Decrease R450 in the ratio 9:8​

A triangle has sides with lengths of 3 feet, 4 feet, and 5 feet which is a right triangle.

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees (a right angle). The side opposite the right angle is called the hypotenuse, while the other two sides are called the legs.

The converse of the Pythagorean theorem states that if a triangle has sides of lengths a, b, and c, and a² + b² = c², then the triangle is a right triangle.

In this case, the triangle has sides of lengths 3 feet, 4 feet, and 5 feet. To determine whether it is a right triangle, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse  is the same as the total of the squares formed by the other two sides' lengths.

If the triangle is a right triangle, then the longest side (the hypotenuse) must satisfy this equation. Thus, we have:

5² = 3² + 4²

25 = 9 + 16

25 = 25

Since this equation is true, we can conclude that the triangle is a right triangle.

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After closely analyzing the line graph, it can be seen that the number of pygmy marmosets weighing more than 1/2 ounce are 8.

To create a line plot, we plot each weight of the each pygmy marmosets on the y-axis and the pygmy marmoset number on the x-axis, which will be ultimately being connected using the points with a line.

After ploting the each mentioned weights (in ounces ) and taking a close study of the plotted line graph,  we can see that there are total 8 pygmy marmosets which are weighing more than 1/2 ounce.

Therefore, the total number of pygmy marmosets with weight more than 1/2 ounces are 8.

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The complete question is :

The table shows the weights of 10 newborn pygmy marmosets make a line plot to display the data. How many pygmy marmosets weigh more than 1/2 ounce ?

Pygmy Marmoset      Weight (ounces)

1                                 0.4

2                                 0.5

3                                 0.6

4                                 0.7

5                                 0.8

6                                 0.9

7                                 1.0

8                                 1.1

9                                 1.2

10                                 1.3

Answer:

The largest knitting needle that the box can hold would have to fit inside the box diagonally from corner to corner.

Using the Pythagorean theorem, we can find the diagonal of the box:

d = sqrt(40^2 + 30^2 + 20^2)

d = sqrt(1600 + 900 + 400)

d = sqrt(2900)

d = 53.85 cm (rounded to two decimal places)

Therefore, the largest knitting needle that the box can hold would have a length of 53.85 cm or less.

The steps which can help in minimizing the variance in the four cards are correct selection of the suit with maximum number of cards.

If you have four cards and were to bet based upon suits while flipping them one at a time while trying to minimize variance, you should follow the below steps:

First, you need to check the suits of the four cards you have.

Now, you need to select the suit that has a maximum number of cards. For instance, if the four cards are 5 of hearts, 7 of spades, 2 of diamonds, and 10 of hearts, you need to select hearts as it has two cards.

After selecting the suit with the maximum number of cards, you need to start flipping the cards one at a time.

Keep a count of the number of times you get the chosen suit. For instance, if you chose hearts, you will count the number of times you get hearts after flipping each card.

Once you get the chosen suit, you can place your bet. The amount of your bet can depend on the number of times you have gotten the chosen suit so far.

So, this process doesn't guarantee that you will win every time.

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Answer:

a) 44 children can safely play in the playground of area 154 m^2.

b) The smallest playground area in which 24 children can play is 84 m^2.

Step-by-step explanation:

We have the ratio 210m^2 : 60.

a) 154/210 is 11/15. Multiplying this scale factor gives the ratio 154 m^2 : 44.

44 is found by multiplying 11/15 by 60.

44 children can safely play in the playground of area 154 m^2.

b) 24/60 is 2/5. Multiplying this scale factor gives the ratio 84 m^2 : 24

84 is found by multiplying 2/5 by 210.

The smallest playground area in which 24 children can play is 84 m^2.

Hope this helps!

The simplified expression using De Morgan's law are  a)x'y'z'u  b)x'y'u c): x'y'u and d)x'y'z'+xy'z'+xyz.

The main idea is to simplify each Boolean expression by repeatedly applying De Morgan's law until each complement operation operates on a single variable.

Then, apply the double complement law to simplify the expression further. In the end, the simplified expression should contain only AND and OR operations without any complement operators acting on multiple variables.

a. 1/ x + yz + u can be simplified using De Morgan's law to: (x'y'z')u'. Then, applying the double complement law, we get the simplified expression as: x'y'z'u.

b. 1/x(y + 2)u can be simplified using De Morgan's law to: x'(y'+2')u'. Then, applying the double complement law, we get the simplified expression as: x'y'u.

c. 1/(x + y)(uv + xy) can be simplified using De Morgan's law to: (x'y')(u' + x'y'). Then, applying the double complement law, we get the simplified expression as: x'y'u.

d. 1/xy + yz + xz can be simplified using De Morgan's law to: (x'+y')(y'+z')(x'+z'). Then, applying the double complement law, we get the simplified expression as: x'y'z'+xy'z'+xyz.

In summary, to simplify Boolean expressions, we can apply De Morgan's law repeatedly and then use the double complement law to remove complement operators acting on a single variable twice.

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Answer:

[tex]=20x^{2} +2x-6[/tex]

Step-by-step explanation:

First step: you're going to multiply each variable from the first bracket to the first and second variable from the second bracket:

=4x(5x)+4x(-2)

=[tex]20x^{2} -8x[/tex]

Second step: now you're going to do the same thing but for the second variable:

=2(5x)+3(-2)

=10x-6

we're going to add the answers that we got from the first two steps to get the final answer:

[tex]=(20x^{2} -8x)+(10x-6)\\[/tex]

[tex]=20x^{2} +(-8x+10x)+(-6)\\=20x^{2} +2x-6[/tex]

The expected value (μ) of the probability distribution of the discrete random variable X is μ = 7.26.

The correct option is B.

In probability theory, the expected value (or mean) is a measure of the central tendency of a random variable. It represents the average value that we would expect to observe if we repeated an experiment or random process many times.

Now, The expected value (μ) is calculated as:

μ = Σ(x P(X = x))

Let's calculate the expected value using the given probability distribution:

= (1 x 0.23) + (3 x 0.09) + (5 x 0.05) + (7 x 0.01) + (9 x 0.30) + (11 x 0.21) + (13 x 0.11)

= 0.23 + 0.27 + 0.25 + 0.07 + 2.7 + 2.31 + 1.43

= 7.26

Therefore, the expected value (μ) of the probability distribution is μ = 7.26.

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The total cost that he has to pay is 80 rupees which is calculated by addition of costs of all the things bought by him.

In math, addition is the act of adding two or more numbers together. The numbers being added are known as addends, and the outcome of the addition process, or the ultimate answer, is known as the sum. It is one of the fundamental arithmetic operations we employ on a daily basis.
we are given that:-
- the price of one pencil= 4.00 rs.
- the price of one pen= 6.00 rs.
- the price of one book = 11.00 rs.
- Mr, James buy : 4 books
- 3 pencils and
- 4 pens.

now, we have to find the total cost he has to pay.
As we are given that the price of one book is 11.00 rs , the price of the 4 books= 11.00*4= 44.00 rs.
The price of one pen= 6 rs
the price of the 4 pens = 6*4= 24 rs.
Similarly the price of one pencil is 4 rs
The price of 3 pencils= 4*3= 12 rs.

Now, add all of these costs that we have calculated above, we will get the total cost:-

44+24+12= 80 rs.

Hence, the total cost which he has to pay is 80 ruperupeeses.

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Answer:

Elise's average pay per hour for the year 2017 was £11.88.

Step-by-step explanation:

First, we need to calculate Elise's monthly salary in 2017 after the 3% increase:

3% of £2100 = 0.03 x £2100 = £63

Elise's new monthly salary in 2017 = £2100 + £63 = £2163

Next, we need to calculate her total earnings in 2017:

Total earnings = monthly salary x number of months worked

Elise worked for 45 weeks in 2017, which is approximately 10.4 months:

Total earnings = £2163 x 10.4 = £22,477.20

Finally, we can calculate her average pay per hour for the year:

Total number of hours worked = 42 hours/week x 45 weeks = 1890 hours

Average pay per hour = total earnings / total number of hours worked

Average pay per hour = £22,477.20 / 1890 hours = £11.88 per hour

Answer:

I got $13.73 per hour

Step-by-step explanation:

If you multiply 2100 by 12 months you get $25200 for an annual salary before percent increase.

Then you include the 3% salary increase each month 25200 x .03 = 756 then add the two 25200 + 756 = 25956 OR you can calculate the longer way by doing each individual month, either way it will be the same answer. 2100 x .03 = 63 then 2100 + 63 = 2163 then 2163 x 12 = 25956

Then you divide 25956 by the number of weeks worked in the year. 25956/45 = 576.8

So, each week $576.8 was earned.

Then to get an hourly rate, you divide this by the number of hours worked in each week. 576.8/42 = 13.73

So, the hourly rate is $13.73.

I hope this helped :)

Using the digits 9, 8, 7, 6, 5, 4, 3, 2, and 1 in that order to create 100 by using the mathematical operations, (9 + 8 + 7) x (6 - 5) x (4 + 3 + 2) - 1 = 100

It is not possible to create 100 using the digits 9, 8, 7, 6, 5, 4, 3, 2, and 1 in that order without using any mathematical operations.

However, it is possible to create 100 by using mathematical operations like addition, subtraction, multiplication, division, and parentheses. One possible way to do this is:

(9 + 8 + 7) x (6 - 5) x (4 + 3 + 2) - 1 = 100

Here, we first add the first three digits (9 + 8 + 7 = 24), subtract the fourth and fifth digits (6 - 5 = 1), add the next three digits (4 + 3 + 2 = 9), and finally multiply all three results and subtract the last digit (24 x 1 x 9 - 1 = 215 - 1 = 100).

So, using this sequence of digits and mathematical operations, we can create 100.

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Problem 1 (3 pts). The nearest neighbor method is used with the following labeled training data in a two class, two feature problem. Show clearly the decision boundaries obtained. Indicate all break points and slopes correctly Class1={(0,0)^T,(0,1)^T}. Class2={(1,0)^T,(0,0.5)^T}.

The decision boundary is the line that separates the two classes. When it comes to the nearest neighbour method, the boundary of a class is determined by the closest data point in the other class.What is the nearest neighbour method?The nearest neighbour method (NNM) is a type of lazy learning method in which the data is stored and the computation is deferred until the classification phase.

The goal of the nearest neighbor technique is to use the pattern of the closest data point to classify a new sample. It's also one of the simplest non-parametric classification methods, and it's based on the assumption that the feature spaces used by each class are continuous regions.For Class 1, there are two labeled training data points: (0, 0)T and (0, 1)T. Similarly, for Class 2, there are two labeled training data points: (1, 0)T and (0, 0.5)T.

To generate decision boundaries, follow the steps below:Step 1: Plot the given labeled training data. They are shown in the figure below. Step 2: Label the data points according to their class: Class 1 and Class 2.Step 3: Now, we need to find the nearest neighbors. Find the nearest neighbor for each of the labeled training data points from the opposite class. The distances are calculated as follows:For the Class 1 data points, the nearest neighbor is Class 2's (1, 0)T data point.

For the Class 2 data points, the nearest neighbour is Class 1's (0, 1)T data point.Step 4: Connect the nearest neighbour's labeled training data points with a line. These are the decision boundaries. The red line represents the decision boundary between Class 1 and Class 2. The green line represents the decision boundary between Class 2 and Class 1. The slopes of the decision boundaries are -1 for the red line and 1 for the green line. These slopes are the result of the negative reciprocal of the nearest neighbour's slope. The break point for the red line is 0.5, while the break point for the green line is 0. A break point is the point at which the decision boundary intersects the y-axis.

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The division of 4x³ + 14x² + 13x + 11 by x + 2 will have a quotient of 4x² + 14x + 1 and a remainder of 8 using synthetic division and can be expressed as (4x² + 14x + 1) + 9/(x + 2).

The procedure for synthetic division involves the following steps:

Divide.

Multiply.

Subtract.

Bring down the next term, and

Repeat the process to get zero or arrive at a remainder.

We shall divide 4x³ + 14x² + 13x + 11 by x + 2 as follows;

4x³ divided by x equals 4x²

x + 2 multiplied by 4x² equals 4x³ + 8x²

subtract 4x³ + 8x² from 4x³ + 14x² + 13x + 11 will give us 6x² + 13x + 11

6x² divided by x equals 6x

x + 2 multiplied by 6x equals 6x² + 12x

subtract 6x² + 12x from 6x² + 13x + 11 will give us x + 11

x divided by x equals 1

x + 2 multiplied by 1 equals x + 2

subtract x + 2 from x + 11 will give result to a remainder of 9.

Therefore by synthetic division, 4x³ + 14x² + 13x + 11 by x + 2 will result to a quotient of 4x² + 14x + 1 and a remainder of 9.

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4 10 + 40 100 is equivalent to 2/5 + 2/5, which simplifies to 4/5.

To find fractions that are equivalent to 4 10 + 40 100, we need to simplify the given fraction to its lowest terms.

First, we can simplify 4 10 by dividing both the numerator and denominator by the greatest common factor, which is 2. This gives us 2/5.

Next, we can simplify 40 100 by dividing both the numerator and denominator by the greatest common factor, which is 20. This gives us 2/5 as well.

Therefore, 4 10 + 40 100 is equivalent to 2/5 + 2/5, which simplifies to 4/5.

Now, to find fractions that are equivalent to 4/5, we can multiply the numerator and denominator by the same non-zero number.

For option A, 80/100 can be simplified to 4/5 by dividing both the numerator and denominator by 20. Therefore, A is equivalent to 4/5.

For option B, 4/5 is already in its simplest form, so B is equivalent to 4/5.

For option C, 20/100 can be simplified to 1/5 by dividing both the numerator and denominator by 20. Therefore, C is not equivalent to 4/5.

For option D, 8/10 can be simplified to 4/5 by dividing both the numerator and denominator by 2. Therefore, D is equivalent to 4/5.

In summary, options A, B, and D are equivalent to 4 10 + 40 100, while option C is not.

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The measure of the angle is x = 25°.

What is angle?

An angle is a geometric figure formed by two rays that share a common endpoint called the vertex. The two rays are called the sides or arms of the angle. Angles are typically measured in degrees, with a full rotation around a point being 360 degrees.

Let the angle be x.

The supplement of the angle is 180° - x.

The complement of the angle is 90° - x.

According to the problem, we have:

180° - x = 2(90° - x) + 25°

Simplifying and solving for x, we get:

180° - x = 180° - 2x + 25°

x = 25°

Therefore, the measure of the angle is x = 25°.

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The explicit formula for the nth term of the sequence 16, 19, 22 is:

[tex]a_n = 16 + 3(n-1)[/tex]

What is an sequence ?

A sequence is an ordered list of numbers or other mathematical objects that follow a particular pattern. Each individual number or object in the sequence is called a term. The pattern that governs the sequence is usually defined by a rule or formula that determines how each term is related to the previous terms. Sequences can be finite or infinite, depending on whether there is a last term or not. They can also be arithmetic, geometric, or have more complicated patterns. The formula for the nth term of a sequence is typically derived from observing patterns or relationships among the given terms.

Finding the explicit formula for the given sequence :

To find the explicit formula for the given sequence 16, 19, 22, we can first observe that each term in the sequence is obtained by adding 3 to the previous term. That is, 19 = 16 + 3 and 22 = 19 + 3. This suggests that the sequence follows an arithmetic pattern, where each term is the sum of the first term and a multiple of a common difference d (in this case, d = 3) multiplied by the position of the term minus 1 (n-1).

Using this pattern, we can write the explicit formula as follows:

[tex]a_n = a_1 + d(n-1)[/tex], where [tex]a_1[/tex] is the first term and d is the common difference.

Plugging in the values for the first term and common difference in the given sequence, we get:

[tex]a_n = 16 + 3(n-1)[/tex]

Therefore, the explicit formula for the nth term of the sequence 16, 19, 22 is:

[tex]a_n = 16 + 3(n-1)[/tex]

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The probability that the ticket picked up is a multiple of 3 or 5 is 0.47 or 47%.

To find the probability that the ticket picked up is a multiple of 3 or 5, we need to count the number of tickets that are multiples of 3 or 5, and then divide by the total number of tickets (100).

Multiples of 3: There are 33 multiples of 3 between 1 and 100 (3, 6, 9, ..., 99).

Multiples of 5: There are 20 multiples of 5 between 1 and 100 (5, 10, 15, ..., 100).

However, we have counted the multiples of 15 twice (since they are multiples of both 3 and 5), so we need to subtract them once to avoid double counting. There are 6 multiples of 15 between 1 and 100 (15, 30, 45, ..., 90).

Therefore, the total number of tickets that are multiples of 3 or 5 is

33 + 20 - 6 ⇒ 47.

The probability that the ticket picked up is a multiple of 3 or 5 is therefore:

P(multiple of 3 or 5) = number of multiples of 3 or 5 / total number of tickets

P(multiple of 3 or 5) = 47 / 100

P(multiple of 3 or 5) = 0.47

So the probability that the ticket picked up is a multiple of 3 or 5 is 0.47 or 47%.

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Based on the following calculator output, the mean of the dataset, rounding to the nearest 100th is 167.14

Mean value is defined as sum of all data values divided by the total number of data values,

using the given data output, we get:

Mean = x = ∑x/n

x = 1170/7

since its given to us that ∑x = 1170; n = 7

therefore, x = 167.14

(Rounded to 2 decimals)

Based on the following calculator output, the mean of the dataset, rounding to the nearest 100th is 167.14

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Answer:

The sum of all four angles of a quadrilateral is 360 degrees.

Let's denote the two unknown angles as 4x and 9x, where x is a constant.

We can set up an equation using the given information:

291 + 17 + 4x + 9x = 360

Simplifying the equation:

308 + 13x = 360

13x = 52

x = 4

So, the two unknown angles are:

4x = 16 degrees

9x = 36 degrees

Therefore, the measures of the two angles in a ratio of 4:9 are 16° and 36°.

The value of x in the right angled triangle is 12.99 in( nearest hundredth)

Trigonometric Ratios are defined as the values of all the trigonometric functions based on the value of the ratio of sides in a right-angled triangle.

The longest side is the hypotenuse and its always opposite to the right angle. The other two sides are the opposite and adjascent.

sin(tetha) = opp/hyp

cos(tetha) = adj/hyp

tan(tetha) = opp/adj

In this case, the hypotenuse = 15

opposite = x

sin 60 = x/15

x = sin60 × 15

x = 12.99 in( nearest hundredth)

therefore the value of x is 12.99 in

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Answer:

13

Step-by-step explanation:

distance formula:

=[tex]\sqrt{(y2-y1)^{2}+(x2-x1)^{2} }[/tex]

=[tex]\sqrt{(5--2)^{2}+(-5-6)^{2} } \\\sqrt{170} \\13[/tex]

The teacher asks her students to bring all the plasticine they have a total of 19 4/9 plasticine jars.

we can convert all the mixed numbers to improper fractions:

Daniela: 8 5/9 = 77/9

Antonio: 3 4/9 = 31/9

Francisco: 1 2/9 = 11/9

Mariana: 5 7/9 = 56/9

Now we can add them up:

77/9 + 31/9 + 11/9 + 56/9 = 175/9

So the total amount of plasticine is 175/9 jars. We can simplify this fraction to a mixed number if desired:

175/9 = 19 4/9

An improper fraction is a type of fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). It is called "improper" because it does not represent a proper or mixed number.

For example, 7/4 is an improper fraction because the numerator (7) is greater than the denominator (4). This can also be written as a mixed number, 1 3/4, where the whole number is the result of dividing 7 by 4 and the remainder (3) is placed over the denominator.

Improper fractions can be converted to mixed numbers, and vice versa, by dividing the numerator by the denominator to get the whole number, and using the remainder as the numerator of the fraction. For example, to convert 11/3 to a mixed number, divide 11 by 3 to get 3 with a remainder of 2, so the mixed number is 3 2/3.

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Complete Question:

For the art class, the teacher asks her students to bring all the plasticine they can.  Daniela has 8 5/9 plasticine jars, Antonio 3 4/9, Francisco 1 2/9 and Mariana 5 7/9

Mails consumed a total of 5000 ml of liquid on the road.

First, let's understand what is meant by "total." Total refers to the sum of all the individual quantities of liquid that Mails consumed. In this case, we are trying to find the total amount of liquid that Mails consumed.

Next, we need to know that 1 liter is equal to 1000 milliliters (ml). This is an important conversion factor that we will use to convert half-liters to milliliters.

Now, we can start to solve the problem. We know that Mails used 10 half-liter hotties of liquid on the road. To convert half-liters to milliliters, we need to multiply by 500 (since half a liter is 500 milliliters). So, we can calculate the total amount of liquid that Mails consumed as follows:

Total amount of liquid = 10 x 500 ml/hottie

Total amount of liquid = 5000 ml

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The answer for question 63 is [tex]\frac{36}{7}[/tex] by using BODMAS rule and the answer for question 33 by simplifying is [tex]4a-27b[/tex].

What is BODMAS ?

BODMAS stands for "Bracket, Order, Division, Multiplication, Addition, and Subtraction," which is the order of operations used in mathematics to determine the sequence in which calculations are performed.

Question 63 [tex][(6+3)\div(-5-2)](-4)[/tex]

Now put  [tex]a=-5[/tex] and [tex]b=6[/tex]

⇒ [tex][(6+3)\div(-5-2)](-4)[/tex]

⇒ [tex][(9)\div(-7)](-4)[/tex]

⇒ [tex][\frac{9}{-7}] (-4)[/tex]

⇒ [tex]\frac{36}{7}[/tex]

Hence, the answer for question 63 is [tex]\frac{36}{7}[/tex] by using BODMAS rule.

Question 33  [tex](-5)*(3b-2a)-6(a+2b)[/tex]

⇒ [tex](-15b+10a)-6a-12b[/tex]

⇒ [tex]4a - 27b[/tex]

Hence, the answer for question 33 by simplifying is [tex]4a-27b[/tex].

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Answer:

The island of Martinique has received $32,000 for hurricane relief efforts. The island’s goal is to fundraise at least y dollars for aid by the end of the month. They receive donations of $4500 each day. Write an inequality that represents this

situation, where x is the number of days.

Step-by-step explanation:

The total amount raised after x days is given by:

Total amount raised = $32,000 + $4,500x

We want this to be at least y by the end of the month. Assuming that there are 30 days in the month, we can write the inequality:

$32,000 + $4,500x ≥ y

Alternatively, if we don't want to assume the number of days in the month, we can use a variable for the number of days:

$32,000 + $4,500x ≥ y

This inequality states that the sum of the initial donation and the donations received each day multiplied by the number of days must be greater than or equal to the fundraising goal y.

Russ James' commission for selling $100,000 worth of chemicals last month is $5,000.

What is the percentage?

A percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%", although the abbreviations "pct.", "pct" and sometimes "pc" are also used. A percentage is a dimensionless number; it has no unit of measurement.

Russ James is paid a commission rate of 5% on all sales. If he sold $100,000 worth of chemicals last month, his commission can be calculated by multiplying the sales amount by the commission rate:

Commission = Sales Amount * Commission Rate

Commission = $100,000 * 0.05

Commission = $5,000

Therefore, Russ James' commission for selling $100,000 worth of chemicals last month is $5,000.

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216 cm2 is the size of the rectangle border.

the translation of the question is

A painting including its frame is 25 cm long and 10 cm wide, what is the area of ​​the frame if it is 4 cm wide?

What is a rectangle's area?

When the dimensions of a rectangle with length and width are multiplied, the area of the rectangle is determined as follows:

A = lw.

The total area is therefore given by:

A = 25 x 10 = 250 cm².

The white region's size is shown by:

A = (25 - 2 x 4) x (10 - 2 x 4) is equal to 17x 2 and 34 cm2.

Hence, the border's area is as follows:

216 cm2 = 250 cm2 - 34 cm2.

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Answer:

Step-by-step explanation:

[tex]\frac{2}{x} -3=\frac{1}{2}[/tex]

      [tex]\frac{2}{x}=\frac{1}{2}+3[/tex]      (-3 both sides)

      [tex]\frac{2}{x}=\frac{7}{2}[/tex]            (added fraction)

      [tex]2=\frac{7x}{2}[/tex]           (×[tex]x[/tex] both sides)

      [tex]4=7x[/tex]           (×2 both sides)

      [tex]x=\frac{4}{7}[/tex]             (÷7 both sides)

so is in essence implicit differentiation

[tex]R(x,y)\implies x^2+9y^2=25\implies \stackrel{ chain~rule }{2x\cdot \cfrac{dx}{dt}}+9\cdot \stackrel{ chain~rule }{2y\cdot \cfrac{dy}{dt}}=0 \\\\\\ x\cdot \cfrac{dx}{dt}+9y\cdot \cfrac{dy}{dt}=0\implies 9y\cdot \cfrac{dy}{dt}=-x\cdot \cfrac{dx}{dt}\implies \cfrac{dy}{dt}=-x\cdot \cfrac{dx}{dt}\cdot \cfrac{1}{9y} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\textit{we also know that at t(0)}\hspace{5em} x'(t(0))=-\cfrac{1}{2}\hspace{5em}(\stackrel{x}{-2}~~,~~\stackrel{y}{1}) \\\\[-0.35em] ~\dotfill\\\\ \left. \cfrac{dy}{dt}=-x\cdot \cfrac{dx}{dt}\cdot \cfrac{1}{9y}\right|_{(-2,1)}\implies -(-2)\left( -\cfrac{1}{2} \right)\cdot \cfrac{1}{9(1)}\implies \boxed{-\cfrac{1}{9}}[/tex]

Answer:

A.) The axis of symmetry:

To find the axis of symmetry, use the formula x = -b/2a, where a and b are the coefficients of the x^2 and x terms, respectively.

In this case, a = 2 and b = -12, so:

x = -(-12) / 2(2) = 3

The axis of symmetry is x = 3.

B.) The vertex:

To find the vertex, plug in the x-coordinate of the axis of symmetry (3) into the function and evaluate:

f(3) = 2(3)^2 - 12(3) + 3 = -33

So the vertex is (3, -33).

C.) The X-intercepts:

To find the x-intercepts, set y (or f(x)) equal to 0 and solve for x:

0 = 2x^2 -12x +3

Using the quadratic formula, we get:

x = (6 ± sqrt(6^2 - 4(2)(3))) / (2(2))

x = (6 ± 3sqrt(2)) / 4

x = (3/2) ± (3/2)sqrt(2)

So the x-intercepts are approximately (-0.68, 0) and (4.18, 0).

D.) The Y-intercept:

To find the y-intercept, set x = 0 and evaluate the function:

f(0) = 2(0)^2 - 12(0) + 3 = 3

So the y-intercept is (0, 3).

E.) The domain and range:

The domain of the function is all real numbers, since there are no restrictions on the values of x that can be plugged into the function.

To find the range, note that the coefficient of the x^2 term (2) is positive, which means that the parabola opens upwards. Therefore, the minimum value of the function occurs at the vertex, and the range is all real numbers greater than or equal to the y-coordinate of the vertex. In this case, the range is (-33, ∞).