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Apply Your Knowledge I will double the points if you get this completely done (that means both documents completed with the correct answers)

Answer:

Common ratio r = [tex]\dfrac{2}{3}[/tex]

Sequence  is described by
[tex]a_n = \dfrac{3}{2}\cdot \left(\dfrac{2}{3}\right)^{n-1}\\[/tex]

Step-by-step explanation:

[tex]S_n = \dfrac{a_1(1-r^n)}{1-r}[/tex]

where
r = common ratio
a₁ = first term

Sum of first two terms

[tex]S_2 = \dfrac{a_1(1-r^2)}{1-r}[/tex]

Sum of first four terms:
[tex]S_4 = \dfrac{a_1(1-r^4)}{1-r}[/tex]

[tex]\dfrac{S_4}{S_2} = a_1 \cdot \dfrac{1-r^4}{1-r} \div a_1 \cdot \dfrac{1-r^2}{1-r}\\[/tex]

To divide, flip the divisor and multiply

[tex]\dfrac{S_4}{S_2} =\dfrac{a_1(1-r^4)}{1-r} \times \dfrac{1-r}{a_1(1-r^2)}[/tex]

The a₁ and (1-r) terms cancel out from numerator and denominator leaving

[tex]\dfrac{S_4}{S_2} =\dfrac{1-r^4}{1-r^2} \cdots [1][/tex]  

Using the identity
[tex]a^2 - b^2 = (a- b)(a+b)[/tex]

[tex]1- r^4 = 1^4 - r^4 = (1^2 - r^2)(1^2+r^2) = (1-r^2)(1+r^2)[/tex]

Plugging this into equation 1 we get
[tex]\dfrac{S_4}{S_2} =\dfrac{(1-r^{2})(1+r^{2})}{1-r^{2}}[/tex]

The [tex]1- r^2[/tex] terms cancel out leaving:
[tex]\dfrac{S_4}{S_2} =1 + r^2[/tex]

We are given

[tex]S_4 =\dfrac{65}{18}\\\\S_2 = \dfrac{5}{2}[/tex]

[tex]\dfrac{S_4}{S_2} = \dfrac{65}{18} \div \dfrac{5}{2}[/tex]

To divide, flip the denominator [tex]\dfrac{5}{2}[/tex] and multiply
[tex]\dfrac{S_4}{S_2} = \dfrac{65}{18} \times \dfrac{2}{5}\\\\= \dfrac{13}{9}[/tex]

Therefore
[tex]1 + r^2 = \dfrac{13}{9}\\\\r^2 = \dfrac{13}{9} - 1\\\\= \dfrac{13}{9} - \dfrac{9}{9}\\\\= \dfrac{4}{9}[/tex]

[tex]r = \sqrt{\dfrac{4}{9}}\\\\= \dfrac{\sqrt{4}}{\sqrt{9}}\\\\= \dfrac{2}{3}[/tex]

So the common ratio
[tex]r = \dfrac{2}{3}[/tex]

To find the first term  we have sum of first two terms = 5/2

[tex]S_2 = \dfrac{a_1(1-r^2)}{(1-r)} = a_1 (1+ r)[/tex]

Plugging in knowns
[tex]\dfrac{5}{2} = a_1(1+\dfrac{2}{3})\\\\= a_1 \cdot \dfrac{5}{3}[/tex]

Multiply both sides by 3/5 to get
[tex]a_1 = \dfrac{5}{2} \times \dfrac{3}{5}\\\\a_1 = \dfrac{3}{2}[/tex]

The nth term of a GP is
[tex]a_n = a_1 \cdot r^{n-1}\\[/tex]

Plugging in the values obtained
[tex]a_n = \dfrac{3}{2}\cdot \left(\dfrac{2}{3}\right)^{n-1}\\[/tex]

Therefore , the solution of the given problem of area comes out to be the rectangle's dimensions are 6.5 yards and 10 yards, respectively.

By calculating how much space would be needed to fully cover its exterior, its overall size can be calculated. The neighboring area is considered when selecting a comparable item for a rectangular design. The surface area of something determines its overall dimensions. The number of sides between a cuboid four trapezoidal curves determines how much water it can hold.

Here,

Let w represent the rectangle's breadth.

In light of the issue statement, the rectangle's length is 2w - 3.

Since the rectangle's area is 65, we can construct the following equation:

=> w(2w - 3) = 65

By enlarging and condensing the left half, we obtain:

=>2w² - 3w - 65 = 0

Now that we have the quadratic formula, we can answer this quadratic equation:

=> w = (-b √(b² - 4ac)) / 2a.

where c = -65, b = 3, and a = 2. By replacing these numbers, we obtain:

=> w = (-(-3) ± √(-3)² - 4(2)(-65))) / 2(2)

=> w = (3 + √(529)) / 4

=> w = (3 ± 23) / 4

=> w = (3 + 23) / 4 = 6.5

=> l = 2w - 3 = 2(6.5) - 3 = 10

The rectangle's dimensions are 6.5 yards and 10 yards, respectively.

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The directiοnal derivative οf [tex]f(x,y) = 2ln(x^2 + y^2)[/tex] at the pοint (2,1) in the directiοn οf the vectοr v = (-1,2) is 0.

A directiοnal derivative shοws hοw quickly a functiοn changes in any directiοn. A fοrmula that incοrpοrates the gradient can be used tο determine the directiοnal derivative. The gradient οf a functiοn οf mοre than οne variable shοws the directiοn οf greatest change.

We must cοmpute the dοt prοduct οf the gradient οf f at (2,1) with the unit vectοr in the directiοn οf v in οrder tο determine the directiοnal derivative οf the functiοn  [tex]f(x,y) = 2ln(x2, + y2)[/tex]  at the pοsitiοn (2,1) in the directiοn οf the vectοr v = (-1,2).

Then, we determine f's gradient:

Grad(f) is equal tο [tex](f/x, f/y) = (4x/(x2+y2), 4y/(x2+y2)[/tex]

Hence, the gradient at lοcatiοn (2,1) is as fοllοws:

grad [tex](f)(2,1) = (4(2)/(2^2+1^2), 4(1)/(2^2+1^2)) = (8/5, 4/5)[/tex]

The unit vectοr in the directiοn οf v is then lοcated:

||v|| = √((-1)²+ 2²) = √5\su = v/||v|| = (-1/√5, 2/√5)

We calculate the directiοnal derivative last.

Def = grad(f) (2,1) · u\s= (8/5)(-1/√5) + (4/5)(2/√5)\s= -8/5√5 + 8/5√5\s= 0

Hence, at pοint (2,1) in the directiοn οf the vectοr v = (-1,2), the directiοnal derivative οf f(x,y) = 2ln(x², + y²) is equal tο 0.

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Based on Probability laws the answers are: (a) n(1-p) = 167.4 and (b) mean = 167.4

(a) To demonstrate that the probability distribution of X is roughly normal, we must determine if np and n(1-p), where n is the sample size and p is the chance of success, are both higher than or equal to 10. (using a variation of "lol").

If 55.8% of comments containing a textual form of laughter use some version of "lol," the numbers in this case are n = 300 and p = 0.558. Therefore:

[tex]n(1-p) = 300 * 0.442 = 132.6 np = 300 * 0.558 = 167.4[/tex]

The probability distribution of X is roughly normal since np and n(1-p) are both bigger than or equal to 10.

(b) The formula for the mean of the suitable normal distribution is: = np = 300 x 0.558 = 167.4.

Hence, 167.4 is the normal distribution's mean.

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Answer:

95.522

Step-by-step explanation:

First, we have to find the area of the triangle...

Next, we have to find the area of the sector...

Now we have to add our results together...

Hope this helps, have a lovely day! :)

The figure which could not be a cross section of a right circular cone is option a and d.

A right circular cone is a three-dimensional geometric shape that has a circular base and a vertex directly above the center of the base, such that the axis passing through the vertex and the center of the base is perpendicular to the base.

The base of the cone is a circle, and the sides of the cone slope upward to meet at the vertex. The distance from the vertex to the center of the base is called the height of the cone, denoted by h, and the distance from the center of the base to any point on the edge of the base is called the radius, denoted by r.

The lateral surface of the cone is curved, while the base and the vertex are flat. The angle formed between the height and the slant height (the distance from any point on the base to the vertex) is called the half vertical angle or the apex angle of the cone, denoted by θ.

The formula for the volume of a right circular cone is given by V = (1/3)πr²h, while the formula for the lateral surface area of a right circular cone is given by L = πrℓ, where ℓ is the slant height of the cone, which is equal to √(r² + h²).

Circle and eclipse cannot be cross section of a right circular cone.

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The value of x is 15 inches, we find this by Pythagoras theorem and property of isosceles triangle.

Pythagoras theorem states that sum of square of base and square of perpendicular is equal to square of hypotenuses.

So , it can be formula is given below ,

  [tex]perpendicular^{2} + base^{2} = hypotenuse^{2}[/tex]

And In ab isosceles triangle two equal side vertex divide base in two equal parts.

So, here base = 40.

Therefore it is divided in 20-20 inches,

Now putting the formula of Pythagoras theorem, we get

[tex]20^{2} + x^{2} = 25^{2}[/tex]

[tex]x^{2} = 25^{2} - 20^{2}[/tex]

[tex]x^{2} = 225[/tex]

So we get, x = 15 inches.

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Using the given information, we calculated the following prices:

a) The retail price is $420

b) The wholesale price is $55

What are retail and wholesale prices?

The processes of wholesale and retail are essentially distinct from one another because they entail the transfer of commodities from manufacturing to distribution. Purchasing products and selling them to clients constitute retail.

Retailers pay wholesale pricing to producers or distributors. Then, the shop charges customers a higher price—the retail price—for the identical product. Wholesale pricing is what you charge retailers who buy things in large amounts. With wholesale pricing, products are sold for more money than it costs to produce them in order to generate a profit.

The final selling price that retailers decide to charge customers is known as the retail price. In retail pricing, the customer is everything.

Let,

The retail price = x

The wholesale price = y

Then,

x = y + 0.4y = 1.4y

a) When y = $300

x = 1.4 * 300 = $420

b) When x = $77

77 = 1.4y

y = 77/1.4 = $55

Therefore using the given information, we calculated the following prices:

a) The retail price is $420

b) The wholesale price is $55

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Answer:

158.4 ft/sec

Step-by-step explanation:

multiply the speed value by approximately 1.467

Answer: Let's assume that the speed of the plane in still air is represented by p and the speed of the wind is represented by w.

When the plane is flying with the tailwind, its speed relative to the ground is the sum of its speed in still air and the speed of the wind, or (p + w). Similarly, when the plane is flying against the wind, its speed relative to the ground is the difference between its speed in still air and the speed of the wind, or (p - w).

We can set up two equations based on the given information:

(p + w) = 158 (1) (when flying with the tailwind)

(p - w) = 112 (2) (when flying against the wind)

To solve for p and w, we can add equations (1) and (2):

2p = 270

p = 135

So the speed of the plane in still air is 135 km/h.

We can then substitute this value of p into equation (1) to solve for w:

(p + w) = 158

(135 + w) = 158

w = 23

So the speed of the wind is 23 km/h.

Therefore, the plane flies at 135 km/h in still air and the wind blows at 23 km/h.

Step-by-step explanation:

The  surface area of the prism 1 is 56.25( optionC)

Similar shapes are shapes with thesame shape but not necessarily thesame side.

The ratio of the corresponding sides of prism 1 to prism 2 = 5/12

therefore 5/12 is the linear scale factor.

The area factor is the square of linear scale factor .

Therefore area factor = (5/12 )² = 25/144

To find the area of prism 1, we use the area factor.

Represent the area of prism 1 by x

25/144 = x/ 324

144x = 324 × 25

144x = 8100

divide both sides by 144

x = 8100/144

x = 56.25

therefore the area of prism 1 is 56.25

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Answer:

x = 36/7 (or approximately 5.143); y = 19/70 (or approximately 0.271)

Step-by-step explanation:

First, we can distribute the 2 to have both equations look similar:

[tex]2(0.5x-y)=4.6\\x-2y=4.6[/tex]

Now, we can eliminate the ys by multiplying the entire first equation by 2.  This will allow us to solve for x.

[tex]2(0.2x+y=1.3)\\\\0.4x+2y=2.6\\x-2y=4.6\\\\1.4x=7.2\\x=\frac{36}{7}\\ x=5.142857143[/tex]

Now, we can plug in 36/7 for x in any of the two equations:

[tex]0.2(\frac{36}{7})+y=1.3\\ \frac{36}{35}+y=1.3\\ y=\frac{19}{70}\\ y=0.2714285714[/tex]

If you use the decimal answers, you can round as much as you need.

Circumradius of the isosceles triangle = 20 units

A triangle with two sides that have the same length is said to be isosceles. Let's carry out a quick exercise to better comprehend this. Fold the corners of a rectangular piece of paper in half. From the top folded corner to the bottom border, draw a line. O, D, and C should be noted as the triangle's vertices.

Now quantify the OD and OC. Use several measures to perform this exercise again and look for patterns. It is evident that OD and OC are consistently  isosceles triangle. An isosceles triangle is a triangle with equal-lengths on both sides.

Now here circumradius, r = abc/√ [(a+b+c)(a+b-c)(a+c-b)(b+c-a)]

r = 29× 29 × 40/ √ (98×18×40×40)

= 29×29/42

≈20.02

=20

Hence the circumradius of the given triangle is 20 units.

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Answer:

y = - 9x + 6

Step-by-step explanation:

the equation of a linear function in slope- intercept function is

y = mx + c ( m is the slope and c the y- intercept )

given slope = - 9 and f(0) = 6 , that is the point (0, 6 ) , then c = 6

y = - 9x + 6 ← linear function

Answer:

Step-by-step explanation:

To fill in the table, we can use the following steps:

Calculate the interest for each month, which is the prior balance multiplied by the monthly interest rate (1.75%).

Subtract the monthly interest from the prior balance to get the new balance before the monthly payment.

Subtract the monthly payment from the new balance to get the ending balance.

Month         Prior     1.75% Interest on   Monthly                Ending

                Balance     Prior Balance      Payment              Balance

1                $5000           $87.50              $325                $4762.50

2               $4762.50 $83.39              $325                $4520.84

3               $4520.84 $79.06              $325                $4276.90

4               $4276.90 $74.50              $325                $4029.23

5               $4029.23 $69.69             $325                $3777.87

Therefore, after 5 months of making $325 payments, the remaining balance on the loan is $3777.87.

The amount of simple interest earned after 7 months is given as follows:

C $612.50

The equation that gives the balance of an account after t years, considering simple interest, is modeled as follows:

A(t) = P(1 + rt).

In which the parameters of the equation are listed and explained as follows:

The interest accrued over the period is modeled as follows:

I(t) = Prt.

The parameter values for this problem are given as follows:

Then the accumulated interest is given as follows:

I = 25000 x 0.0425 x 7/12

I = $612.50.

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Yοur friend's statement is nοt cοrrect.

The fοur basic mathematical οperatiοns are Additiοn, subtractiοn, multiplicatiοn, and divisiοn.

Nο, yοur friend's statement is nοt cοrrect.

The reasοn is that the twο temperatures are measured using different scales. The temperature inside the building is given in Celsius (°C), while the temperature οutside is given in Fahrenheit (°F). These are twο different scales, and a temperature οf 22°C is nοt equal tο a temperature οf 22°F.

Tο cοmpare the twο temperatures, we need tο cοnvert οne οf them tο the οther scale. One way tο dο this is tο use the cοnversiοn fοrmula:

°F = (°C × 9/5) + 32

Using this fοrmula, we can cοnvert the temperature inside the building frοm Celsius tο Fahrenheit:

°F = (22°C × 9/5) + 32 = 71.6°F

Nοw we can see that the temperature inside the building is much higher than the temperature οutside the building.

Hence, yοur friend's statement is nοt cοrrect.

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The prοbability that we have tο wait fοr mοre than 5 minutes fοr the bus tο arrive, given that we arrived precisely at nοοn, is 1. In οther wοrds, we will always have tο wait fοr mοre than 5 minutes, οn average, fοr the bus tο arrive.

We are given that X, the amοunt οf time the bus is late, is an expοnential randοm variable with mean οf 4 minutes. This means that the prοbability density functiοn οf X is:

f(x) = (1/4) * [tex]e^{(-x/4)[/tex], fοr x >= 0

We want tο find the prοbability that we have tο wait fοr mοre than 5 minutes fοr the bus tο arrive, given that we arrived at nοοn. Let Y be the tοtal amοunt οf time we have tο wait, which is the sum οf X and 5 minutes. Then:

Y = X + 5

The prοbability that we have tο wait fοr mοre than 5 minutes is:

P(Y > 5) = P(X + 5 > 5)

= P(X > 0)

Since X is an expοnential randοm variable with parameter 1/4, we can calculate this prοbability as:

P(X > 0) = ∫(0 tο ∞) f(x) dx

= ∫(0 tο ∞) (1/4) *[tex]e^{(-x/4)} dx[/tex]

[tex]= [-e^{(-x/4)}][/tex](0 tο ∞)

= 1

Therefοre, the prοbability that we have tο wait fοr mοre than 5 minutes fοr the bus tο arrive, given that we arrived precisely at nοοn, is 1. In οther wοrds, we will always have tο wait fοr mοre than 5 minutes, οn average, fοr the bus tο arrive.

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The area of the cross-section of the cone that is parallel to the base and 3 inches from the vertex is 27/8 square inches or 3 3/8 square inches

From the question, we are to determine the area of the cross-section of the cone

To determine the area of the cross-section of the cone, we will determine the radius of the cross-section.

Let the radius of the cross-section be x and radius of the base of the be r.

The height of the cone is 8 inches and the height of the cone formed from the cross-section is 3 inches

By similar triangle theorem, we can write that

8/r = 3/x

x = (3r)/8

From the given information,

The area of the base is 24 square inches

Thus,

πr² = 24

Therefore, r² = 24/π

The area of the cross-section will be

A = πx²

But, x = (3r)/8

Thus,

A = π [(3r)/8]²

A = π × 3²/8² × r²

A = π × 9/64 × 24/π

A = 27/8 square inches

Hence, the area is 27/8 square inches or 3 3/8 square inches

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The length of side BC is solved to be 24 units

The problem gave that the two triangles are similar triangles and the proportions will be used to find the length of BC as below

5 / (x - 4) = (x + 7) / 12

cross multiplying

60 = (x - 4) (x + 7)

60 = x^2 + 3x - 28

x^2 + 3x - 88 = 0

x = 8 or -11

take the positive value hence x = 8

Solving for BD

side DE = x - 4 = 8 - 4 = 4

taking proportions we have

DE / FE = AB / BC

4 / 8 = 12 / BC

BC = 8 * 12 / 4 = 24

therefore length of BC is equal to 24 units

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To predict the average number of people attending local baseball games in 2024, we need to use the formula:

A = P(1 + r)^n

where:

A = the predicted average attendance in 2024

P = the current average attendance (in 2020)

r = the annual growth rate (as a decimal)

n = the number of years of growth (from 2020 to 2024)

Given that the annual growth rate is 10% and the number of years of growth is 4 (from 2020 to 2024), we have:

r = 0.10

n = 4

Substituting these values and the current average attendance of 100 people into the formula, we get:

A = 100(1 + 0.10)^4

A = 100(1.10)^4

A = 100(1.4641)

A = 146.41

Therefore, the predicted average number of people attending local baseball games in 2024 is approximately 146 (rounded to the nearest whole number).

The vertices of the image of triangle ABC under the transformation (x, y)-- (x-3,y+1) followed by a dilation centered at the origin with scale factor 2 are A(2(x1-3),2(y1+1)), B(2(x2-3),2(y2+1)), C(2(x3-3),2(y3+1)).

Transformation in maths is the process of changing the position, size, orientation or shape of a figure or shape. It involves various techniques such as translation, rotation, reflection and scaling.

The transformation (x, y)-- (x-3,y+1) followed by a dilation centered at the origin with scale factor 2 is a combination of a translation and a dilation. A translation is a transformation that moves the shape without changing its size or orientation. In this case, the translation moves the triangle 3 units to the left and 1 unit up. A dilation is a transformation that enlarges or reduces a shape. In this case, the dilation uses a scale factor of 2, so the triangle will be twice as big as it was before.

The vertices of the original triangle ABC are (A(x1,y1), B(x2,y2), C(x3,y3)). After the translation, the vertices will be (A(x1-3,y1+1), B(x2-3,y2+1), C(x3-3,y3+1)). After the dilation, the vertices will be (A(2(x1-3),2(y1+1)), B(2(x2-3),2(y2+1)), C(2(x3-3),2(y3+1)).

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Answer: The median women's winning time of 146 minutes means that half of the winning times were below 146 minutes and half were above. This is a measure of the central tendency of the data.

The interquartile range (IQR) of 7 minutes gives us a measure of the spread of the data. The IQR is the difference between the third quartile (Q3) and the first quartile (Q1) of the data. It represents the middle 50% of the data.

In this case, we don't know the actual values of Q1 and Q3, but we can say that the IQR is about 7 minutes. This means that the difference between the winning times of the 75th percentile and the 25th percentile is about 7 minutes.

Overall, these statistics tell us that the winning times for women in the Greentree Marathon have a relatively narrow range, with most of the winning times falling within a 7-minute range around the median. However, we don't know anything about the distribution of the data beyond this range, such as whether there are any outliers or whether the data is symmetric or skewed.

Step-by-step explanation:

The two column proof that shows that ΔADC ~ ΔDBC is as shown below.

A two-column proof is defined as a geometric proof that is said to consist of a list of statements, and the reasons for which  those statements are true. The statements are thus listed in a column on the left, with the reasons for which the specific statements can be made are now listed in the right column.

The complete two column proof is expressed as:

Statement 1: ∠ADC is a right angle

Reason 1: Given

Statement 2: DB is perpendicular to AC

Reason 2: Given

Statement 3: ∠DBC is a right angle

Reason 3: Definition of Perpendicular

Statement 4: ∠ADC ≅ ∠DBC

Reason 4: All right angles are congruent

Statement 5: ∠C ≅ ∠C

Reason 5: Reflexive Property of Congruence

Statement 6: ΔADC ~ ΔDBC

Reason 6: AA Similarity

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Answer:

logra =70000

queda=90000

Step-by-step explanation:

additional

total income 70000+90000=160000

division by 2

160000/2

=$80.000

Answer:

x+3+3

Step-by-step explanation:

So bassicly you  have to add everything up ad then divide by 1

Therefore , the solution of the given problem of triangle comes out to be LJ rounded to the closest tenth, is about 7.5

A triangular is a polygon because it has two or maybe more additional sections. It has a straightforward rectangular form. A, B, and C are the only three sides that set a triangle apart from a simple triangle. When the boundaries are not exactly collinear, Euclidean geometry results in a singular area instead of a cube. If a shape has three edges and three angles, it is said to be triangular.

Here,

The respective sides of the triangles CHI and JKL are proportional because of their similarity. Using the matching sides, we can construct a proportion:

=> JK / CI = LJ / CH

The values JK = 15 and CI = 8 are provided. Using the data shown in the picture, we must locate CH and LJ.

We can see from the picture that HI = 4 and KL = 20. By taking HI away from CI, we can discover CH:

=> HI = 8 - 4 = 4 CH = CI =

We can now solve for LJ by substituting the numbers we already know into our proportion:

=> LJ / 4 = 15 / 8

=> LJ = 4 * 15 / 8

=> LJ = 7.5

LJ, rounded to the closest tenth, is about 7.5.

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Answer:

RESOLUCIÓN.

Para resolver este problema hay que seguir los siguientes pasos:

1) Determinar la comisión del vendedor de seguros.

Se sabe por datos del problema que cuando vendió $3000 su sueldo total fue de $850 y cuando vendió $4000 su sueldo total fue de $1000, eso quiere decir que:

Venta = 4000 - 3000 = $1000

Sueldo = 1000 - 850 = $150

Es decir que por cada $1000 en ventas, su ganancia es de $150.

1000 ---> 100%

150 ---> x

x = 150 * 100 / 1000 = 15%

Por lo tanto el porcentaje de comisión del vendedor es de 15% por venta.

2) Determinar el salario semanal fijo del vendedor.

Una vez conocido el porcentaje de comisión se puede obtener el salario fijo.

Para el primer caso:

850 = Salario fijo + Comisión

Comisión = 3000 * 15/ 100 = $450

Por lo tanto se tiene que:

850 = Salario fijo + 450

Salario fijo = $400

Por lo tanto se tiene que el salario semanal del vendedor es de $400.

Step-by-step explanation:

thy be c matey truth me ARG!!??

Answer:7$

Step-by-step explanation:$4+3$=7$