z = 12

z = (39/10) - (19/10)y

Finally, substituting these expressions for x and z into any of the original equations (let's use the first one), we can solve for y:

2((-8y - 3)/5) + 3y - ((39/10) - (19/10)y) = 1

Simplifying and solving for y, we get:

y = 2

x = (-8(2) - 3)/5 = -19/5

z = (39/10) - (19/10)(2) = 1/10

In conclusion, the solution to the system of equations is:

x = -19/5, y = 2, z = 1/10

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The solution to the given system of linear equations is

x = 3, y = -1 and z = 2

From the question, we are to solve the given system of linear equations.

The given system of linear equations is

2x + 3y − z = 1

3x + y + 2z = 12

x + 2y − 3z = −5

Multiply the third equation by 2

x + 2y − 3z = −5 ) ×2

2x + 4y - 6z = -10

Subtract the first equation from the resulting equation

  2x + 4y - 6z = -10

- (2x + 3y − z = 1

----------------------------------------

y - 5z = -11     ---------- (1)

Now,

Multiply the third equation by 3

x + 2y − 3z = −5 ) ×3

3x + 6y -9z = -15

Subtract the second equation from the resulting equation

 3x + 6y -9z = -15

- (3x + y + 2z = 12

-----------------------------------------

5y - 11z = -27     ----------- (2)

Solve equations (1) and (2) simultaneously

y - 5z = -11         ------------ (1)

5y - 11z = -27     ------------ (2)

Multiply equation (1) by 5

y - 5z = -11  ) ×5

5y - 25z = -55

Subtract the resulting equation from equation (2)

  5y - 11z = -27

-( 5y - 25z = -55

-----------------------------

14z = 28

Divide both sides by 14

14z/14 = 28/14

z = 2

Substitute the value of z into equation (1)

y - 5z = -11

y - 5(2) = -11

y - 10 = -11

Add 10 to both sides of the equation

y - 10 + 10 = -11 + 10

y = -1

Substitute the value of y and z into the third equation

x + 2y − 3z = −5

x + 2(-1) − 3(2) = −5

x - 2 - 6 = -5

x - 8 = -5

Add 8 to both sides of the equation

x - 8 + 8 = -5 + 8

x = 3

Hence,

The solution is

x = 3, y = -1 and z = 2

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The result of adding -3 (row 1) to row 2 is determined as (0  10) |-14.

The result of the row multiplication in the matrix is calculated by applying the following method;

row 1 in the given matrix = [1  -4] | 8

To multiply row1 by -3, we will multiply each entity by -3 as shown below;

= -3(1  -4) | 8

= (-3  12) | -24

To add the result to 3;

(-3  12) | -24  +  (3   -2)|10

= (0  10) |-14

Thus, the result of the row multiplication is determined by multiplying each entry in row 1, by - 3.

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The best percentage bar graph that represents Camryn's data is:

C Winter | Spring | Summer | Fall

0% 20% 40% 60% 80% 100%

A bar graph's purpose is to visually convey relational information quickly. The bars show the value for a specific type of data.

This is because the graph correctly shows the percentage of students who prefer each season, with the bar for each season extending to the appropriate percentage mark on the graph. It also clearly labels each season under the corresponding bar.

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Answer: 1.75

Step-by-step explanation:

Answer:

6

Step-by-step explanation:

Since the garden bed looks like the shape of a trapezoid the equation to find the area of a trapezoid is A = 1/2h (base 1 + base 2)

The area is already given so you can just plug it in. Also you can plug in the bases.

21 = 1/2h (3 + 4)

21 = 1/2h (7)  rewrite

21 = 1/2(7)h

21 = 3.5h  Divide both sides by 3.5

6 = h

the total number of boxes with at least 20 but fewer than 60 broken crayons is  33.To answer this question, we need to use the stem-and-leaf plot to count the number of boxes with at least 20

what is at least ?

"At least" is a phrase that is used to indicate a minimum quantity or degree of something. It means that the stated quantity or degree is the smallest amount that is acceptable or possible. For example, if someone says "I need at least 10 dollars,"

In the given question,

To answer this question, we need to use the stem-and-leaf plot to count the number of boxes with at least 20 but fewer than 60 broken crayons.

From the stem-and-leaf plot, we can see that the stems 2, 3, 4, and 5 have leaves that add up to at least 20 but fewer than 60:

Stem 2 has leaves 3 and 8, which add up to 11.

Stem 3 has leaf 3, which adds up to 3.

Stem 4 has leaves 0 and 9, which add up to 9.

Stem 5 has leaves 1 and 9, which add up to 10.

Therefore, the total number of boxes with at least 20 but fewer than 60 broken crayons is 11 + 3 + 9 + 10 = 33.

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The given input-output relation is a nonlinear function of the input signal x(t), since it involves squaring the signal at every point in time. A linear system must satisfy the property of superposition, which states that the output to a linear combination of inputs must be the same as the linear combination of the outputs to each individual input.

To determine whether the system is time-invariant, we need to check if a time shift in the input signal produces the same time shift in the output signal. That is, if: [tex]y(t) = T{x(t)}[/tex] for all values of t. Then we want to check [tex]y(t - τ) = T{x(t - τ)}[/tex] for all values of [tex]τ[/tex] and [tex]t[/tex].

We shall the consider the case where [tex]τ > 0[/tex], then we have:

[tex]y(t - τ) = (x(t - τ))^2[/tex]

[tex]T{x(t - τ)} = (x(t - τ))^2[/tex]

Since [tex]y(t - τ) = T{x(t - τ)}[/tex], we can conclude that the given system is time-invariant.

Therefore, the given system is nonlinear but time-invariant.

Johannes will need to pay $8.61 for 5 pounds of ground beef

Since Johannes needs 5 pounds of ground beef, and the available pounds are 1.5, 2, 3, and 3.75 pounds.

To reduce the cost, he need to select the package with the lowest cost per pound.

The cost per pound for each package is:

Package 1: $5.58 / 1.5

              = $3.72 per pound

Package 2: $7.44 / 2

              = $3.72 per pound

Package 3: $1.16 / 3

              = $0.39 per pound

Package 4: $13.95 / 3.75

                = $3.72 per pound

Therefore, the package with the lowest cost per pound is Package 3 and package 2  which will give us 5 pounds

So he can do it as:

$1.16 + $7.44

= $8.6

Or

(1 package of 3 pounds x $0.39 per pound) + (1 package of 2 pounds x $3.72 per pound)

= $1.17 + $7.44

= $8.61

Therefore, Johannes will need to pay $8.61 for 5 pounds of ground beef.

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I hope this helps you.

The most precise description of quadrilateral ABCD is a square.

We have,

Quadrilateral ABCD.

We see that,

AB is parallel to CD

AB = CD _____(1)

AD is parallel to BC.

AD = BC _____(2)

Now,

AB = AD ______(30

From (1), (2), and (3).

AB = BC = CD = AD

This means,

The Quadrilateral ABCD is a square.

Thus,

The most precise description of quadrilateral ABCD is a square.

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From the circle graph of the survey, 52 more people prefer pigs than chickens.

A sort of data visualisation tool used to depict data as a circular chart is a circle graph, commonly referred to as a pie chart. Each slice of the circle graph, also known as a sector, represents a certain category or chunk of the entire. Each slice's size is proportional to how much data or what percentage it represents, and the sum of all slices is 100%.

Circle graphs are frequently used to depict percentages or categorical data, and they are especially helpful for comparing and illustrating relative proportions among several categories within a data collection.

From the circle graph we see that the percentage of people who prefer chickens are 29%.

Thus, 29/100 (400) = 116 people

Also, 16% people prefer pig:

Thus, 16/100 (400) = 64 people

Thus, the difference is:

116 - 64 = 52

Hence, from the circle graph of the survey, 52 more people prefer pigs than chickens.

Learn more about circle graph here:

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