50 POINTS

Which of the following can be determined from the table above?
A.

Events B and C are independent.
B.

Events D and A are independent.
C.

Events C and A are independent.
D.

Events E and B are independent.

Answer:

[tex]9\ years[/tex]

Step-by-step explanation:

Let

P----> the initial price of the ticket

y ---> the price of the ticket after t years

t---> the time in years

we know that

100%+8%=108%=108/100=1.08

so

[tex]y=P(1.08)^{t}[/tex] ----> equation A

If the price is doubled

then

[tex]y=2P[/tex] -----> equation B

equate equation A and equation B and solve for t

[tex]2P=P(1.08)^{t}[/tex]

Simplify

[tex]2=(1.08)^{t}[/tex]

Apply log both sides

[tex]log(2)=t*log(1.08)[/tex]

[tex]t=log(2)/log(1.08)=9\ years[/tex]

It will take approximately 9 years for the price of theater tickets to double at an annual increase of 8%.

To solve this problem, we can use the Rule of 70, which is a quick and easy way to estimate the number of years required for a quantity to double at a constant growth rate. The Rule of 70 is given by the formula:

[tex]\[ \text{Years to double} \approx \frac{70}{\text{Annual growth rate}} \][/tex]

Given that the annual growth rate is 8%, we can apply this formula:

[tex]\[ \text{Years to double} \approx \frac{70}{8} \] \[ \text{Years to double} \approx 8.75 \][/tex]

Since we cannot have a fraction of a year in this context, we round to the nearest whole number. Therefore, it will take approximately 9 years for the price to double.

Answer:

  Equations A and C are equivalent, and of those, equation C is in the form most useful ...

Step-by-step explanation:

In standard form, the equations are ...

Equation A: y = 3x² -6x +21

Equation B: y = 3x² -6x +18

Equation C: y = 3x² -6x +21 . . . . equivalent to A

Equation D: y = 3x² -6x +24

__

Equation C is in vertex form, so the vertex (extreme value) can be read directly from the equation. It is (x, y) = (1, 18).

Equations A and C are equivalent; equation C is most useful for finding the vertex.

Answer with explanation:

→→Two equations or two polynomials are said to be equivalent, if written in distinct ways, and real value of variables is substituted in both equivalent and Original Polynomial, the numerical value of both the polynomials are Same.

→→The four Quadratic Polynomials are:

[tex]1.\text{Equation} A: y = 3x^2 - 6 x + 21\\\\2.\text{Equation} B: y = 3x^2 - 6 x + 18\\\\3.\text{Equation} C: y = 3(x-1)^2 +18=3(x^2-2 x +1)+18\\\\y=3x^2 - 6 x +18+3\\\\y=3x^2 - 6 x +21\\\\3.\text{Equation} C: y = 3(x-1)^2 +21[/tex]

→→If you will look at Equation A and Equation C, both the equation are Quadratic, Coefficient of x², Coefficient of x, as well as , constant term is same in both the equation.So, Equation A, and Equation C, are equivalent.

→→If you will look at the function,

[tex]y=3\times (x-1)^2+18\\\\y-18=3\times(x-1)^2[/tex]

at, x=1, y=18, which is extreme value of the function, as at vertex of the parabola , Parabola attains it's Maximum value.

⇒⇒Equations A and C,

[[tex]1.\text{Equation} A: y = 3x^2 - 6 x + 21\\\\3.\text{Equation} C: y = 3(x-1)^2 +18[/tex]]

are equivalent, and of those, equation is in the form most useful for identifying the extreme value is [tex]3.\text{Equation} C: y = 3(x-1)^2 +18[/tex]]  function it defines.

Answer: A, it is not plausible that 50% of the population had taken a yoga class because the data shows that a sample proportion of 40% is unlikely.  

Answer:

Option A is correct.

Step-by-step explanation:

Kendra surveyed a random sample of 100 members of a local gym. She found that 40% of the gym members surveyed had taken a yoga class.

Kendra performed 100 trials of a simulation. Each trial simulated a sample of 100 gym members under the assumption that 50% of the population had taken a yoga class.

The best conclusion based on the plot is - A. It is not plausible that 50% of the population had taken a yoga class because the data shows that a sample proportion of 40% is unlikely.

Looks like the paraboloid has equation

[tex]z=2-x^2-y^2[/tex]

and [tex]S[/tex] is the part of this surface with [tex]0\le x\le1[/tex] and [tex]0\le y\le1[/tex]. Parameterize [tex]S[/tex] by

[tex]\vec s(u,v)=u\,\vec\imath+v\,\vec\jmath+(2-u^2-v^2)\,\vec k[/tex]

with [tex]0\le u\le1[/tex] and [tex]0\le v\le1[/tex]. Take the normal vector to [tex]S[/tex] to be

[tex]\vec s_u\times\vec s_v=2u\,\vec\imath+2v\,\vec\jmath+\vec k[/tex]

Then the flux of [tex]\vec F[/tex] across [tex]S[/tex] is

[tex]\displaystyle\iint_S\vec F\cdot\mathrm d\vec S[/tex]

[tex]\displaystyle=\int_0^1\int_0^1(uv\,\vec\imath+v(2-u^2-v^2)\,\vec\jmath+u(2-u^2-v^2)\,\vec k)\cdot(2u\,\vec\imath+2v\,\vec\jmath+\vec k)\,\mathrm du\,\mathrm dv[/tex]

[tex]\displaystyle=\int_0^1\int_0^1(2u^2v+(2v+1)u(2-u^2-v^2))\,\mathrm du\,\mathrm dv=\boxed{\frac{293}{180}}[/tex]

Answer:

  16/(3(x+1))

Step-by-step explanation:

We can factor the denominator of the second term, which lets us see how to combine terms:

[tex]\displaystyle\frac{7}{x+1}-\frac{5}{3x+3}=\frac{7}{x+1}-\frac{5}{3(x+1)}\\\\=\frac{3\cdot 7}{3(x+1)}-\frac{5}{3x+3}=\frac{21-5}{3(x+1)}=\frac{16}{3(x+1)}[/tex]

Answer:

  A(2/9, 1/3)

  B(2/3, -1/3)

  slope = -1.5

Step-by-step explanation:

A graph can show the coordinates of interest: A(2/9, 1/3); B(2/3, -1/3).

Rearranging the equation to slope-intercept form, we have ...

  3y = -4.5x +2

  y = -1.5x +2/3

The slope is -1.5.

Final answer:

After solving for the missing coordinates, point A is (2/9, 1/3), and point B is (2/3, -1/3). Calculating the slope using these two points gives us a slope (m) of -3/2 for the line.

Explanation:

To find the missing coordinates for point A, we plug y=1/3 into the equation 4.5x+3y=2 and solve for x. Here is how:

4.5x + 3(1/3) = 2

4.5x + 1 = 2

4.5x = 1

x = 1 / 4.5

x = 2/9

Therefore, A(2/9, 1/3)

To find the missing coordinate for point B, we plug x=2/3 into the equation 4.5x+3y=2 and solve for y. Here is how:

4.5(2/3) + 3y = 2

3 + 3y = 2

3y = -1

y = -1/3

Therefore, B(2/3, -1/3)

To find the slope of the line, we use the two points A(2/9, 1/3) and B(2/3, -1/3). The slope formula is (y2 - y1) / (x2 - x1), which gives:

m = (-1/3 - 1/3) / (2/3 - 2/9)

m = (-2/3) / (4/9)

m = (-2/3) * (9/4)

m = -3/2, which is the slope of the line.

Answer:

Step-by-step explanation:

Let t represent the time driving at 50 mi/h. Then her total distance driven (in miles) is ...

  distance = speed · time

  290 = 50t + 45(6-t)

  20 = 5t . . . . . . . . . . . subtract 270, collect terms

  4 = t . . . . . . divide by the coefficient of t

Taylor drove 4 hours at 50 miles per hour, then 2 hours at 45 miles per hour.

Answer: D) 0, one real solution

A quadratic function is given of the form:

[tex]ax^2+bx+c=[/tex]

We can find the roots of this equation using the quadratic formula:

[tex]x_{12}=\frac{-b \pm \sqrt{b^2-4ac}}{2a}[/tex]

Where [tex]\Delta=b^2-4ac[/tex] is named the discriminant. This gives us information about the roots without computing them. So, arranging our equation we have:

[tex]4a^2-4a-6=-7 \\ \\ Adding \ 7 \ to \ both \ sides \ of \ the \ equation: \\ \\ 4a^2-4a-6+7=-7+7 \\ \\ 4a^2-4a+1=0 \\ \\ Then \ the \ discriminant: \\ \\ \Delta=(-4)^2-4(4)(1) \\ \\ \Delta=16-16 \\ \\ \boxed{Delta=0}[/tex]

Since the discriminant equals zero, then we just have one real solution.

Answer: D) -220, no real solution

In this exercise, we have the following equation:

[tex]-r^2-2r+14=-8r^2+6[/tex]

So we need to arrange this equation in the form:

[tex]ax^2+bx+c=[/tex]

Thus:

[tex]-r^2-2r+14=-8r^2+6 \\ \\ Adding \ 8r^2 \ to \ both \ sides \ of \ the \ equation: \\ \\ -r^2-2r+14+8r^2=-8r^2+6+8r^2 \\ \\ Associative \ Property: \\ \\ (-r^2+8r^2)-2r+14=(-8r^2+8r^2)+6 \\ \\ 7r^2-2r+14=6 \\ \\ Subtracting \ 6 \ from \ both \ sides: \\ \\ 7r^2-2r+14-6=6-6 \\ \\ 7r^2-2r+8=0[/tex]

So the discriminant is:

[tex]\Delta=(-2)^2-4(7)(8) \\ \\ \Delta=4-224 \\ \\ \boxed{\Delta=-220}[/tex]

Since the discriminant is less than one, then there is no any real solution

Answer: C) 144

What we need to find is the value of [tex]c[/tex] such that:

[tex]x^2+24x+c=0[/tex]

is a perfect square trinomial, that are given of the form:

[tex]a^2x^2\pm 2axb+b^2[/tex]

and can be expressed in squared-binomial form as:

[tex](ax\pm b)^2[/tex]

So we can write our quadratic equation as follows:

[tex]x^2+2(12)x+c \\ \\ So: \\ \\ a=1 \\ \\ b=12 \\ \\ c=b^2 \therefore c=12^2 \therefore \boxed{c=144}[/tex]

Finally, the value of [tex]c[/tex] that completes the square is 144 because:

[tex]x^2+24x+144=(x+12)^2[/tex]

Answer: C) [tex]\frac{121}{4}[/tex]

What we need to find is the value of [tex]c[/tex] such that:

[tex]z^2+11z+c=0[/tex]

So we can write our quadratic equation as follows:

[tex]z^2+2\frac{11}{2}z+c \\ \\ So: \\ \\ a=1 \\ \\ b=\frac{11}{2} \\ \\ c=b^2 \therefore c=\left(\frac{11}{2}\left)^2 \therefore \boxed{c=\frac{121}{4}}[/tex]

Finally, the value of [tex]c[/tex] that completes the square is [tex]\frac{121}{4}[/tex] because:

[tex]z^2+11z+\frac{121}{4}=(x+\frac{11}{2})^2[/tex]

In this problem, we need to find the length and width of a rectangle. We are given the area of the rectangle, which is 45 square inches. We know that the formula of the area of a rectangle is:

[tex]A=L\times W[/tex]

From the statement we know that the length of the rectangle is is one inch less than twice the width, this can be written as:

[tex]L=2W-1[/tex]

So we can introduce this into the equation of the area, hence:

[tex]A=L\times W \\ \\ \\ Where: \\ \\ W:Width \\ \\ L:Length[/tex]

[tex]A=(2W-1)(W) \\ \\ But \ A=45: \\ \\ 45=(2W-1)(W) \\ \\ Distributive \ Property:\\ \\ 45=2W^2-W \\ \\ 2W^2-W-45=0 \\ \\ Quadratic \ Formula: \\ \\ x_{12}=\frac{-b\pm \sqrt{b^2-4ac}}{2a} \\ \\ W_{1}=\frac{-(-1)+ \sqrt{(-1)^2-4(2)(-45)}}{2(2)} \\ \\ W_{1}=\frac{1+ \sqrt{1+360}}{4} \therefore W_{1}=5 \\ \\ W_{2}=\frac{-(-1)- \sqrt{(-1)^2-4(2)(-45)}}{2(2)} \\ \\ W_{2}=\frac{1- \sqrt{1+360}}{4} \therefore W_{2}=-\frac{9}{2}[/tex]

The only valid option is [tex]W_{1}[/tex] because is greater than zero. Recall that we can't have a negative value of the width. For the length we have:

[tex]L=2(5)-1 \\ \\ L=9[/tex]

Finally:

[tex]The \ length \ is \ 9 \ inches \\ \\ The \ width \ is \ 5 \ inches[/tex]

The distance in miles between mars and a satellite is given by the equation:

[tex]d=-9t^2+776[/tex]

where [tex]t[/tex] is the number of hours it has fallen. So we need to find when the satellite will be 452 miles away from mars, that is, [tex]d=452[/tex]:

[tex]d=-9t^2+776 \\ \\ 452=-9t^2+776 \\ \\ 9t^2=776-452 \\ \\ 9t^2=324 \\ \\ t^2=\frac{324}{9} \\ \\ t^2=36 \\ \\ t=\sqrt{36} \\ \\ \boxed{t=6h}[/tex]

Finally, the satellite will be 452 miles away from mars in 6 hours.

Answer:

294.5 sq meters

Step-by-step explanation:

I found the area of the circle, subtracted away the area of the sector, then had to add back in the area of the triangle.  The areas for each is as follows:

[tex]A_{c}=\pi (11.1)^2[/tex]

A = 387.0756 sq m

[tex]A_{s}=\frac{130}{360}*\pi (11.1)^2[/tex]

A = 139.7773

[tex]A_{t}=\frac{1}{2}(11.1)(11.1)sin(130)[/tex]

A = 47.1922

Now taking the area of the circle - area of sector + area of triangle:

387.0756 - 139.7773 + 47.1922 = 294.5 sq m

Answer:

294.5

Step-by-step explanation:

Answer:

Alyssa walked 200 ft more than Miki.

Step-by-step explanation:

According to the Pythagorean theorem formula if we square the a(400) and b(300) and add them both we would get 250,000. From then you square root it to 500. So Miki walked 500ft and Alyssa walked 700ft (400+300). Subtract 500 from 700 and you would get 200ft.

Answer:

D

Step-by-step explanation:

9.6 is a floating-point number, therefore, not a integer.

Answer:

d

Step-by-step explanation:

Answer:

see below

Step-by-step explanation:

[tex]\sqrt[4]{162b^8}=\sqrt[4]{2\cdot 3^4b^8}=\sqrt[4]{2(3b^2)^4}\\\\=3b^2\sqrt[4]{2}[/tex]

Answer: regular hexagon

Step-by-step explanation:

The polygon shown is a regular hexagon.

A regular hexagon is a six-sided polygon with all its sides of equal length and all its interior angles of equal measure. The term "regular" signifies the uniformity and symmetry of this geometric shape. Each side of a regular hexagon is congruent to the others, and each interior angle measures 120 degrees.

Regular hexagons can be found in various contexts, from nature to man-made structures. Honeycombs, for instance, are often composed of hexagonal cells. The six-sided structure allows for efficient packing and maximizes space utilization, which is why it's a prevalent shape in the natural world.

In geometry, a regular hexagon can be divided into equilateral triangles, demonstrating its versatility and ease of partitioning. This feature makes it a fundamental shape in tessellation patterns and various geometric designs.

Additionally, regular hexagons possess rotational symmetry. You can rotate a hexagon by 60 degrees, and it will align perfectly with its original position, making it a key element in tessellation art and certain engineering applications.

Overall, the regular hexagon is an elegant, balanced, and harmonious shape with numerous real-world and mathematical applications, valued for its regularity and aesthetic appeal.

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Is it 12 chairs per three tables? I might be wrong..

Answer: 12 chairs per 3 tables

Step-by-step explanation:

Answer:

-4y = x², or y = - x²/4, or y = -(1/4)x²

Step-by-step explanation:

Because the focus is beneath the directrix, this vertical parabola opens down.  The general formula is 4py = x².  

Because the distance between focus and directrix is 2 units, p = -1 here.  The negative sign shows that the parabola opens down.

4py = x² becomes 4(-1)y = x², or -4y = x², or y = - x²/4

A fraction is closer to 1/2 than to 0 or 1 if its numerator is more than half of 0 and less than half of the denominator. For instance, 3/5 is closer to 1/2 than to 0 or 1 because its numerator is more than half of 0 and less than half of 5.

To determine which fraction is closer to 1/2 than to 0 or 1, consider the numerical value of the fractions in relation to 1/2. As a rule of thumb, if a fraction has a numerator that is half the denominator, it equals 1/2. When the numerator is less than half the denominator, the value is less than 1/2; when the numerator is over half the denominator, the value is greater than 1/2.

For example, if we take 1/3, it is clear that this value is less than 1/2 because the numerator, 1, is less than half of the denominator, 3. Comparatively, the fraction 2/3 is closer to 1 than to 1/2 since the numerator, 2, is greater than half of the denominator, 3.

Another example would be comparing 3/5 and 4/5 to 1/2. The fraction 3/5 has a numerator that is more than half of the denominator, making it closer to 1/2 than to 0 or 1, while 4/5 is closer to 1. Therefore, 3/5 is the fraction that is closer to 1/2 than to 0 or 1.

A fraction closer to 1/2 than to 0 or 1 falls in the range (> 1/3 but < 2/3); 5/8 is an example of such a fraction. An intuitive understanding and the use of a common denominator can aid in identifying and comparing these fractions.

Determining which fraction is closer to 1/2 than to 0 or 1 involves understanding the number line and how fractions fall on it.

To identify such a fraction, it should be evident that any fraction greater than 1/2 will naturally be closer to 1, while any fraction less than 1/2 is closer to 0.

Therefore, we look for a fraction in the range (> 1/3 but < 2/3) to ensure it is closer to 1/2.

By using the intuitive sense of fractions, which is like having an understanding of fractions through visualization or practical examples, we can gauge the closeness to 1/2.

For instance, we know that 1/3 is less than 1/2, and similarly, 2/3 is more than 1/2.

Following this line of reasoning, the addition of fractions (1/3 + 1/6 = 1/2) indicates that 1/6 is the gap needed to reach from 1/3 to 1/2.

A fraction such as 5/8 would be a good example of a fraction closer to 1/2.

This fraction is more than 1/2 (4/8) but less than 3/4 (6/8), placing it comfortably closer to 1/2 on the number line.

Employing the common denominator strategy also helps to compare fractions effectively, by aligning them to a unifying reference point.

Answer:

$112

Step-by-step explanation:

Reduced price of $89.60 = 80% (100% - 20%)

You can get the original price (100%) by dividing $89.60 with 80% = $112

The original price is $112.

Given:

A store has 20% off sale.

Reduced price of an item = $89.60

We have to find the original price of an item.

Now, the store has 20% off sale. Hence, the reduced price of an item $89.60 is the 80% of the original price.

Form the given information we can find the original price of an item.

⇒ (80/100) × original price = 89.60

Multiply both sides by 100, we get:

⇒ 80 × original price = 8960

Divide both the sides by 80, we get:

⇒ Original Price = $112

Therefore, the original price of an item is $112.

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The volume of a sphere with radius [tex]r[/tex] is [tex]V=\dfrac43\pi r^3[/tex]. Sphere B has a volume of [tex]64\pi[/tex], so

[tex]V_B=\dfrac43\pi{r_B}^3\implies r_B=\sqrt[3]{\dfrac{64\pi}{\frac43\pi}}=\sqrt[3]{48}[/tex]

Now,

[tex]\dfrac{r_A}{r_B}=\dfrac25\implies r_A=\dfrac{2r_B}5[/tex]

so sphere A has volume

[tex]V_A=\dfrac43\pi\left(\dfrac{2r_B}5\right)^3=\dfrac{512}{125}\pi[/tex]

Answer:

First choice listed

Step-by-step explanation:

Pick 2 points from the table and find the slope between them.  I chose (30,12) and (20,8).  Apply the slope formula to find the cost per CD:

[tex]\frac{Cost}{CD}= \frac{30-20}{12-8}=\frac{5}{2}[/tex]

The function then is C = 5/2d. or C = 2.5d, first choice

C = 2.5d. Plugging in for d and C we get the same solution on each side

Answer:

19.5

Step-by-step explanation:

Answer:

Area of the base is 19.5 unit square.

Step-by-step explanation:

The volume of the cone is given as = 52 cubic inches

Height or h = 8 inches

Volume of the cone is given as : [tex]\pi r^{2} \frac{h}{3}[/tex]

[tex]52=3.14\times r^{2} \times\frac{8}{3}[/tex]

[tex]52= r^{2} \times 8.37[/tex]

[tex]r^{2} =6.212[/tex]

r = 2.492 inches

Now, area of the base is given as [tex]\pi r^{2}[/tex] because it is a circle.

Area = [tex]3.14\times2.492\times2.492[/tex]

= 19.499 ≈ 19.5 unit square.

Answer:

  FALSE

Step-by-step explanation:

Area is a two-dimensional measure, so is measured in linear units that have a power of 2 (not 3).

_____

Please note that the exponent applies to linear units, such as meters or inches. Area can also be measured in area units with no exponent, such as barns or acres. (A "barn" is equal to 10^-28 m^2. It is used in nuclear physics to measure cross sectional areas of atomic particles. I think of it as one of the jokes built into modern physics, having its origins in the saying "can't hit the broad side of a barn.")

Final answer:

The statement that surface area is measured in cubic units is false. Surface area is measured in square units, whereas cubic units are used for volume. For example, the surface area of a cube is calculated as 6 times the area of one face, whereas volume is the cube of the side length.

Explanation:

The statement that surface area is measured in cubic units or units³ is false. Surface area is actually measured in square units (units²), not cubic units. Cubic units are used to measure volume, not surface area.

For example, the surface area of a cube that has a side length of 4 units would be 6(4x4) = 96 units², because a cube has 6 sides and each side would be a square with an area of 4x4.

On the other hand, the volume of the same cube would be 43 = 64 units³, because volume is calculated by multiplying length by width by height.

For example, if you take a large cube with a side length of 3 units, its total surface area would be 3 x 3 x 6 = 54 units² and the volume would be 33 = 27 units³.

However, if this cube is replaced with 27 small cubes, each with a side length of 1 unit, the combined surface area becomes much larger at 162 units², while the total volume remains the same at 27 units³.

Answer:

  7.  x=8

  8.  x=7

Step-by-step explanation:

7.  The segment marked 30 is bisected by the segment marked x. So, you have a right triangle with legs x and 15 and hypotenuse 17. The Pythagorean theorem applies:

  x^2 + 15^2 = 17^2

  x^2 = 289 -225 = 64

  x = √64 = 8

__

8. The arc subtended by the chord x is 360° -230° -65° = 65°. Since this is the same measure as the arc subtended by the chord of length 7, x will also be of length 7.

___

In fact, this geometry is impossible. The combination of circle radius, arc measure, and chord length cannot be obtained all in the same circle. The answer you get will depend on how you work the problem.

Answer:

D

12²

Step-by-step explanation:

Given in the question, an integer = 144

To find the exponential form 144 we will do factorisation

The prime factorisation of 144 is

2 × 2 × 2 × 2 × 3 × 3

So the exponential form is 2[tex]^{4}[/tex] × 3²

The other way is

√144 = 12

which means

12² = 144

Answer: 12^2=144

Step-by-step explanation:

Which means (12)(12)=144

Answer:

- 3/8

Step-by-step explanation:

- 3/4  - (-3/8)

= -3/4 + 3/8

= -6/8 + 3/8

= - 3/8

Explanation:

Each row in Pascal's triangle is a listing of the values of nCk = n!/(k!(n-k)!) for some fixed n and k in the range 0 to n. nCk is the number of combinations of n things taken k at a time.

If you consider what happens when you multiply out the product (a +b)^n, you can see where the coefficients nCk come from. For example, consider the cube ...

  (a +b)^3 = (a +b)(a +b)(a +b)

The highest-degree "a" term will be a^3, the result of multiplying together the first terms of each of the binomials.

The term a^b will have a coefficient that reflects the sum of all the ways you can get a^b by multiplying different combinations of the terms. Here they are ...

Adding these three products together gives 3a^2b, the second term of the expansion.

For this cubic, the third term of the expansion is the sum of the ways you can get ab^2. It is essentially what is shown above, but with "a" and "b" swapped. Hence, there are 3 combinations, and the total is 3ab^2.

Of course, there is only one way to get b^3.

So the expansion of the cube (a+b)^3 is ...

  (a +b)^3 = a^3 + 3a^2b +3ab^2 +b^3 . . . . . with coefficients 1, 3, 3, 1 matching the 4th row of Pascal's triangle.

__

In short, the values in Pascal's triangle are the values of the number of combinations of n things taken k at a time. The coefficients of a binomial expansion are also the number of combinations of n things taken k at a time. Each term of the expansion of (a+b)^n is of the form (nCk)·a^(n-k)·b^k for k =0 to n.

The values in Pascal's triangle are the coefficients in the binomial expansion of (x+y)^n. Each row in the triangle represents the coefficients for a certain power of the binomial being expanded.

In mathematics, Pascal's triangle is a triangular array of numbers where each number is the sum of the two immediately above it. Each row of Pascal's Triangle corresponds to the coefficients in the binomial expansion of (x+y)n. For example, the third row of the Pascal's triangle is 1, 2, 1, which are the coefficients for the expansion of (x+y)2, yielding x2 + 2xy + y2. To put it differently, the values in Pascal's triangle provide the coefficients for each term in the binomial expansion. The index of the row in the triangle indicates the power of the binomial being expanded. So, effectively, the values in Pascal’s triangle connect directly to the coefficients in an algebraic binomial expansion.

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Answer:

  about 0.254

Step-by-step explanation:

The light is red for 0.3 of the period, so that is the probability one car is stopped. Probability 3 cars are stopped and 5 are not is 0.3^3·0.7^5, about 0.004538. In the group of 8 cars, there are 56 different ways 3 of the cars can be stopped, so your overall probability could be 56·0.004538 ≈ 0.254.

_____

Comment on the question

Many factors go into a driver's decision to stop at a light. Many factors go into the distribution of arrival times at a light. Here, the problem is only tractable if we assume that cars arrive at the light individually and at random times with respect to the light's fixed 50-second cycle. (This is possibly the case only early in the morning hours when traffic is at its lightest (not associated with bar closings or night shift changes).)

Answer:

I think C

Step-by-step explanation:

if you want to find out how money you have left then you are supposed to subtract the spent money from the total money.

$42.69-$9.45= $33.24

$33.24- $14.35= $18.89

you have $18.89 of your money left and you have spent $23.80.

Hope this helps

D. it is correct because the length of the unknown side is the difference of the squares of the sides.

Answer:

It is correct because the length of the unknown side is the difference of the squares of the sides.

Step-by-step explanation:

Answer:

The answer is 3 8/15 miles.

Hope I helped ; )

Answer:

your answer will be 3 8/15

Step-by-step explanation: