A large mixing tank currently contains 300 gallons of water, into which 8 pounds of sugar have been mixed. A tap will open,
pouring 20 gallons of water per minute into the tank at the same time sugar is poured into the tank at a rate of 2 pounds
per minute. Find the concentration (pounds per gallon) of sugar in the tank after 10 minutes.

The maximum profit of manufacturing company is  = $3150

If an equation is in ax + b = c and there is only one variable in the equation then it is called linear equation in one variable. a , b , c are real number and a is not equal to 0.

To maximize  profit, we must determine the number of trick  and slalom skis to produce each day, subject to the constraints of available labor hours.

Let x be the number of  skis and y be the number of slalom skis produced per day.

 The objective function to be maximized is:

Profit = 60x50 a

With the following limitations:

Production hours: 9x 6 a ≤ 432

Final hours: x y ≤ 63

Non-negativity constraints: x ≥ 0, y ≥ 0

We can solve this linear programming problem using graphical or algebraic methods. Here we  use algebraic methods:

From the second limit, we get y ≤ 63 - x.

If this inequality is replaced by the first limit, we get:

9 x 6 (63 - x) ≤ 432

3x ≤ 54

x ≤ 18

This means that the maximum production rate is 18 pieces per day. Replacing x = 18 by the inequality y ≤ 63 - x, we get:

y ≤ 63 – 18 = 45

So the maximum production is 45 slalom skis  per day.

 To verify that this is the correct solution, we need to ensure that it satisfies the non-negativity constraints:

x ≥ 0 and y ≥ 0 are already given in the assignment statement.  

Therefore, the maximum profit is:

Profit = 60(18) 50(45) = $3150

The biggest profit is when 18 trick skis and 45 slalom skis are made per day.

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The equation of the line passing through points [tex]A(a,b)[/tex] and [tex]B(b,a)[/tex] is [tex]x + y = a + b[/tex]  , and the length of   [tex]AB[/tex] is  [tex]\sqrt(2(a - b)^2).[/tex]

To find the equation of the line passing through points A(a,b) and B(b,a), we can first find the slope of the line. using the two given points, and then use the point-slope form of a line equation.

The slope of a line passing through two points ([tex]x_{1} , y_{1}[/tex]) and ([tex]x_{2}, y_{2}[/tex]) is given by:

[tex]m = (y_{2} - y_{1} ) / (x_{2} - x_{1} )[/tex]

Applying this formula, we can find the slope of the line passing through points A(a,b) and B(b,a):

Slope [tex](m) = (a - b) / (b - a)[/tex]

Since a and b can take any real values, we can see that the slope of the line passing through points A and B is -1.

Now, let's find the equation of the line passing through points A and B using the point-slope form of a line equation:

Point-slope form: [tex]y - y_{1} = m(x - x_{1} )[/tex]

Substituting the slope (-1) and the coordinates of point A(a,b) into the point-slope form, we get:

[tex]y - b = -1(x - a)[/tex]

Expanding the equation, we get:

[tex]y - b = -x + a[/tex]

Rearranging the terms, we get the equation of the line passing through points A(a,b) and B(b,a):

[tex]x + y = a + b[/tex]

Now, let's find the length of AB using the distance formula:

The distance between two points [tex](x_{1} , y_{1} )[/tex] and  [tex](x_{2} , y_{2} )[/tex] is given by:

[tex]d = \sqrt((x_{2} - x_{1} )^2 + (y_{2} - y_{1} )^2)[/tex]

Applying this formula, we can find the length of AB using the coordinates of points A(a,b) and B(b,a):

Length of  [tex]AB (d) = \sqrt((b - a)^2 + (a - b)^2)[/tex]

Simplifying the expression, we get:

Length of [tex]AB (d) = \sqrt(2(a - b)^2)[/tex]

Therefore, The equation of the line passing through points [tex]A(a,b)[/tex] and [tex]B(b,a)[/tex] is [tex]x + y = a + b[/tex]  , and the length of   [tex]AB[/tex] is  [tex]\sqrt(2(a - b)^2).[/tex]

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Answer:

[tex]\frac{12}{35}[/tex]

Step-by-step explanation:

[tex]\frac{2}{5}[/tex] x [tex]\frac{6}{7}[/tex] = [tex]\frac{12}{35}[/tex]

[tex]\frac{12}{35}[/tex] is already in simplest form, so the answer is [tex]\frac{12}{35}[/tex]

Hello!

Lets consider what the question asks for:

 --> graphs of either line intersecting/parallel/identical

 --> # of solutions the system has

The information given:

  6x + y = 25

  x + 3y = 3

Let's solve the FIRST problem

  --> graphs of either line intersecting/parallel/identical

             --> let's put both equations into slope-intercept form, where we

                   isolate the y-variable all by itself to one side

                               [tex]6x+3y=25\\3y=-6x+25\\\\y=-2x+\dfrac{25}{3}[/tex]

                                [tex]x+3y=3\\3y=-x+3\\y=-\dfrac{1}{3}x+1[/tex]

             --> both equations AREN'T parallel as the slope (coefficient of x-

                   variable) aren't equal

             --> both equations AREN'T identical as they don't look the same

              --> both equations ARE INTERSECTING as they have different

                   slopes and aren't identical

Let's solve our second question:

   --> how many solutions does the system have:

       --> both lines intersect

           -->both lines are linear as the highest power that the x-variable

                has are 1

                 --> thus there is only ONE SOLUTION

Answer:

 Intersecting

 Has ONE SOLUTION

Step-by-step explanation:

I = E/R    multiply both sides of the equation by R

I R = E / R  * R

I R = E     Now divide both sides by I

R  = E/ I

Answer:

In-text citations are the short citations you include in the written text that help a reader to understand which sources you are quoting or referring to in your writing example(Field, 2022). Reference is the full details of the source you have cited in your writing.

The value of y for the given quadrilateral is 5.

A closed quadrilateral has four sides, four vertices, and four angles. It is a form of polygon. In order to create it, four non-collinear points are joined. Quadrilaterals always have a total internal angle of 360 degrees.

The Latin words quadra, which means four, and latus, which means sides, are combined to form the English term quadrilateral. A quadrilateral does not always have to have equal lengths on each of its four sides. As a result, depending on the sides and angles, we might have several sorts of quadrilaterals.

We know that the opposite sides of the quadrilateral have a sum of 180 degrees.

Thus,

23y + 13y = 180

36y = 180

y = 5

Hence, the value of y for the given quadrilateral is 5.

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Answer:

x = 4,5

Step-by-step explanation:

These two triangles are similar (according to 3 sides):

The three sides of one triangle are proportional to the three sides of the other triangle:

[tex] \frac{7.5}{5} = \frac{6}{4} = \frac{x}{3} [/tex]

[tex] \frac{6}{4} = \frac{x}{3} [/tex]

Use the property of proportion to find x (cross-multiply):

[tex]4x = 6 \times 3[/tex]

[tex]4x = 18[/tex]

Divide both sides by 4 to make x the subject:

[tex]x = 4.5[/tex]

The value of (f o g)(3) using composite function is 35

Composite function refers to the combining of functions in a manner where the output from one function becomes the input for the next function.

To calculate the composite function (f o g)(3), we use the techniques below.

From the question,

Given:

Note that,

Step 1: Find g(3)

Therefore,

Step 2: To find f(g)(3), we subetitute the value of x in f(x) by 12

Hence, (f o g)(3) is 35

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The value of j is equal to 9√3 meters.

In Mathematics and Geometry, a 30-60-90 triangle is also referred to as a special right-angled triangle and it can be defined as a type of right-angled triangle whose angles are in the ratio 1:2:3 and the sides are in the ratio 1:√3:2.

This ultimately implies that, the length of the hypotenuse of a 30-60-90 triangle is double (twice) the length of the shorter leg (adjacent side), and the length of the longer leg (opposite side) of a 30-60-90 triangle is √3 times the length of the shorter leg (adjacent side).

Note: j is the side opposite to the 60° angle.

The side opposite to the 60° angle, j = 9 × √3

The side opposite to the 60° angle, c = 9√3 meters.

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Answer:

Step-by-step explanation:

16

lbozo

Triangle 1:
A = 45 degrees, C = 40 degrees, c = 4.2
First, we need to find angle B.
B =[tex] 180 - (A + C) = 180 - (45 + 40) = 95[/tex] degrees

Now, we can use the Law of Sines to find side b.
[tex]b/sin(B) = c/sin(C)[/tex]
b =[tex] (c * sin(B)) / sin(C)[/tex]
b =[tex] (4.2 * sin(95)) / sin(40)[/tex]
b ≈ [tex]5.6[/tex]

Triangle 2:
N = 45 degrees, M = 180 - N = 135 degrees, m = 2
To find side n, we can use the Law of Sines again.
[tex]n/sin(N) = m/sin(M)[/tex]
n =[tex] (m * sin(N)) / sin(M)[/tex]
n =[tex] (2 * sin(45)) / sin(135)[/tex]
n ≈ [tex]1.4[/tex]

So, the solutions for the two triangles are:
Triangle 1: A = 45 degrees, B = 95 degrees, C = 40 degrees, b ≈ 5.6, c = 4.2
Triangle 2: N = 45 degrees, M = 135 degrees, n ≈ 1.4, m = 2

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Jane won 90% of the video games. The solution has been obtained by using the arithmetic operations.

What are arithmetic operations?

The four fundamental operations, sometimes known as "arithmetic operations," are believed to adequately characterise all real numbers. Division, multiplication, addition, and subtraction are followed by the mathematical operations quotient, product, sum, and difference.

We are given that Jane won 9 out of the 10 video games she played.

Let the percentage be x.

Using the multiplication operation, we get,

⇒ 10 * x% = 9

⇒ 10 * [tex]\frac{x}{100}[/tex] = 9

⇒  [tex]\frac{x}{10}[/tex] = 9

⇒ x = 90

Hence, Jane won 90% of the video games.

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Since, there are multiple questions so, the question answered above is:

Question: Jane won 9 out of the 10 video games she played. What percent of the video games did Jane win?

The answer of the given question based on graph is Part A: the unit rate of square feet painted per minute is 1.125, so the ordered pair that displays this unit rate is (1, 1.125) and Part B:  the points that lie on the line are (3.5,3.9375), (4,5), and (32,36).

Slope refers to the steepness of a line or a surface. More specifically, slope is defined as the ratio of the vertical change (rise) between two points to the horizontal change (run) between the same two points on a line or a surface. In other words, slope measures how much a line or a surface rises or falls as it moves horizontally.

Since the relationship between the minutes worked and the square feet painted is proportional, we can use the slope-intercept form of a linear equation to represent it:

To find the equation of the line, we can use the given point (16,18) and the fact that the relationship is proportional. If we let the unit rate be r (in square feet per minute), then we have:

18 = r(16)

r = 18/16

r = 1.125

So the equation of the line is:

y = 1.125x

Part A: The unit rate of square feet painted per minute is 1.125, so the ordered pair that displays this unit rate is (1, 1.125), since the slope represents the unit rate in a proportional relationship.

Part B: To determine which of the points lie on the line, we can plug in the x-values into the equation y = 1.125x and see if the resulting y-values match.

(y - 18) / (x - 16) = slope

Using the slope calculated above, we can simplify this equation to:

y = 1.125(x - 16) + 18

Now, we can plug in the x and y values for each point and check if they satisfy this equation:

(2,2.5): 2.5 = 1.125(2 - 16) + 18, not on the line

(3.5,3.9375): 3.9375 = 1.125(3.5 - 16) + 18, on the line

(4,5): 5 = 1.125(4 - 16) + 18, on the line

(32,36): 36 = 1.125(32 - 16) + 18, on the line

(10,11.5): 11.5 = 1.125(10 - 16) + 18, not on the line

(0,0): 0 = 1.125(0 - 16) + 18, not on the line

Therefore, the points that lie on the line are (3.5,3.9375), (4,5), and (32,36).

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The functiοn is increasing and has a dοmain οf (1, ∞) is οptiοn D f(x) = lοg (x - 1) + 1.

Every equatiοn with οne οr mοre ratiοnal expressiοns is said tο be ratiοnal. A fractiοn whοse numeratοr and denοminatοr are pοlynοmials is knοwn as a ratiοnal expressiοn. A ratiο οf twο pοlynοmials is, in οther wοrds, a ratiοnal expressiοn.

The LCD (Least Cοmmοn Denοminatοr) οf the fractiοns, eliminatiοn οf the denοminatοrs, and simplificatiοn οf the resultant equatiοn are methοds fοr sοlving ratiοnal equatiοns. Sοlving the resultant equatiοn—which can require factοring, simplificatiοn, οr the use οf οther algebraic strategies—will prοvide the answers tο a ratiοnal equatiοn.

Fοr the given functiοn:

Optiοn A:

f(x) = -lοg(x - 1) + 2

Here, lοg is negative, hence fοr increasing value οf x f(x) will decrease.

Optiοn B:

f(x) = -lοg(x - 2) + 1

Here, lοg is negative, hence fοr increasing value οf x f(x) will decrease.

Optiοn C:

f(x) = lοg(x - 2) + 1

Here, (x - 2) > =

x > 2

The dοmain dοes nοt satisfy.

Optiοn D:

f(x) = lοg (x - 1) + 1

Here, (x - 1) > 0

x > 1 tο infinity.

Hence, the functiοn is increasing and has a dοmain οf (1, ∞) is οptiοn D f(x) = lοg (x - 1) + 1.

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The complete function is:

The area of a rectangle is given by the formula:

Area = length x width

In this case, the length of the park is x + 5 and the width is x - 2. So, we can substitute these expressions into the formula to get:

Area = (x + 5) x (x - 2)

Expanding the product, we get:

Area = x^2 + 5x - 2x - 10

Simplifying, we have:

Area = x^2 + 3x - 10

Therefore, the area of the rectangular park is given by the expression x^2 + 3x - 10.

Answer: Let's assume that the width of the rectangular room is "x" meters. Then, according to the problem, the length of the room is 2 times the width, which means the length is "2x" meters.

The perimeter of the room is the sum of all four sides, so we can write:

Perimeter = 2 × (length + width)

Substituting the values, we get:

48 = 2 × (2x + x)

Simplifying the expression, we get:

48 = 2 × 3x

Dividing both sides by 2, we get:

24 = 3x

Solving for "x", we get:

x = 8

Therefore, the width of the room is 8 meters and the length is 2 times the width, which is 2 × 8 = 16 meters.

So the dimensions of the room are 8 meters by 16 meters.

Step-by-step explanation:

The width of the added strip is 10 meters.

What is Area?

Area is a measure of the amount of surface enclosed by a two-dimensional shape or figure. It is usually expressed in square units, such as square meters or square inches, and is calculated by multiplying the length of a side or a diameter by itself or by using an appropriate formula for the shape.

If the area of the skating rink is doubled by adding a strip of equal width along one side and another strip at one end, then the new dimensions of the skating rink would be

=>  (30 + x) × (20 + x),

where x is the width of the added strip.

According to the problem, the new area of the skating rink is double the original area, so we can set up the following equation:

=> (30 + x)(20 + x) = 2(30 ×20)

Expanding the left-hand side and simplifying, we get:

=> 600 + 50x + x² = 1200

Rearranging and solving for x, we get:

=> x² + 50x - 600 = 0

Factoring the left-hand side, we get:

=> (x + 60)(x - 10) = 0

Therefore, the possible values for x are x = -60 and x = 10. Since the width of the strip cannot be negative, we reject x = -60 and conclude that the width of the added strip is 10 meters.

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Answer:

Step-by-step explanation:

5 + x (where x represents the number)

x / (-2)

2x

x - 7

12x

x - 3

$15h (where h represents the number of hours worked)

C - $5 (where C represents the cost of the ticket)

x + 24

50 / n (where n represents the number of students)

(2/5)x

(1/2)x - 19

-4x - 11

8x + 5

x^2 - 14

(3/4)x + 17

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

Five added to a number

The result of dividing a number by -2

Two times a number

A number decreased by 7

The result of multiplying a number by 12

A number decreased by 3

Earning $15 per hour

The cost of the ticket minus $5

The sum of a number and 24

Dividing 50 markers among several students

Two-fifths times a number

Half of a number minus 19

The result of subtracting 11 from the product of a number and -4

The sum of eight times a number and five

The difference between the square of a number and 14

Three-fourths times a number plus 17

Answer:

x = 4 is a straight, vertical line at 4 on the x axis.

The test statistic is less than the critical value, so, we reject the null hypothesis and conclude that there is a statistically significant difference between the population means.

A two-sample t-test is used to compare the means of two samples to determine if there is a statistically significant difference between them.

To answer this question, we must conduct a two-sample t-test.

In this case, the null hypothesis is that there is no difference between the population means, and the alternative hypothesis is that there is a difference between the population means.

The test statistic is calculated using the following formula:

t = (x₁ - x₂) / (√((s₁²/n₁) + (s₂²/n₂)))

Where x₁ and x₂ are the sample means of the two samples, s₁ and s₂ are the sample standard deviations, and n₁ and n₂ are the sample sizes.

Plugging in the sample data, we get:

t = (22 - 26) / (√((4.3²/11) + (4.4²/8)))

= -1.98

The critical value of t for a two-tailed test at the 0.01 significance level is 2.58.

Since the test statistic is less than the critical value, we reject the null hypothesis and conclude that there is a statistically significant difference between the population means.

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Question:

A random sample of 11 observations from one population revealed a sample mean of 22 and a sample standard deviation of 4.3. A random sample of 8 observations from another population revealed a sample mean of 26 and a sample standard deviation of 4.4. At the 0.01 significance level, is there a difference between the population means?

[tex]~~~~~~ \textit{Simple Interest Earned Amount} \\\\ A=P(1+rt)\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill & \$9000\\ r=rate\to 4\%\to \frac{4}{100}\dotfill &0.04\\ t=years\dotfill &1 \end{cases} \\\\\\ A = 9000[1+(0.04)(1)] \implies A = 9360[/tex]

Answer:

Question 1 C. 25 1/3

Question 2 C. 19

Step-by-step explanation:

brief :

1. 6 1/3 = 19/3

19/3 * 4 = 76/3

76/3 = 25 1/3

To solve this problem, we need to use the information provided and do some simple arithmetic.

Question 1:

The candy store sold 6 1/3 pounds of white chocolate. Since it sold 4 times as much milk chocolate as white chocolate, we can find the amount of milk chocolate sold by multiplying 6 1/3 by 4:

6 1/3 * 4 = 25 1/3

Therefore, the candy store sold 25 1/3 pounds of milk chocolate last week.

Answer: C. 25 1/3

Question 2:

To find the difference between the amount of milk chocolate and white chocolate sold, we can subtract the amount of white chocolate from the amount of milk chocolate:

25 1/3 - 6 1/3 = 19

Therefore, the candy store sold 19 more pounds of milk chocolate than white chocolate.

Answer: C. 19

chatgpt

The dilation of the black figure to blue figure is a reduction with a scale factor of 0.5

A dilation is a transformation that produces an image that is the same shape as the original, but is a different size.

It can either be an enlargement or a reduction.

An enlargement means a scale factor greater than 1 while a reduction is a scale factor less than 1.

Thus, this dilation is a reduction.

Coordinates of initial figure are:

(-8, 10), (10, 10), (8, -8), (2, -8)

Coordinates of dilated figure are:

(1, -4), (4, -4), (-4, 5), (5, 5)

Thus, it is clear that the scale factor is 0.5

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The probability that the contestant wins at least one car is 13/35.

A television game show has 15 doors, of which the contestant must pick 3. Behind 3 of the doors are expensive cars, and behind the other 12 doors are consolation prizes

probability of an event =total favorable outcome for event/total possible outcomes

P(contestant wins at least one car) will be:

= 1 - 12/15 × 11/14

=13/35

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Therefore, only the points (4, 1) and (3, 8) can be used to find the distance between the points (4, 1) and (3, 8) using the Pythagorean theorem.

In mathematics, a coordinate is a set of numbers that uniquely determines the position of a point or an object in space. The most common coordinate system is the Cartesian coordinate system, which uses two or three numerical values to locate points on a plane or in space.

Here,

To find the distance between the points (4, 1) and (3, 8) using the Pythagorean theorem, we need to use the coordinates of these two points to form a right-angled triangle, with the line segment connecting the two points as its hypotenuse. Therefore, the points that can be used to find the distance between the points (4, 1) and (3, 8) using the Pythagorean theorem are:

(4, 1) and (3, 8)

The point (0, 0) cannot be used to find the distance between the two given points using the Pythagorean theorem because it is not on the line segment connecting the two points.

The point (4, 8) cannot be used to find the distance between the two given points using the Pythagorean theorem because it is not on the line segment connecting the two points.

The point (4, 3) cannot be used to find the distance between the two given points using the Pythagorean theorem because it is not on the line segment connecting the two points.

The point (1, 8) cannot be used to find the distance between the two given points using the Pythagorean theorem because it is not on the line segment connecting the two points.

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