3. [1 points] a random variable x ∼ n(μ,σ2) is gaussian distributed with mean μ and variance σ2. given thatforanya,b∈r,wehavethaty =ax bisalsogaussian,finda,bsuchthaty ∼n(0,1).

The area of the rectangle is 72 cm^2.

Let's call the side length of each square s.

Since the perimeter of each square is 24 cm, then each side measures 24 cm / 4 = 6 cm.

So the length of the rectangle is 2s = 2 * 6 cm = 12 cm.

And the width of the rectangle is s = 6 cm.

Therefore, the area of the rectangle is A = l * w = 12 cm * 6 cm = 72 cm^2.

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The area of the rectangle is 72 square centimeters.

The rectangle is formed by placing two identical squares side by side the perimeter of each square is 24 cm.

Remember that for a square of sidelength S, the perimeter is P = 4*S

Then the sidelength of the squares is

24cm = 4*S

24cm/4 = 6cm = S

When we place these two squares together, we will have a rectangle of 6cm by 12cm, and the area is the product of the dimensions, then the area is:

A = 6cm*12cm = 72cm²

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Answer: [tex]\sqrt{585}[/tex] units or [tex]3\sqrt{65}[/tex] units

Step-by-step explanation:

The formula for volume of a cone is π[tex]r^{2}[/tex][tex]\frac{h}{3}[/tex]. You can find the height by plugging in 72π for volume and 3 for radius.

diameter = radius x 2, so 6/2=3 units for radius.

72π = π([tex]3^{2}[/tex])([tex]\frac{h}{3}[/tex])

72=9*[tex]\frac{h}{3}[/tex]

8 = [tex]\frac{h}{3}[/tex]

h = 24 units

The formula for slant height is [tex]\sqrt{{r}^2 + {h}^2 }[/tex]

Plug in 3 for r and 24 for h. You should get [tex]\sqrt{(9+576)}[/tex]=[tex]\sqrt{585}[/tex] or 3[tex]\sqrt{65}[/tex] as your answer.

The two equations are y = - 1 and y = (1/5)x - 12/5 and the solution to the system is (7, - 1).

We know lines have various types of equations, the general type is

Ax + By + c = 0, and the equation of a line in slope-intercept form is

y = mx + b.

Where slope = m and b = y-intercept.

the slope is the rate of change of the y-axis with respect to the x-axis and the y-intercept is the (0,b) where the line intersects the y-axis at x = 0.

The first line passes through (- 1, 0) and (2, - 1).

Slope(m) = (- 1 + 1)/(2 - 0).

Slope(m) = 0.

- 1 = 0(2) + b.

b = - 1.

y = - 1.

A horizontal line at y = - 1.

The second line passes through (4, - 2) and (- 3, - 3).

Slope(m) = (- 3 + 2)/(- 3 - 4).

Slope(m) = - 1/- 5.

Slope(m) = 1/5.

- 3 = (1/5)(- 3) + b.

b = 3/5 - 3.

b = (3 - 15)/5.

b = - 12/5.

y = (1/5)x - 12/5.

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To convert cups to pounds, you can use the following conversion factor   1 cup = 0.5 pounds

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The total number of arrangements possible if A and B must not sit together:

=  3600

Now, According to the question:

The no. of circular arrangements of n distinct items = (n-1)!

If A,B,C,D,E,F,G and H are the 8 persons to be seated around the table and A,B the 2

particular persons who must not sit together.

The total no.of circular arrangements for 8 persons = (8-1)! = 7! = 5040 --(1)

The no. of circular arrangements possible if A and B were to sit together = 6! (considering A and B as a single pair).

But, A and B can interchange their positions.

Hence, the no. of arrangements =6!×2=720×2=1440−−−−>(2)

So, the total number of arrangements possible if A and B must not sit together, = (1) - (2)

= 5040-1440

= 3600

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There are 640 students are expected to choose chocolate as their favorite ice cream.

How many students are expected to choose chocolate as their favorite ice cream flavor if there are 700 students in the school?

Total number of student =700

no of students are expected to choose chocolate as their favorite ice cream be x

school are randomly selected and asked about their favorite ice cream flavor =60

60+x=700

x=700-60=640

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There are 55 senators and 235 representatives are in this state's legislature.

An equation is a mathematical statement that shows the equality of two expressions. Equations can be used to represent relationships between variables and to solve problems. They are written using an equal sign (=) and can include numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division.

For example, the equation 2x + 3 = 7 is an equation that states that the expression on the left side (2x + 3) is equal to the expression on the right side (7). The goal is to find the value of the variable x that satisfies the equation. In this case, x = 2 is a solution to the equation.

Let's call the number of senators in the state legislature "s" and the number of representatives "r". We know the following:

s + r = 290 (the combined number of senators and representatives is 290)

r - s = 180 (the difference of the numbers is 180)

We can use the first equation to solve for one of the variables in terms of the other:

r = 290 - s

Next, we can substitute this expression for r into the second equation:

290 - s - s = 180

Simplifying the left-hand side:

290 - 2s = 180

Adding 2s to both sides:

290 = 180 + 2s

Subtracting 180 from both sides:

110 = 2s

Dividing both sides by 2:

s = 55

So there are 55 senators in the state legislature. To find the number of representatives, we can use the first equation:

r = 290 - s = 290 - 55 = 235

So there are 235 representatives in the state legislature.

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The angle relations in a cyclic quadrilateral are discussed above.

Given is that ABCD is a cyclic quadrilateral.

A cyclic quadrilateral is a four sided shape that can be inscribed into a circle.

Each vertex of the quadrilateral lies on the circumference of the circle and is connected by four chords.

The opposite angles of a cyclic quadrilateral have a total of 180°.

For a cyclic quadrilateral with successive sides a, b, c, d, semi - perimeter s, and angle A between sides a and d, the angle relations for a cyclic quadrilateral as -

        [tex]$\tan {\frac {\theta }{2}}={\sqrt {\frac {(s-b)(s-d)}{(s-a)(s-c)}}}.[/tex]

       [tex]${\displaystyle \cos {\frac {\varphi }{2}}={\sqrt {\frac {(s-b)(s-d)(b+d)^{2}}{(ab+cd)(ad+bc)}}}}[/tex]

Therefore, the angle relations in a cyclic quadrilateral are discussed above.

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Answer:

Below

Step-by-step explanation:

See diagram below   solve the angles in order 1-2-3-4-5-6

as shown in the diagram:

Answer:

85 degrees

Step-by-step explanation:

Okay, let's have a look at the triangle on the left.
We can see an exterior angle of 115 degrees, adjacent to the interior angle.
Thus, as they both form a line (180 degrees), Interior angle = 180 - 115
= 65 degrees
We know that all the angles of the triangle equal 180.
Now to find the remaining angle of the triangle on the left,
180 - (85 + 65) = 30 degrees

Moving towards the center, we have our interior triangle angle (30 degree), a 90 degree angle and an interior angle of the triangle on the right. So, as they all form a line, the interior angle of the triangle on the right will be
180 - (90 + 30) = 60 degrees

Also, on top of the triangle on the right, we have a 115 degree exterior angle again.
Repeating our first step, the interior angle would be 65 degree

Now to find the remaining angle of the triangle on the right,
180 - (30 + 65) = 95 degrees

This angle forms a line with another angle exterior to this one, so again,
180 - 95 = 85 degrees

Hope this helps

The probability that a randomly selected female high school basketball player will be drafted by a WNBA team is 0.000351 or approximately 0.04%.

The probability of a female high school basketball player being drafted into the WNBA can be taken as a combination of two probabilities.

The probability that she will play basketball in the NCAA, which is equal to 3.9%, and the probability that she will be drafted into the WNBA given that she plays in the NCAA, which is equal to 0.9%, are both equal.

The probability can be calculated by using the formula:

P(WNBA draft) = P(NCAA) × P(WNBA draft | NCAA)

Where, P(NCAA) = 3.9 ÷ 100 which is equal to 0.039
and, P(WNBA draft | NCAA) = 0.9 ÷ 100 which is equal to 0.009

Hence, by substituting the values in the equation we get

P(WNBA draft) = 0.039 × 0.009

Which is equal to 0.000351 or approximately 0.04%

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The Sun's diameter is 864,000 miles. 864,000 x 108 = 93,000,000 kilometers separate the earth from the sun. This is equivalent to 109 times the size of Earth.

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Option is not mentioned, The ratio of the diameter of the earth to the sun is di = 1.8 × 10-3m.

The Sun is 1,391,000 kilometres (864,400 miles) across. It is the same as Earth being 109 times its size. The Sun's mass is roughly 333,000 times that of the Earth.

The Sun's diameter is equal to the diameter of 108 Earths. The Sun is 864,000 kilometres across. The earth is 93,000,000 kilometres from the sun, or 864,000 times 108 kilometres. It is the same as Earth being 109 times its size.

The Sun is 864,000 kilometres across. The earth is 93,000,000 kilometres from the sun, or 864,000 times 108 kilometres. It is the same as Earth being 109 times its size.

The Sun is 1,391,000 kilometres (864,400 miles) across. This is the same as 109 times the size of Earth. .

The angle of reflection is θ.

D is the distance between the mirror's centre and the sun's centre, where d is the sun's diameter. d/D the equation is equal to 0.009

Angle of incidence = angle of reflection according to the rules of reflection. Consequently, θ2 = θ, = θ, d D The ratio tan θ;=0.009 tan θ =0.009 is another option.

We will assume that because the sun is far away from the mirror, the image will form at the mirror's focus. Therefore, distance OB = f, where f is the concave mirror's focal length

The focal length of a spherical concave mirror is equal to half of its radius of curvature. Given that the radius of the curveturer is 0.4 m, f= 42 = 0.4 m 2 = 0.2 m.

Taking into account the image's diameter as d2, tan 0, di f tan 0 di 0.2 = Taking the value of tan from above, 0.009 di 0.2 = di 0.009 x 0.2 = As a result, the image's diameter is d; = 0.0018m di = 1.8 × 10-3m.

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[tex]y = 4-2x , y = 0 , x = 0[/tex] The region is a triangle with the upper right corner at (2, 4).

1. Start by drawing the three lines: the line [tex]y=4−2x[/tex], the line y=0, and the line x=0.

2. Find the coordinates of the upper right corner of the region. To do this, we need to find the point where the two lines y=4−2x and x=0 intersect. We can solve this by setting y=4−2x and x=0, which gives us the point (0, 4). This is the upper right corner of the region.

3. The region is a triangle with the upper right corner at (2, 4).

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By applying the algebra concept, it can be concluded that Leona ordered 25 boxes of pens.

Algebra is a branch of mathematics that uses symbols and mathematical operations, such as addition, subtraction, multiplication, and division to solve problems.

We have this information:

The price of a box of pens = $8

The price of a box of rubber bands = $4

Total order = 30 boxes

Total payment = $220

Let p symbolize the number of pens and r symbolize the number of rubber bands. Now we have the following equations:

p + r = 30 ....................... (1)

8p + 4r = 220 ............... (2)

To find the value of p and r, we can do it by simplifying the first equation as follows:

p + r = 30

r = 30 - p

Then we can substitute this value into the second equation:

         8p + 4r = 220

8p + 4(30 - p) = 220

8p + 120 - 4p = 220

                 4p = 220 - 120

                      = 100

                   p = 100/4

                      = 25 boxes of pens

So we can calculate the value of r as well:

r = 30 - p

 = 30 - 25

 = 5 boxes of rubber bands

Thus, it can be concluded that Leona ordered 25 boxes of pens.

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The given function is [tex]f(x,y) = x^2 + y^2[/tex].

To find the gradient of f at the point (x0, y0), we take the partial derivatives of f with respect to x and y:

[tex]$$\nabla f(x0,y0) = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right) = \left(2x0, 2y0\right)$$[/tex]

The rate of increase of $f$ at the point (x0, y0) in the direction (u,v) is given by the dot product of the gradient and the direction vector:

[tex]$$\nabla f(x0,y0) \cdot (u,v) = 2x0u + 2y0v$$[/tex]

The direction of maximum rate of increase at (x0, y0) is the direction of the gradient, which is (2x0,2y0). The rate of increase in this direction is the length of the gradient vector, which is given by:

[tex]$$||\nabla f(x0,y0)|| = \sqrt{(2x0)^2 + (2y0)^2} = 2\sqrt{x0^2 + y0^2}$$[/tex]

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The new interest rate for his mortgage would be  3.75%.

The amount that the lender charges the borrower over and beyond the principal amount is referred to as the interest rate. A person who deposits money in a bank or other financial institution also gets additional income in terms of the recipient, known as interest, taking into account the time value of money.

Purchasing a point simply implies negotiating for a lower rate of interest, which involves paying 1% of the loan in lieu of 1% interest reduction.

A point implies a reduction in interest rate by 0.25% while 3 points would reduce the interest rate by 0.75%(0.25%*3)

New interest rate=4.5%-0.75%=3.75%

Thus, The new interest rate for his mortgage would be  3.75%.

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The student has 36 different combinations of breakfast options to choose from (4 options x 9 days).

To determine this:

The student has four different options for breakfast each day for 9 days, so the number of different selections could be calculated as 4 options x 9 days = 36 different selections.

This means that the student has 36 different combinations of breakfast options to choose from, as they can choose one option each day for 9 days. The options are meat bowl, vegetarian bowl, cereal bowl, and taco bowl, and they can choose any combination of these options for their 9 days of breakfast.

For example, they can choose a meat bowl on day 1, a vegetarian bowl on day 2, a cereal bowl on day 3, a taco bowl on day 4, and so on, or they can choose the same option every day, or they can mix and match their options in any combination they like.

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Answer:

B. y = -3x -1

Step-by-step explanation:

Is this one because of the part of maths that studies the way in which words and the groups they form are combined to express meanings, as well as the relationships that are established between all these units. A set of rules defining the correct sequences of the elements of a programming science and maths formulas.

The value of the double integral is [tex]$-\frac{\pi}{12}$[/tex].

We are given that;

∫∫D^y^dA where d is the set of points (x, y)

0 ≤ 2x/ π ≤ y, y ≤ sin x

Now,

We can see that x varies from 0 to [tex]\pi[/tex], and for each fixed x, y varies from [tex]\frac{2x}{\pi} to \sin x[/tex]. Therefore, we can write the double integral as:

[tex]$$\iint_D y \, dA = \int_0^\pi \int_{\frac{2x}{\pi}}^{\sin x} y \, dy \, dx$$[/tex]

Now we can evaluate the inner integral with respect to y:

[tex]$$\int_{\frac{2x}{\pi}}^{\sin x} y \, dy = \frac{y^2}{2} \bigg|_{\frac{2x}{\pi}}^{\sin x} = \frac{\sin^2 x}{2} - \frac{2x^2}{\pi^2}$$[/tex]

Plugging this result into the outer integral, we get:

[tex]$$\iint_D y \, dA = \int_0^\pi \left( \frac{\sin^2 x}{2} - \frac{2x^2}{\pi^2} \right) dx = \frac{1}{4} \int_0^\pi (1 - \cos 2x) dx - \frac{2}{\pi^2} \int_0^\pi x^2 dx$$[/tex]

Using the antiderivatives of [tex]\cos 2x[/tex] and x^2, we get:

[tex]$$\iint_D y \, dA = \frac{1}{4} (x - \frac{1}{2} \sin 2x) \bigg|_0^\pi - \frac{2}{\pi^2} (\frac{x^3}{3}) \bigg|_0^\pi =\frac{\pi}{4} - 0 - 0 + 0 - \frac{2}{\pi^2} (\frac{\pi^3}{3}) + 0 =\frac{\pi}{4} - \frac{2\pi}{3} =-\frac{\pi}{12}$$[/tex]

Therefore, by integral the answer will be [tex]$-\frac{\pi}{12}$[/tex].

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The equation to see how far Zach travels is y = 21x and Zach traveled 336 miles using 16 gallons of gas.

Number of miles traveled Zach's car traveled= 21 miles

Quantity of gallons used = 1 gallon

y = kx

Where,

x = total gallons used

k = unit rate of miles per gallon

y = Number of miles traveled

So,

21 = k × 1

21 = k

y = 21x

Hence,

if he uses 16 gallons of gas

y = kx

y = 21 × 16

y = 336 miles

In conclusion, Zach's car traveled 336 miles.

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The two column proof showing that m∠AKG = m∠HKB is as below and explained.

A two-column proof is defined as a geometric proof consists of a list of statements, and the reasons that we know those statements are true.

The two column proof showing the vertical angles are explained below.

Statement 1: Segment GH Intersects Segment AB at K

Reason 1: Given

Statement 2: m∠AKG + m∠GKB = 180°

m∠GKB + m∠HKB = 180°

Reason 2:  Definition of supplementary angles

Statement 3: m∠AKG + m∠GKB = m∠GKB + m∠HKB

Reason 3: Substitution property

Statement 4: m∠AKG = m∠HKB

Reason 4: Subtraction property

Now, supplementary angles are defined as angles that sum up to 180 degrees and that is why m∠AKG + m∠GKB = 180° and m∠GKB + m∠HKB = 180°

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The product of AB and CD in a regular hexagon with an area of 60 is (40 / √3) / 2.

To find the product of the length of AB and CD in a regular hexagon, we have to initially calculate the length of one side of the hexagon and later use that information to find the length of AB and CD.

Let us say that the length of one side of the hexagon = s.

W.k.t the area of the hexagon = 60, so we can use the formula for the area of a regular hexagon:

Area = (3 * √3) / 2 * [tex]s^2[/tex]

Making this equal to 60 and solving s:

60 = (3 * √3) / 2 *    [tex]s^2[/tex]

120 = 3 * √3 * [tex]s^2[/tex]

40 = √3 * [tex]s^2[/tex]

(40 / √3) = [tex]s^2[/tex]

Taking the square root of both sides:

s = (40 / √3)^(1/2)

Now that we found the length of one side of the hexagon, So we can use that to find the length of AB and CD.

AB = the length of one side of the hexagon times the square root of 3,  CD = the length of one side of the hexagon divided by 2.

So,

AB = s * √3

CD = s / 2

Now, we can find the product of AB and CD by multiplying them together:

AB * CD = s * √3 * s / 2

AB * CD = (s^2 * √3) / 2

AB * CD = (40 / √3) / 2

So the product of AB and CD in a regular hexagon with an area of 60 is equal to (40 / √3) / 2.

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a) P(X<25) = 0.7135         P(X>15) = 0.4724        P(15<X<25)=0.1859 and The 40th percentile is : 10.2165

The exponential distribution is a continuous probability distribution used in probability theory and statistics that frequently addresses the amount of time until a certain event occurs. It refers to a system in which things proceed continuously and separately at such an average speed that is constant.

Let X = The time (in minutes) that it takes a mechanic to change oil

We have,

X = Exp (mean = 20)

The distribution function of X is: F(x) = P (X < x) = 1 - e{-x/20}

a) Consider, P (X < 25) = F(25)

= 1 - e{-25/20}

= 0.7135

Consider, P( X > 15)

= 1 - P (X < 15)

= 1 -  1 - e{-15/20}

= 0.4724

Consider, P(15 < X < 25)

= P (X < 25) - P (X < 15)

=  1 - e{-25/20} - 1 - e{-15/20}

= 0.1859

Hence,

P(X<25) = 0.7135         P(X>15) = 0.4724        P(15<X<25)=0.1859

b) We want to find x such that P(X<x) =0.40

Consider, P (X < x) = 0.40

1 - e{-x/20} = 0.40

x = 10.2165

Hence, The 40th percentile is : 10.2165

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The probability of drawing 1 black pencil and 2 yellow pencils without replacement is 0.035714.

The probability of drawing 1 black pencil and 2 yellow pencils without replacement can be calculated by multiplying the individual probabilities of drawing each pencil.

The first step is to calculate the probability of drawing a black pencil which is 5/8. Then, the probability of drawing the first yellow pencil is 4/7 and the probability of drawing the second yellow pencil is 3/6.

To find the probability of drawing all three pencils, you multiply these three probabilities together:

5/8 × 4/7 × 3/6 = 0.035714.

This result represents the probability of drawing 1 black pencil and 2 yellow pencils in that order without replacement.

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We can see that p-q = 1, which implies that p and q are directly related.

(a) If x and y are inversely prοpοrtiοnal, then we knοw that xy = k fοr sοme cοnstant k.

If [tex]x^p[/tex] and [tex]y^q[/tex] are alsο inversely prοpοrtiοnal, then we can write [tex](x^p)(y^q) = k'[/tex] fοr sοme cοnstant k'. Using the inverse prοpοrtiοnality relatiοn, we get:

[tex](x^p)(y^q) = k'[/tex]

[tex](xy)^pq = k'[/tex]

[tex](k)^pq = k'[/tex]

[tex]k^(pq-1) = k'[/tex]

Frοm this equatiοn, we can see that pq-1 = 0, which implies that pq = 1. Therefοre, p and q are inversely related.

(b) If x and y are inversely prοpοrtiοnal, then we knοw that xy = k fοr sοme cοnstant k.

If [tex]x^p[/tex] and [tex]y^q[/tex] are directly prοpοrtiοnal, then we can write [tex](x^p) = k'(y^q)[/tex] fοr sοme cοnstant k'. Using the direct prοpοrtiοnality relatiοn, we get:

[tex](x^p) = k'(y^q)[/tex]

[tex](xy)^p = k'(y^q)(k)[/tex]

[tex]k^p = k'(y^{(q-1)})[/tex]

[tex]k^{(p-q)} = k'[/tex]

Frοm this equatiοn, we can see that p-q = 1, which implies that p and q are directly related.

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Answer:

So first solve the equation given

X+2y=-14

-x        -x

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

2y=-x-14

----  ---------

2        2

y=- 1/2x-7

this would be your parallel line equation because y=-1/2x-7

Parallel lines have the same slope

Perpendicular lines have a negative reciprocal slope (so -2 would be positive 1/2)

Hope this helps!

Please leave a comment if I made a mistake, I will correct it!

The supply curve shifts as a result of every factor but a change in market price. The movement along the supply curve and the change in price are correlated.

The supply curve can change depending on a number of variables, such as shifts in production costs (such as raw material and labor prices), advancements in technology, the level of competition and the number of sellers/producers, as well as the regulatory and tax environment.

Although a shift in the quantity supplied or along the supply curve for a given commodity or service is frequently brought about by a change in price, the supply curve itself does not alter as a result.

Input prices, weather conditions, technological advancements, as well as governmental taxes, regulations, and subsidies, can all modify the supply curve for goods and services, resulting in a different quantity being delivered for a given price.

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Answer:

4a⁴ + 5a² - 3a + 11

Step-by-step explanation:

5a² + 5 - a + 6 - 2a + 4a⁴

Combine like terms.

5a² + 11 - 3a + 4a⁴

4a⁴ + 5a² - 3a + 11

Answer:

4a⁴ + 5a² - 3a + 11

Step-by-step explanation:

Now we have to,

→ Simplify the given expression.

The expression is,

→ (5a² + 5 - a) + (6 - 2a + 4a⁴)

Let's simplify the expression,

→ (5a² + 5 - a) + (6 - 2a + 4a⁴)

→ 5a² + 5 - a + 6 - 2a + 4a⁴

→ (4a⁴) + (5a²) + (-a - 2a) + (5 + 6)

→ 4a⁴ + 5a² + (-3a) + (11)

→ 4a⁴ + 5a² - 3a + 11

Therefore, this is the answer.