HELP ASAP FOR BRAINLIEST:

Find the sixth term of a geometric sequence with t5 = 24 and t8 = 3

Thank you sooo much! Show work!

Answer:

Time = 9.0252 *10^12 s

Step-by-step explanation:

Given:

- The distance from Earth to Alpha Centauri = 4.3 light years

- The distance from Earth to Sirius = 8.6 light years

- Speed of the probe is V = 18.03 km/s

- 1 AU = 1.58125 x 10-5 light-years

Find:

How long will the probe have been traveling from when it first leaves Earth to when it arrives at Sirius?

Solution:

- We will track probe for each destination it reaches one by one:

                       Earth ------> Alpha Centauri d_1 = 4.3 light years

                       Alpha Centauri ------> Earth d_2 =4.3 light years

                       Earth ------> Sirius                  d_3 = 8.6 light years  

                       Total distance                       D    = 17.2 light years.

- Now we know the total distance traveled by the probe is D. We will convert the distance into km SI units:

                          1 AU    ------------------> 1.58125 x 10-5 light-years

                          x AU   ------------------> 17.2 light years.

- Using direct proportions

                           x = 17.2 / (1.58125 x 10-5) = 1087747.036 AU

Also,

                          1 AU  ---------------------> 149597870700 m

                           1087747.036 AU  ---->    D m

- Using direct proportions

                            D =   1087747.036*149597870700  = 1.62725*10^17 m

- Now use the speed - distance - time formula to compute the total time taken:

                            Time = Distance traveled (D) / V

                            Time = 1.62725*10^17 / 18.03*10^3    

Answer:               Time = 9.0252 *10^12 s

Answer:

Therefore,

[tex]m\angle R = 69.4\°[/tex]

Step-by-step explanation:

Given:

In Right Angle Triangle PQR at ∠Q = 90° such that

RQ = 3     ....Adjacent side of angle R

PQ = 8     ....Opposite side of angle R

To Find:

m∠R = ?

Solution:

In Right Angle Triangle PQR, Tan Identity,

[tex]\tan R= \dfrac{\textrm{side opposite to angle R}}{\textrm{side adjacent to angle R}}[/tex]

Substituting the values we get

[tex]\tan R= \dfrac{PQ}{QR}=\dfrac{8}{3}=2.6666\\\\\angle R=\tan^{-1}(2.666)=69.439=69.4\°[/tex]

Therefore,

[tex]m\angle R = 69.4\°[/tex]

Answer: Extraneous variable

Step-by-step explanation:

In an experiment , Independent variable can be manipulated by the experimenter to see the change in the dependent variable or response variable. But there are some variable known as extraneous variable that is attached to independent variable and make experimenter confuse.  

Extraneous variables is defined as :

∴ A variable that is systematically linked with the factor you believe is causing the overall effect in your research is called the extraneous variable variable. The presence of such a variable can prevent you from knowing what is really causing the effect.

Answer:

6 meters

Step-by-step explanation:

Intersecting Chord Theorem: When two chords intersect each other inside a circle, the products of their segments are equal.

One chord is divided into two segments with lengths of 15 m and 2 m.

Anothe chord is divided into two segments with measures of 5 m and n m.

Therefore,

[tex]5\cdot n=15\cdot 2\\ \\5n=30\\ \\n=6\ m[/tex]

Answer:

6 m

Step-by-step explanation:

just took the test

Answer:

Side length = [tex]\sqrt{\frac{A}{3} }[/tex] cm ,   Height =  [tex]\frac{1}{2} \sqrt{\frac{A}{3} }[/tex] cm  ,  Volume = [tex]\frac{A\sqrt{A}}{6\sqrt{3} }[/tex]  cm³

Step-by-step explanation:

Assume

Side length of base = x

Height of box = y

total material required to construct box = A ( given in question)

So it can be written as

A = x² + 4xy

4xy = A - x²

Volume of box = Area x height

V = x² ₓ y

V = x² ₓ ( [tex]\frac{A - x^{2} }{4x}[/tex] )

V =  [tex]\frac{Ax - x^{3} }{4}[/tex]

To find max volume put V' = 0

So taking derivative equation becomes

[tex]\frac{A - 3 x^{2} }{4} = 0[/tex]

A = 3 [tex]x^{2}[/tex]

[tex]x^{2}[/tex] = [tex]\frac{A}{3}[/tex]

x = [tex]\sqrt{\frac{A}{3\\} }[/tex]

put value of x in equation 1

y = [tex]\frac{A - \frac{A}{3} }{4\sqrt{\frac{A}{3} } }[/tex]  

y = [tex]\frac{2 \sqrt{\frac{A}{3} } }{4 \sqrt{\frac{A}{3} } }[/tex]

y = [tex]\frac{1}{2} \sqrt{\frac{A}{3} }[/tex]

So the volume will be

V = [tex]x^{2}[/tex] × y

Put values of x and y from equation 2 & 3

V = [tex]\frac{A}{3} (\frac{1}{2} \sqrt{\frac{A}{3} } )[/tex]

V = [tex]\frac{A\sqrt{A}}{6\sqrt{3} }[/tex]

The side length is [tex]\mathbf{l =\sqrt{ \frac A3}}[/tex], the height is [tex]\mathbf{h = \frac{1}{2}\sqrt{\frac A3}}[/tex] and the maximal volume is  [tex]\mathbf{V = \frac A6 \sqrt{\frac A3}}[/tex]

Let the dimensions of the box be l and h, where l represents the base length and h represents the height.

The volume is calculated as:

[tex]\mathbf{V = l^2h}[/tex]

The surface area is:

[tex]\mathbf{A= l^2 + 4lh}[/tex]

Make h the subject

[tex]\mathbf{h = \frac{A- l^2}{4l}}[/tex]

Substitute [tex]\mathbf{h = \frac{A- l^2}{4l}}[/tex] in [tex]\mathbf{V = l^2h}[/tex]

[tex]\mathbf{V = l^2 \times \frac{A- l^2}{4l}}[/tex]

[tex]\mathbf{V = l \times \frac{A- l^2}{4}}[/tex]

[tex]\mathbf{V = \frac{Al- l^3}{4}}[/tex]

Split

[tex]\mathbf{V = \frac{Al}{4}- \frac{l^3}{4}}[/tex]

Differentiate

[tex]\mathbf{V' = \frac{A}{4}- \frac{3l^2}{4}}[/tex]

Set to 0

[tex]\mathbf{\frac{A}{4}- \frac{3l^2}{4} = 0}[/tex]

Multiply through by 4

[tex]\mathbf{A- 3l^2 = 0}[/tex]

Add 3l^2 to both sides

[tex]\mathbf{3l^2 = A}[/tex]

Divide both sides by 3

[tex]\mathbf{l^2 = \frac A3}[/tex]

Take square roots

[tex]\mathbf{l =\sqrt{ \frac A3}}[/tex]

Recall that: [tex]\mathbf{h = \frac{A- l^2}{4l}}[/tex]

So, we have:

[tex]\mathbf{h = \frac{A - \frac{A}{3}}{4\sqrt{A/3}}}[/tex]

[tex]\mathbf{h = \frac{\frac{2A}{3}}{4\sqrt{A/3}}}[/tex]

Divide

[tex]\mathbf{h = \frac{2\sqrt{A/3}}{4}}[/tex]

[tex]\mathbf{h = \frac{\sqrt{A/3}}{2}}[/tex]

Rewrite as:

[tex]\mathbf{h = \frac{1}{2}\sqrt{\frac A3}}[/tex]

Recall that:

[tex]\mathbf{V = l^2h}[/tex]

So, we have:

[tex]\mathbf{V = \frac A3 \times \frac{1}{2}\sqrt{\frac A3}}[/tex]

[tex]\mathbf{V = \frac A6 \sqrt{\frac A3}}[/tex]

Hence, the side length is [tex]\mathbf{l =\sqrt{ \frac A3}}[/tex], the height is [tex]\mathbf{h = \frac{1}{2}\sqrt{\frac A3}}[/tex] and the maximal volume is  [tex]\mathbf{V = \frac A6 \sqrt{\frac A3}}[/tex]

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Answer:b also d

Step-by-step explanation:

Adrian's recipe has a greater ratio of sauce to dough, and the ratios can be compared by comparing 15 to 18 because 30 is in both dough columns.

In Adrian's recipe, the ratio of sauce to dough is 15:30, which simplifies to 1:2.

In Juliet's recipe, the ratio of sauce to dough is 18:30, which also simplifies to 1:2.

Since both ratios simplify to the same value of 1:2, it means that Adrian's and Juliet's recipes have equal ratios of sauce to dough.

Therefore, the statement "Adrian's and Juliet's recipes have equal ratios of sauce to dough" is true.

The reason we can compare the ratios by looking at 15 and 18 is because both of these values represent the amount of sauce, and 30 represents the amount of dough in both recipes.

By comparing the sauce portions directly, we can determine that the ratios are indeed equal.

It's important to note that the absolute values of the sauce and dough quantities are less relevant in this context than their proportions, which are expressed in the ratios.

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Answer:

10.2 feet.

Step-by-step explanation:

We have been given that Derek places  a standard 12-inch ruler next to the goal post and measures the shadow of the ruler and the backboard. If the ruler has a shadow of 10 inches. We are asked t find the height of the backboard, if the backboard has a shadow of 8.5 feet.

We will use proportions to solve our given problem as ratio between sides ruler will be equal to ratio of sides of background.

[tex]\frac{\text{Actual height of ruler}}{\text{Shadow of ruler}}=\frac{\text{Actual height of backboard}}{\text{Shadow of backboard}}[/tex]

[tex]\frac{12}{10}=\frac{\text{Actual height of backboard}}{8.5}[/tex]

[tex]\frac{12}{10}*8.5=\frac{\text{Actual height of backboard}}{8.5}*8.5[/tex]

[tex]1.2*8.5=\text{Actual height of backboard}[/tex]

[tex]10.2=\text{Actual height of backboard}[/tex]

Therefore, the actual height of the back-board is 10.2 feet.

Answer:

Step-by-step explanation:

Since the backboard and the ruler are both vertical and the sun is at the same position in the sky, the triangle made by the backboard and its shadow is similar to the triangle made by the ruler and its shadow.

The ratio of the corresponding sides of the triangles are equal and the height of the backboard can be determined by solving the following proportion.

Therefore, the top of the backboard is 7.64 feet high.

There are 85 candy bars in each box, and he will have 340 candy bars left after he gives three boxes to a friend.

Answer: he has 340 candy bars left.

Step-by-step explanation:

The total number of candy bars that

Miguel ordered is 595.

They come in 7 boxes. Assuming each box contains equal number of candy bars. This means that the number of candy bars in each box would be

595/7 = 85 candy bars

If he gives 3 boxes of candy bars to his friend, it means that the number of candy bars that he gave to his friend is

85 × 3 = 255 candy bars

Therefore, the number of candy bars that he has left is

595 - 255 = 340

Answer: the average weight of the remaining 2 non-blue items is 81 pounds

Step-by-step explanation:

The formula for determining average is expressed as

Average = sum of each item/ total number of items

Let x represent the average weight of the remaining 2 non-blue items.

The average weight of 5 items is 24 pounds. If the total weight of 3 blue items is 39 pounds, it means that

(x + 39)/5 = 24

x + 39 = 5 × 24 = 120

x = 120 - 39

x = 81

Jim's first error occurred when he incorrectly applied the cube to three-fifths. He made the mistake of not applying the power to the denominator of the fraction. The answer is  54/125

The question revolves around a mathematical operation where Jim is attempting to determine the cube of 2 times three-fifths. Jim's first error lies in how he applied the power of three. According to the rules of exponentiation for fractions, you should apply the power to both the numerator and denominator. Hence, Jim's first mistake was that he did not apply the power to the denominator of three-fifths. Instead of merely cubing the numerator (3), Jim should have also cubed the denominator (5). He should have performed the calculation as follows:

= 2 x (3/5)^3

= 2 x (3^3/5^3)

= 2 (27/125).

= 54/125

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Jim first errored by failing to apply the exponent to both the numerator and denominator of the fraction before multiplying by two. The correct calculation for cubing a fraction should involve cubing both the numerator and the denominator, which would have resulted in 54/125 instead of 54/5.

The first error that Jim made was not applying the power to both the numerator and the denominator of the fraction before multiplying by two. When cubing a number in fractional form, such as three-fifths, one must cube both the numerator (33) and the denominator (53) to correctly apply the exponent to the entire fraction. The correct method is to first cube three-fifths, which results in 27/125, and then multiply that result by 2, leading to the final answer of 54/125, not 54/5 as Jim calculated.

It's important to follow the correct order of operations and apply exponents before multiplication. In this case, to find the value of 2 × (three-fifths) cubed, the calculation should be 2 × (3/5)3, which simplifies to 2 × (27/125), and then to 54/125.

Answer:

A. three planes intersecting at a point

Step-by-step explanation:

Whenever three planes intersect at a point, then we have an ordered triplet (x,y,z) that is a solution to the associated system of equation.

This is a unique solution and it is finite.

However three planes intersecting at a line gives infinitely many solutions.

Three or two parallel planes may have no solution or or infinitely many solutions when they coincide.

Final answer:

After Sam eats 1/2 and Jack eats 1/4 of the remaining chocolate bar, 3/8 of the original chocolate bar is left uneaten.

Explanation:

Sam eats 1/2 of a chocolate bar. Jack then eats 1/4 of the remaining chocolate bar. To determine how much of the chocolate bar is left, we need to perform a couple of simple calculations. After Sam eats his portion, 1/2 of the bar is left. Jack eats 1/4 of what remains after Sam, which is 1/4 of 1/2, or 1/2 * 1/4 = 1/8 of the original bar.

This means that Jack's portion is 1/8 of the entire chocolate bar. Now, the total amount eaten by Sam and Jack together is 1/2 (Sam's portion) + 1/8 (Jack's portion). To add these fractions, they need to have a common denominator, which in this case is 8. This makes Sam's portion 4/8 when expressed with the common denominator. Now, add 4/8 (Sam's adjusted portion) + 1/8 (Jack's portion) = 5/8.

Therefore, the total amount eaten by both is 5/8 of the bar, leaving 3/8 of the original chocolate bar uneaten.

Answer:

Step-by-step explanation:

This is an exponential function:

[tex]y=16(\frac{2}{3})^x[/tex] that tells us that the initial height of the ball is 16 feet and that after each successive bounce the ball comes up to 2/3 its previous height.  We are looking for y when x = 2, so

[tex]y=16(\frac{2}{3})^2[/tex] and

[tex]y=16(\frac{4}{9})[/tex] and

[tex]y=\frac{64}{9}[/tex] so

y = 7.1 feet

Answer:

226.08 mm³

Step-by-step explanation:

The figure below shows a capsule mad of;

To determine the amount of vitamin that can be enclosed in the capsule, we determine the volume of the capsule;

Volume of a hemisphere = 2/3 πr³

Therefore; Taking π to be 3.14

Then;

Volume of the hemisphere = 2/3 × 3.14 × 3³

                                             = 56.52 mm³

But, since there are two equal hemispheres;

Then;

Volume of hemispheres = 56.52 mm³ × 2

                                        = 113.04 mm³

Volume of the cylinder is given by;

Volume = πr²h

Height = (10 mm - (2× 3mm)

           = 4 mm

Thus, taking π to be 3.14

Then;

Volume of the cylinder = 3.14 × 3² × 4

                                      = 113.04 mm³

Thus, Volume of the capsule = 113.04 mm³ + 113.04 mm³

                                                = 226.08 mm³

Answer:

Step-by-step explanation:

Let x represent the number of nights for which you rent a suite.

Resorts-R-Us charges $ 125 a night to rent a suite. If you choose this plan, the total cost of renting the suite for x nights would be

125 × x = 125x

If you purchase their one year membership fee for $350, you only pay $75 a night. This means that the total cost of renting a suite for x nights would be

75x + 350

For the plan involving membership to be a better deal,

75x + 350 < 125x

350 < 125x - 75x

350 < 50x

50x > 350

x > 350/50

x > 7

After 7 nights, the membership option would be a better deal. If you intend to rent the suite for lesser than 7 nights in a year, then the first option is a better deal.

Answer:

x= 7.5 mile/h

y= 3.5 miles / h

Step-by-step explanation:

Given x and y

so, x+y = speed down river

and x-y= speed up river

using

travel time = distance / speed for each case;

we get;

22/x+y=2  ---(1)

and

22/x-y=5.5 -----(2)

Solving equations simultaneously

eq 1 gives

2x+2y=22  ---(3)

eq 2 gives

5.5x-5.5y=22 ----(4)

Multiply eq 3 by 2.75

==> 5.5x+5.5y=60.5 --- (5)

adding eq 4 with eq 5

==> 11x =82.5

==> x= 7.5 miles/h

put this value in eq 3

==> 2(7.5)+2y=22

==> 2y=22-19.8

==> 2y=7

==> y = 3.5

Once setting up two equations based on downstream and upstream velocities, using variables x for Pratap's rowing speed and y for the current's speed, you can solve for these variables. The results show Pratap can row at a speed of 7.5 mph in still water, while the rate of the current is 3.5 mph.

Firstly, we have to understand that Pratap's speed with the current and against the current will be affected. Let's say x is the speed of Pratap rowing in still water, and y is the speed of the current. When going downstream (in the direction of the current), Pratap's speed is (x + y) since the current helps him. He covered a distance of 22 miles in 2 hours, so the rate (velocity) can be found by dividing distance by time. Hence, x + y = 22/2 = 11 miles per hour.

On the other hand, when rowing upstream (against the current), Pratap's speed is (x - y), because now the current hinders his rate. He covered the same distance in 5 1/2 hours, so x - y = 22/(5.5) = 4 miles per hour.

Now, we have two equations: x + y = 11 and x - y = 4. Solving these equations (by adding them), we find that 2x = 15 hence x = 7.5. Pratap can row in still water at a speed of 7.5 miles per hour. Substituting x=7.5 into either of the original equations we get y = 3.5. Hence, the rate of the current is 3.5 miles per hour.

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Answer:

C. f(x) = 2^x+2 - 3

Step-by-step explanation:

Looking at the graph and answer choices, it's obvious that the only change is the vertical shift. Since we know that the smallest value of 2^x+2 is 0, we can infer that the vertical shift will be -3 to match the horizontal asymptote of y=-3.

Answer:

summation of five times negative three to the power of n from n equals zero to infinity

Step-by-step explanation:

Summation Notation

It represents the sum of a finite or infinite number of terms. Let's analyze the terms of the given succession:

5-15+45-135+...

If we take 5 as a common factor, we have

5(1-3+9-27+...)

The parentheses contain the alternate sum/subtraction of powers of 3. The odd terms are positive, the even terms are negative, thus the exponent must be n starting from 0 or n-1 starting from 1

The summation is then represented by

[tex]\sum_{n=0}^{\infty}5(-3)^n[/tex]

This corresponds with the option:

summation of five times negative three to the power of n from n equals zero to infinity

Final answer:

The correct summation notation for the series 5 - 15 + 45 - 135 + ... is the summation of five times negative three to the power of n from n equals zero to infinity.

Explanation:

The given sequence is 5 - 15 + 45 - 135 + ... which can be seen as a geometric series with a pattern of alternating signs and a common ratio of -3. The first term of the sequence is 5 (when n=0), and each subsequent term is multiplied by -3. Therefore, to write this pattern using summation notation, the nth term can be represented as 5 × (-3)^n. So, the summation notation for the entirety of the series from n equals 0 to infinity is:

Σ [5 × (-3)^n], from n = 0 to infinity

This matches the option: summation of five times negative three to the power of n from n equals zero to infinity.

Answer:

The answer to your question is 5[tex]\sqrt{3}[/tex]

Step-by-step explanation:

Data

Proportion 1:2

Hypotenuse of the larger triangle = 20

length of the longer leg of the smaller triangle = ?

Process

1.- Remember the proportions of a 30- 60 - 90 triangle

hypotenuse = 2x

short leg = x

long leg = x[tex]\sqrt{3}[/tex]

2.- Use the previous information to find the lengths of the larger triangle

hypotenuse = 20 = 2x

short leg = x = 10

large leg = 10[tex]\sqrt{3}[/tex]

3.- Use the previous information to find the lengths of the smaller triangle

Proportion 1:2

hypotenuse = 10

short leg = x = 5

long leg = 5[tex]\sqrt{3}[/tex]

Answer:

It’s 5 radical 3

Step-by-step explanation:

The recommended number of classes is 5, and the class interval is 10. The first class's lower limit would be 50. To create a relative frequency distribution, divide each class frequency by the total

First, we have to calculate the range of the data by subtracting the smallest number from the highest number (98-51 = 47). A typical choice for the number of classes might be between five and 20. The

square root rule

suggests a round number near the square root of the number of observations, which is approx 4.47 in this case; you can round to 5. Now we divide the range by the number of classes (47/5 approximately 9). So, the class interval would be 10 (we round up as it does not make sense to have a fraction of an oil change). If the smallest number is 51, the lower limit for the first class would be 50. Hence, the

frequency distribution

would break down like this: 50-59, 60-69, 70-79, 80-89, 90-99. To create a relative frequency distribution, divide each class frequency by the total number of observations and multiply by 100 to convert to percentages. As for the shape, you would need to plot the data to see, but it may have a normal distribution as service levels often do.

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Answer:

a) (5,0)

b) (2,0)

Step-by-step explanation:

(a) A particle that lies 5.0 m directly above the origin would have its x-coordinate be 5 and its y-coordinate be 0. So (5,0).

(b) A particle that lies 2.0 m directly below the origin would have its x-coordinate be 2 and its y-coordinate be 0. So (2,0).

In a coordinate system with the positive x-axis directed upward, a particle located 5.0 m directly above the origin has a position of +5.0 m, and a particle 2.0 m below the origin has a position of -2.0 m.

In this coordinate system, the position of a particle is indicated by its vertical position relative to the origin. Therefore:

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Answer:

Step-by-step explanation:

If I'm understanding this correctly, the rate of decay function is

[tex]f(x)=1000(.5)^{\frac{x}{30}}[/tex]

and we want to solve for the amount of element left after x = 40 years.  That would make our equation

[tex]f(40)=1000(.5)^{1.33333333333}[/tex]

Multiply the repeating decimal by .5 to get

f(40) = 1000(.3968502631)

and f(40) = 396.85

So no, it's not safe for human habitation.

f(40) ≈ 793.7 kilograms. The area will not be safe for human habitation by 2025.

For the decay, to find f(40), we substitute x=40 into the function f(x)=1000(0.5)x/30.

f(40)=1000(0.5)40/30

f(40)=1000(0.5)4/3

f(40) ≈ 1000(0.7937)

f(40) ≈ 793.7 kilograms

Since the question asks if the area will be safe for human habitation by 2025, we need to check if f(40) is less than or equal to 100 kilograms.

But the value of f(40) is about 793.7 kilograms, which is greater than 100 kilograms.

Therefore, the area will not be safe for human habitation by 2025.

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Answer: the cost of one apple is $1

Step-by-step explanation:

Let x represent the cost of one apple.

Let y represent the cost of one Pomegranate.

Donata bought 3 Apples and 5 Pomegranates at the local supermarket for a total of $16.50. This means that

3x + 5y = 16.5 - - - - - - - - - - - - -1

Meaghan bought 6 Apples and 11 Pomegranates at the same store for a total of $35.70. This means that

6x + 11y = 35.7- - - - - - - - - - - - -2

Multiplying equation 1 by 2 and equation 2 by 1, it becomes

6x + 10y = 33

6x + 11y = 35.7

Subtracting, it becomes

- y = - 2.7

y = 2.7

Substituting y = 2.7 into equation 1, it becomes

3x + 5 × 2.7 = 16.5

3x + 13.5 = 16.5

3x = 16.5 - 13.5 = 3

x = 3/3 = 1

Answer:

[tex]C. \: 9 \times 7[/tex]

Explanation:

The Law of Sines is your friend, as is the Pythagorean theorem.

Label the unmarked slanted segments "a" and "b" with "b" being the hypotenuse of the right triangle, and "a" being the common segment between the 45° and 60° angles.

Then we have from the Pythagorean theorem ...

  b² = 4² +(2√2)² = 24

  b = √24

From the Law of Sines, we know that ...

  b/sin(60°) = a/sin(θ)

  y/sin(45°) = a/sin(φ)

Solving the first of these equations for "a" and the second for "y", we get ...

  a = b·sin(θ)/sin(60°)

and ...

  y = a·sin(45°)/sin(φ)

Substituting for "a" into the second equation, we get ...

  y = b·sin(θ)/sin(60°)·sin(45°)/sin(φ) = (b·sin(45°)/sin(60°))·sin(θ)/sin(φ)

So, we need to find the value of the coefficient ...

  b·sin(45°)/sin(60°) = (√24·(√2)/2)/((√3)/2)

  = √(24·2/3) = √16 = 4

and that completes the development:

  y = 4·sin(θ)/sin(φ)

Answer:

(a) 0.4096

(b) 0.64

(c) 0.7942

Step-by-step explanation:

The probability that the player wins is,

[tex]P(W)=0.80[/tex]

Then the probability that the player losses is,

[tex]P(L)=1-P(W)=1-0.80=0.20[/tex]

The player is playing the video game with 4 different opponents.

It is provided that when the player is defeated by an opponent the game ends.

All the possible ways the player can win is: {L, WL, WWL, WWWL and WWWW)

(a)

The results from all the 4 opponents are independent, i.e. the result of a game played with one opponent is unaffected by the result of the game played with another opponent.

The probability that the player defeats all four opponents in a game is,

P (Player defeats all 4 opponents) = [tex]P(W)\times P(W)\times P(W)\times P(W)=[P(W)]^{4} =(0.80)^{4}=0.4096[/tex]

Thus, the probability that the player defeats all four opponents in a game is 0.4096.

(b)

The probability that the player defeats at least two opponents in a game is,

P (Player defeats at least 2) = 1 - P (Player losses the 1st game) - P (Player losses the 2nd game) = [tex]1-P(L)-P(WL)[/tex]

                                    [tex]=1-(0.20)-(0.80\times0.20)\\=1-0.20-0.16\\=0.64[/tex]

Thus, the probability that the player defeats at least two opponents in a game is 0.64.

(c)

Let X = number of times the player defeats all 4 opponents.

The probability that the player defeats all four opponents in a game is,

P(WWWW) = 0.4096.

Then the random variable [tex]X\sim Bin(n=3, p=0.4096)[/tex]

The probability distribution of binomial is:

[tex]P(X=x)={n\choose x}p^{x} (1-p)^{n-x}[/tex]

The probability that the player defeats all the 4 opponents at least once is,

P (X ≥ 1) = 1 - P (X < 1)

             = 1 - P (X = 0)

             [tex]=1-[{3\choose 0}(0.4096)^{0} (1-0.4096)^{3-0}]\\=1-[1\times1\times (0.5904)^{3}\\=1-0.2058\\=0.7942[/tex]

Thus, the probability that the player defeats all the 4 opponents at least once is 0.7942.

Final answer:

The probability of defeating all four opponents in one game is 40.96%. The probability of defeating at least two opponents is 87.04%. Playing the game three times, the probability of defeating all four opponents at least once is 88.47%.

Explanation:

To calculate the probability of different outcomes when playing a video game against four opponents, we'll use basic probability rules and assumptions given in the question.

A. Probability of Defeating All Four Opponents

The probability of defeating each opponent is 80% or 0.8. Since the fights are independent, to find the probability of defeating all four, we multiply the probabilities together:
0.8 × 0.8 × 0.8 × 0.8 = 0.4096 or 40.96% chance of defeating all four opponents.

B. Probability of Defeating At Least Two Opponents

To find the probability of defeating at least two opponents, we must consider all possible combinations of winning 2, 3, or 4 opponents' fights, and add those probabilities together. Let W represent a win and L represent a loss:

We already calculated P(WWWW) as 0.4096. The other probabilities can be calculated using similar multiplication of respective individual probabilities (0.8 for W, 0.2 for L). After calculation, we sum these to find the cumulative probability, which is 0.8704 or 87.04%.

C. Probability of Defeating All Four Opponents At Least Once Over Three Games:

The probability of not defeating all four opponents in one game is 1 - P(WWWW) = 0.5904. The events in each game are independent, so the probability of not defeating all four opponents in all three games is (0.5904)^3. Therefore, by subtracting this result from 1, we get the probability of defeating all opponents at least once: 1 - (0.5904 × 0.5904 × 0.5904) = 0.8847 or 88.47%.

Answer:

The average weight of the newborn otter is  [tex]\frac{71}{40}\ kg.[/tex]

Step-by-step explanation:

Given:

A baby otter was born 3/4 of a month early at first it's weight was 7/8 kg which is 9/10 kg less than the average weight of newborn otter in the aquarium.

Now, to find the average weight of the newborn otter.

Let the average weight of the newborn otter be [tex]x.[/tex]

It's weight was = [tex]\frac{7}{8} \ kg.[/tex]

It's weight is less than the average weight of newborn by = [tex]\frac{9}{10} \ kg.[/tex]

According to question:

[tex]x-\frac{9}{10}=\frac{7}{8}[/tex]

Adding both sides by [tex]\frac{9}{10}[/tex] we get:

[tex]x=\frac{9}{10} +\frac{7}{8}[/tex]

[tex]x=\frac{36+35}{40}[/tex]

[tex]x=\frac{71}{40} \ kg.[/tex]

Therefore, the average weight of the newborn otter is [tex]\frac{71}{40}\ kg.[/tex]

Answer:

Each interior angle = 135 degrees,

Step-by-step explanation:

The exterior angles of all convex polygons add up to 360 degrees.

So for a regular octagon each exterior angle = 360 / 8

= 45 degrees.

Therefore each interior angle = 180 - 45 = 135 degrees,

Answer:

6

Step-by-step explanation:

8^2*(TS)^2=10^2

(TS)^2=36

TS=6

Answer:

Step-by-step explanation:

Triangle RST is a right angle triangle.

From the given right angle triangle,

RS represents the hypotenuse of the right angle triangle.

With m∠R as the reference angle,

RT represents the adjacent side of the right angle triangle.

ST represents the opposite side of the right angle triangle.

To determine m∠R, we would apply

the cosine trigonometric ratio.

Cos θ = adjacent side/hypotenuse. Therefore,

Cos m∠R = 8/10 = 0.8

m∠R = Cos^-1(0.8)

m∠R = 36.9 to one decimal place.

Answer:

         [tex]\large\boxed{\large\boxed{3\times 10^8meters}}[/tex]

Explanation:

One of the applications of scientific notation is to make estimations rounding the whole part of the numbers, i.e. the digits before the decimal point, to the nearest integer, and adding the exponents of the powers of base 10.

Here, you must estimate this product of two numbers written is scientific notation:

          [tex](3.21\times 10^3)(1.13\times 10^5)[/tex]

Then, for an estimation you round 3.21 to 3 and 1.13 to 1, then multiply 3 × 1 = 3. That will be the coefficient of your new power of 10.

The power or exponent will be the sum of the powers of the numbers that are being multiplied, i.e. 3 + 5 = 8.

And the result is [tex]3\times 10^8[/tex]

The unit is meters, so you write your answer as: [tex]3\times 10^8meters[/tex]

Answer:

b. 3*10^8

Step-by-step explanation: