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The answer of the given question based on the linear equalities is the system of linear inequalities that represents the constraints is:

6x + 3y ≤ 600 , 3x + 6y ≤ 60 , x ≥ 0, y ≥ 0 .

constraints refer to a set of limitations or restrictions on the variables in a problem. These constraints can be in the form of equations or inequalities, and they define the feasible region of the problem.

The goal of solving a system of linear equations subject to constraints is to find a solution that satisfies all of the constraints while also solving the system of equations. The constraints limit the possible solutions to a subset of the full solution space, and finding a feasible solution that satisfies all of the constraints can be more challenging than finding a solution to the system of equations alone.

Let's first identify the variables in the problem:

x: number of Type 1 TV stands produced

y: number of Type 2 TV stands produced

Next, let's identify the constraints:

Labor constraint: the total labor hours used for producing the TV stands cannot exceed 600 hours.

6x + 3y ≤ 600

Single drawer pull constraint: the total number of single drawer pulls used for producing the TV stands cannot exceed 60 pulls.

3x + 6y ≤ 60

Non-negative constraint: the number of TV stands produced cannot be negative.

x ≥ 0, y ≥ 0

Therefore, the system of linear inequalities that represents the constraints is:

6x + 3y ≤ 600

3x + 6y ≤ 60

x ≥ 0, y ≥ 0

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Where the objective function is: Profit = 200x + 150y.

What is inequalities?

In mathematics, an inequality is a mathematical statement that indicates that two expressions are not equal. It is a statement that compares two values, usually using one of the following symbols: "<" (less than), ">" (greater than), "≤" (less than or equal to), or "≥" (greater than or equal to).

Let x represent the number of Type 1 TV stands produced and y represent the number of Type 2 TV stands produced. Then, the constraints for the available labor and single drawer pulls can be represented as follows:

Labor constraint:

3x + 2y ≤ 600

Single-drawer pulls constraint:

3x + 6y ≤ 60

Non-negative constraint:

x, y ≥ 0

We also need to consider the practical constraint that we cannot produce a negative number of TV stands. Therefore, we must add the non-negative constraint as shown above.

Note that the coefficients 3, 2, 3, and 6 in the above inequalities are based on the information given in the problem statement.

Finally, to represent the objective function for maximizing profit, we use the following expression:

Profit = 200x + 150y

Thus, the system of linear inequalities that represents the given problem is:

3x + 2y ≤ 600

3x + 6y ≤ 60

x, y ≥ 0

Therefore, where the objective function is: Profit = 200x + 150y.

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The equation of the line passing through (-2,-4) and (4,4) is y = (4/3)x - 4/3.

The slope intercept formula, y = mx + b, is used when the slope of the line under study is known and the specified point is also the y intercept (0, b). In the equation, the term b stands in for the y value of the y intercept point.

Using the point-slope form of the equation, we can determine the equation of the line across two points:

y - y1 = m(x - x1)

where m is the slope of the line, and (x1, y1) is one of the provided locations.

Let's start by determining the line's slope:

m = (y2 - y1)/(x2 - x1)

where (x1, Y1) equals (-2, -4), and (x2, Y2) equals (4, 4)

m = (4 - (-4)) / (4 - (-2)) = 8 / 6 = 4 / 3

The equation of the line can now be determined using any of the supplied points:

Making use of (x1, y1) = (-2, -4)

y - (-4) = (4/3)(x - (-2))

y + 4 = (4/3)(x + 2)

y = (4/3)x + (8/3) - 4

y = (4/3)x - 4/3

Hence, y = (4/3)x - 4/3 is the equation for the line through (-2,-4) and (4,4).

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Answer:

D. 9/4

Step-by-step explanation:

Answer: D

Step-by-step explanation:

-1 1/4 + -3 2/4

Cross out the negative sign because we're working with negatives in this equation. Due to taking away the negative sign, we would subtract.

3 2/4 - 1 1/4 = 9/4

Have a good day :D

If 0 is the radian measure of a central angle of a circle and r is the radius of the circle, the area of sector is 0.

The area of a sector of a circle is given by the formula:

A = (θ/2π) x πr^2

where θ is the central angle of the sector in radians and r is the radius of the circle.

In this case, the central angle is 0 radians, which means that the sector is actually a point at the center of the circle. Therefore, the area of the sector is 0.

Alternatively, we can think of the sector as being the entire circle, since a central angle of 0 radians covers no portion of the circle. In that case, the area of the sector would be:

A = (θ/2π) x πr^2

A = (0/2π) x πr^2

A = 0

Either way, the area of the sector is 0, since a central angle of 0 radians either forms a point or covers no portion of the circle.

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The t-test, contributed to the development of Monte Carlo methods by using it to study the normal distribution in small sample sizes.

Monte Carlo methods can be used to test the effect of violating statistical tests' assumptions. This technique is primarily used in creating new statistical methods, including the t test for independent means.The t-test is a statistical hypothesis test used to determine whether there is a significant difference between the means of two groups, which may be related in certain characteristics. The t-test is one of the most frequently used tests in statistics for testing hypotheses about the mean of a population based on a sample mean.

Assumptions are essential for a t-test because they help researchers make logical, trustworthy assumptions based on their statistical data. These assumptions may include the normality assumption, homogeneity of variance, and independent random sampling.

It is important to meet these assumptions to get the correct statistical results. Violating these assumptions can affect the accuracy of the results.Monte Carlo methods are used to simulate the probability of different outcomes in a process that cannot be easily predicted because of random variables. They are used to study systems with many variables that are not easily solved. Egon Pearson and Karl Pearson invented these methods to study Brownian motion. "Student" (Gosset), who is known for creating the t-test, contributed to the development of Monte Carlo methods by using it to study the normal distribution in small sample sizes.

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The expression that was not found for the pair of students is given as follows:

5x + 3.

To add or subtract polynomials, we must combine the like terms on the polynomials.

The polynomials for this problem are given as follows:

Hence the addition of these polynomials is given as follows:

3x + 4 - 2x - 1 = x + 3.

(addition keeps the signal of each term in the second polynomial).

The subtraction is given as follows:

3x + 4 + 2x + 1 = 5x + 5.

(subtraction changes the signal of each term in the second polynomial).

For the product, we multiply all terms then combine the like terms, hence:

(3x + 4)(-2x - 1) = -6x² - 3x - 8x - 4 = -6x² - 11x - 4.

5x + 3 is not a solution of any of the operations, hence it is the answer.

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An effective indicator for assessing retail store efficiency, spotting market trends, and making strategic business decisions is annual earnings per employee for profits.

The retail sales industry's annual profits per employee are shown in the data. The data's mean or median is most likely about $4,000,000, according to the assumption. You may compare the productivity of several retail stores by looking at the annual profits per employee. Businesses that generate more annual earnings per employee are typically more productive than those that do not.

The information can also be utilized to spot patterns in the retail sector, such as which businesses are operating better than others. For instance, if a business has been steadily increasing its annual profit per employee, it can be a sign that it is growing or streamlining its operations. In contrast, a business with declining annual profits per employee might be having trouble or seeing a drop in the demand for its goods.

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Answer:

x + y > 30

3x + 0.5y > 40.00

Step-by-step explanation:

Let x be the number of xylophones and y be the number of yo-yos sold.

The total number of toys sold must be greater than 30, so we have:

x + y > 30

The total amount earned must be greater than $40.00 to earn a profit, so we have:

3x + 0.5y > 40.00

Therefore, the system of inequalities is:

x + y > 30

3x + 0.5y > 40.00

Libby requires an additional 8 blue beads to make a bracelet combing white beads and blue beads in a ratio of 1:3.

The ratio refers to the relative size of one value or quantity compared to another.

Ratios are expressed in percentages, fractions, or decimals and can also be expressed in the standard ratio forms (:) to show the comparative relationship one value bears to another.

The ratio of white beads to blue beads = 1:3

The sum of ratios = 4 (1 + 3)

The number of white beads = 6

The total number of beads = 24 (6 ÷ ¹/₄)

The number of blue beads required = 18 (24 - 6)

The number of blue beads Libby has = 10

The additional number of blue beads required = 8 (18 - 10)

Thus, based on the ratio of white beads to blue beads, Libby requires additional 8 blue beads to make a bracelet.

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How many more blue beads does Libby require?

If a painter mixed 56 ounces of blue and yellow paint. The ratio is 3 to 11, 12 ounces of blue paint were utilized by the painter.

To solve the problem, we can use the ratio of blue to the total amount of paint to determine how many ounces of blue paint were used. The ratio of blue to the total paint is 3 to 11. This means that for every 3 parts of blue paint, there are 11 parts of total paint. We can use this information to set up a proportion:

3/14 = x/56

Where x represents the number of ounces of blue paint used. To solve for x, we can cross-multiply and simplify:

356 = 14x

168 = 14 × x

x = 12

Therefore, the painter used 12 ounces of blue paint.

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In triangle ABC, AB + BC > AC is the correct option.

Three straight sides and three angles make up the two-dimensional geometric shape known as a triangle. they can be of many types .

I think that the question is related to the triangle,

Choose the correct option for ∆ABC,

a. AB + BC > AC

b. AB + BC < AC

c. AB + AC < BC

d. AC + BC < AB

Given, ABC is a triangle.

We are aware that the length of a triangle's third side is always greater than the sum of its other two sides.

Let the two sides of ∆ABC is AB and BC and the third side is AC.

So, AB + BC > AC

Therefore, option a is true.

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What are the possible orderings of the sides of triangle ABC, given the three sides are AB, BC, and AC?

a: BC, AB, AC

b: AB ,BC, AC

c: AC ,AB, BC

d: AB ,AC, BC

Answer: The board is 54 inches wide

The width of a whiteboard, in inches, is 54.

Multiplication is a type of mathematical operation. The repetition of the same expression types is another aspect of the practice.

For instance, the expression 2 x 3 indicates that 3 has been multiplied by two.

We know that the width of the whiteboard is 4.5 feet.

However, if we need to express this width in inches, we have to convert feet to inches, as the two units are not directly comparable.

There are 12 inches in a foot.

Therefore, to convert a value in feet to inches, we can simply multiply the value by 12. In other words, 1 foot is equal to 12 inches.

So, to convert the width of the whiteboard from feet to inches, we can simply multiply the width (4.5 feet) by the conversion factor of 12 inches/foot as follows:

4.5 feet × 12 inches/foot = 54 inches

Therefore, the whiteboard is 54 inches wide.

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Answer:

The total amount of space Devonte uses to write notes on a single sheet of paper is 2 sides x 1 sheet of paper = 2/1 = 2.

And, for each event, he uses 1/8 of a side of the paper. Therefore, the number of events he writes notes for can be found by dividing the total amount of space he used by the amount of space used for each event:

Number of events = Total space used / Space used per event

We can substitute the values we know:

Number of events = 2 / (1/8)

Simplifying this expression by multiplying the numerator and denominator by 8, we get:

Number of events = 2 x 8/1

Number of events = 16

Therefore, Devonte made notes for 16 historical events. Option D, 16, is the correct answer.

Step-by-step explanation:

The steps he can use in solving the quadratic equations are [tex]x = \frac{-12 \± \sqrt{12\² - 4 \cdot 2 \cdot -3}}{4}[/tex], (x + 3)² = 21/2 and 2(x² + 6x + 9) = 3 + 18

The equation is given as

2x² + 12x - 3 = 0

Assuming he uses the quadratic formula method, which is represented as

[tex]x = \frac{-b \± \sqrt{b\² - 4ac}}{2a}[/tex]

So, we have

[tex]x = \frac{-12 \± \sqrt{12\² - 4 \cdot 2 \cdot -3}}{2 \cdot 2}[/tex]

[tex]x = \frac{-12 \± \sqrt{12\² - 4 \cdot 2 \cdot -3}}{4}[/tex]

If he uses completing the square, then we have

2x² + 12x = 3

So, we have

x² + 6x = 3/2

x² + 6x + 9 = 3/2 + 9

So, we have

(x + 3)² = 21/2

If he factorizes, then the expression is

2(x² + 6x + 9) = 3 + 18

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The graph has x-intercepts at 1/2, 3, and 4/3, a y-intercept at (0, 12), and is symmetric about x = 1.833. The left end goes up, and the right end goes down.

What is the function?

A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range) such that each input is associated with exactly one output. In other words, a function takes an input value, performs a specific operation on it, and produces a unique output value.

To graph the function, we can start by finding its intercepts, symmetry, and behavior at infinity.

The x-intercepts occur when the function is equal to zero, so we have already found them to be 1/2, 3, and 4/3.

The y-intercept occurs when x is zero, so we have:

f(0) = (2(0)-1)(-0+3)(3(0)-4) = 12

So the y-intercept is (0, 12).

To find the axis of symmetry, we can use the fact that the function is symmetric about the vertical line x = (1/2 + 3 + 4/3) / 3 = 1.833.

To determine the end behavior, we can look at the leading term of the function, which is -6x³. As x goes to negative infinity, the leading term becomes very negative, so the graph goes down to negative infinity on the left side. As x goes to positive infinity, the leading term becomes very positive, so the graph goes up to positive infinity on the right side.

The graph has x-intercepts at 1/2, 3, and 4/3, a y-intercept at (0, 12), and is symmetric about x = 1.833. The left end goes up, and the right end goes down.

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Answer:

R 270

Step-by-step explanation:

Amount paid for one hour =  [tex]\frac{360}{12}[/tex]

Amount paid for one hour = Rs 30

Amount paid for 9 hours = 9 × 30

Amount paid for 9 hours = Rs 270

The painter will be paid Rs 270 for 9 hours of work.

If the sequence is 36 , 32 , 28, then the explicit formula for term aₙ , the nth term is 40 - 4n.

The given sequence is 36, 32, 28, which is a decreasing arithmetic sequence with a common difference of -4.

To find the nth term of an arithmetic sequence, we can use the formula:

aₙ = a₁ + (n - 1)d

where a₁ is the first term, d is the common difference, and n is the term we want to find.

In this case, a₁  = 36, d = -4, and we want to find the nth term.

So, the formula for the nth term of the sequence is:

aₙ = 36 + (n - 1)(-4)

Simplifying this expression:

aₙ = 36 - 4n + 4

aₙ = 40 - 4n

Therefore, the explicit formula for the nth term of the given sequence is aₙ = 40 - 4n.

This formula can be used to find any term of the sequence by substituting the desired value of n. For example, to find the 6th term, we would plug in n = 6:

a₆ = 40 - 4(6) = 16

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The point οn the graph of [tex]f(x) = 5-x^2[/tex] that is clοsest to the point (1, 0) is (0.492, 4.592).

What is a graph ?

Graph can be defined as visual representation of data using given οrdered pairs.

To find the pοint on the graph οf [tex]f(x) = 5-x^2[/tex] that is closest to the point (1, 0), we need to minimize the distance between the two points. The distance between two points (x1, y1) and (x2, y2) is given by the formula:

d = sqrt((x2 - x1)² + (y2 - y1)²)

Sο, in this case, we want tο minimize the distance between (1, 0) and a pοint οn the graph οf f(x) = 5 − x², which can be written as (x, f(x)).

The distance between these twο pοints is:

d = sqrt((x - 1)² + (f(x) - 0)²)

We want tο minimize this distance, sο we need tο find the value οf x that minimizes this expressiοn. Tο dο this, we can use Newtοn's Methοd.

Let's define a functiοn g(x) as:

g(x) = (x - 1)² + (f(x) - 0)²

We want tο find the value οf x that minimizes g(x). Tο dο this, we need tο find the derivative οf g(x):

g'(x) = 2(x - 1) - 2f(x)f'(x)

We can substitute f(x) = 5 - x² and f'(x) = -2x intο this equatiοn:

g'(x) = 2(x - 1) + 4x(5 - x²)

Simplifying this expressiοn, we get:

g'(x) = 12x² - 8x - 8

Nοw we can use Newtοn's Methοd with the initial guess x0 = 0:

x1 = x0 - g(x0)/g'(x0)

x1 = 0 - g(0)/g'(0)

x1 = -0.6667

Using x1 as οur new guess, we repeat the prοcess:

x2 = x1 - g(x1)/g'(x1)

x2 = -0.6667 - g(-0.6667)/g'(-0.6667)

x2 = 0.4293

We cοntinue this prοcess until we reach a desired level οf accuracy. Let's say we want tο rοund οur final answer tο three decimal places. Then we can stοp οnce we find an xn such that |xn - xn-1| < 0.001.

Using a spreadsheet οr calculatοr, we can cοntinue this prοcess and find that:

x3 = 0.4873

x4 = 0.4915

x5 = 0.4916

Since |x5 - x4| < 0.001, we can rοund οur final answer tο three decimal places:

(x, y) = (0.492, 4.592)

Therefοre, the pοint οn the graph οf f(x) = 5 − x² that is clοsest tο the pοint (1, 0) is (0.492, 4.592).

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The unit rate is the cost of the certain brand of nuts per ounce, which is $0.20. This is the same as 20 cents per ounce.

The unit rate is a measure of a quantity, such as the cost of a product, in relation to a single unit of that quantity. In this case, the cost of the nuts is measured in dollars and the quantity is measured in ounces.

To find the unit rate, we need to divide the total cost by the total quantity. In this problem, the total cost of the nuts is given as $3.20 and the total quantity is given as 16 ounces. Dividing these two numbers, we get:

Unit rate = Total cost / Total quantity

= $3.20 / 16 ounces

= $0.20 per ounce

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Tan has a value of 3/4. Trigonometric theory has been used to arrive at this value.

This is due to the fact that trigonometry is still regarded as a relatively challenging and abstract branch of mathematics in comparison to other branches. When learning trigonometry, students frequently run across mistakes, misunderstandings, and barriers.

Calculus is the most difficult math topic, and very few students, whether in junior high or elsewhere, succeed in it. In vector space, linear algebra is a subset of abstract algebra. Matrix technology makes things less abstract & easier to understand because it is more concrete.

tan α [tex]= \frac{15}{20}[/tex]

⇒tan α [tex]=\frac{3}{4}[/tex]

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This function has zeros at x = -2, x = -5, and x = 7, since when any of these values are plugged into the function, the corresponding factor becomes zero, making the entire expression zero.

An expression is a combination of numbers, variables, and operators that represents a mathematical quantity or relationship. It can consist of a single number or variable, or it can be a more complex combination of terms.Expressions can be used to represent a wide range of mathematical ideas, such as functions, equations, inequalities, and identities.

In the given question,

A possible function that has zeros at -2, -5, and 7 is:

f(x) = a(x+2)(x+5)(x-7)

where a is a constant that determines the scale of the function.

Expanding the expression, we get:

f(x) = a(x³ + 5x² - 14x - 70)

This function has zeros at x = -2, x = -5, and x = 7, since when any of these values are plugged into the function, the corresponding factor becomes zero, making the entire expression zero.

Note that there are other possible functions that could have the same zeros, but this is one example.

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(5+4) divided bye 3  is not equals 9 divided by 4

Division is a basic arithmetic operation used to find how many times one quantity is contained within another quantity. It involves splitting a number or quantity into equal parts or groups. The result of a division is called the quotient.

In division, the number being divided is called the dividend, the number by which it is divided is called the divisor, and the result is the quotient. For example, in the division problem 10 ÷ 2 = 5, 10 is the dividend, 2 is the divisor, and 5 is the quotient.

Given 5+4=9

To prove;  (5+4) divided bye 3 = 9 divided by 4

Taking LHS,

(5+4) divided bye 3

=9/3

=3

Taking RHS,

9 divided by 4

=9/4

=2.25

LHS is not equal to RHS

Hence, given statement is false.

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Answer:

Point A represents -1 1/3.

Point A is correct.

0.4103

Answer:GivenA fair die is rolled six times.The probability that it comes up 4 at least once is to be determined.We have to find the probability that it comes up 4 atleast once which means that it can come up more than once as well.Therefore we have to consider all the cases i.e 1,2,3,4,5,6Case 1 : Number of times that it comes up 4 is 1The remaining number of times can be any of the remaining 5 numbersTherefore the probability of the event is(1/6) * (5/6)^5Case 2 : Number of times that it comes up 4 is 2The remaining number of times can be any of the remaining 5 numbersTherefore the probability of the event is(5C2 * (1/6)^2 * (5/6)^4)Case 3 : Number of times that it comes up 4 is 3The remaining number of times can be any of the remaining 5 numbersTherefore the probability of the event is(5C3 * (1/6)^3 * (5/6)^3)Case 4 : Number of times that it comes up 4 is 4The remaining number of times can be any of the remaining 5 numbersTherefore the probability of the event is(5C4 * (1/6)^4 * (5/6)^2)Case 5 : Number of times that it comes up 4 is 5The remaining number of times can be any of the remaining 5 numbersTherefore the probability of the event is(5C5 * (1/6)^5 * (5/6)^1)Now we will add all the cases to get the probability that it comes up 4 atleast once.P(A) = (1/6) * (5/6)^5 + (5C2 * (1/6)^2 * (5/6)^4) + (5C3 * (1/6)^3 * (5/6)^3) + (5C4 * (1/6)^4 * (5/6)^2) + (5C5 * (1/6)^5 * (5/6)^1)On solving we get P(A) = 0.4103Therefore the required probability is 0.4103 or 41.03% (approximately)Answer : 0.4103

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Nοam made a tοtal οf 7.5 quarts οf applesauce in all.

The quart is an English unit οf vοlume equal tο a quarter gallοn. Three kinds οf quarts are currently used: the liquid quart and dry quart οf the US custοmary system and the imperial quart οf the British imperial system.

All are rοughly equal tο οne litter. It is divided intο twο pints οr (in the US) fοur cups. Histοrically, the exact size οf the quart has varied with the different values οf gallοns οver time and in reference tο different cοmmοdities.

Given that,

Nοam made - 5 quarts

There are 2 pints in 1 quart, sο 5 pints is equal tο 5/2 = 2.5 quarts.

Therefοre, Nοam made a tοtal οf 5 + 2.5 = 7.5 quarts οf applesauce in all.

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The number of rectangles in Josh's drawing should be 6.

A three-dimensional solid shape that has six rectangular faces is a rectangular prism. It has eight vertices, or corners, and 12 edges connecting the vertices.

To draw a net of a rectangular prism, Josh needs to represent the three-dimensional object on a two-dimensional surface by drawing each face of the prism separately. A rectangular prism has six faces, and every face is a rectangle.

Josh needs to draw all six rectangles that make up the six faces of the prism. The six faces consist of a pair of congruent rectangles for the top and bottom faces, and four rectangles for the sides. Josh can arrange the six rectangles in different ways to create a net that will fold into a rectangular prism.

It is important that Josh draws the rectangles accurately and with correct dimensions to ensure that the net can be assembled into a rectangular prism.

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The amount of simple interest after 17 months is $72.21.

To calculate the amount of simple interest after 17 months, given that Joanne has $4408 in an account earning 3.7% simple interest per annum, the following steps are involved:

Calculation of simple interest

Simple interest can be calculated using the formula;

Simple interest = P × R × T

where, P = principal amount, R = rate of interest, and T = time in years.To calculate the amount of simple interest for Joanne, we have;

P = $4408R = 3.7% per annum

T = 17 months ÷ 12 months/year

T = 1.4167 years

Substituting the values into the formula for simple interest;

Simple interest = P × R × T= $4408 × 3.7% × 1.4167= $72.21.

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To convert a quadratic expression [tex]ax^2 + bx + c[/tex] to the form [tex](x+p)^2 + q[/tex], where p=b/2a and [tex]q=(b^2-4ac)/4a[/tex], complete the square by adding and subtracting [tex](b/2)^2[/tex] and simplify the resulting expression.

To express the quadratic expression [tex]x^2 + 12x + 11[/tex] in the form [tex](x+p)^2 + q[/tex], we need to complete the square by adding and subtracting a suitable constant term.

First, we take half of the coefficient of x (which is 12) and square it, giving [tex](12/2)^2 = 36[/tex]. We then add and subtract this value inside the bracket:

[tex]x^2 + 12x + 11 = x^2 + 12x + 36 - 36 + 11[/tex]

Now we can group the first three terms and write them as a square of a binomial:

[tex]x^2 + 12x + 36 = (x + 6)^2[/tex]

Simplifying the remaining terms, we have:

[tex]x^2 + 12x + 11 = (x + 6)^2 - 25[/tex]

Therefore, we have expressed the quadratic expression [tex]x^2 + 12x + 11[/tex] in the form [tex](x+p)^2 + q[/tex], where p = 6 and q = -25.

In general, to express a quadratic expression of the form [tex]ax^2 + bx + c[/tex] in the form [tex](x+p)^2 + q[/tex], where a, b, and c are constants and p and q are to be determined, we can follow these steps:

Take half of the coefficient of x (which is b/2) and square it, giving [tex](b/2)^2[/tex].

Calculate this value inside the bracket:

[tex]ax^2 + bx + c = ax^2 + bx + (b/2)^2 - (b/2)^2 + c[/tex]

Group the first three terms and write them as a square of a binomial:

[tex]ax^2 + bx + (b/2)^2 = a(x + b/2a)^2[/tex]

Simplify the remaining terms:

[tex]ax^2 + bx + (b/2)^2 + c - (b/2)^2 = a(x + b/2a)^2 + (4ac-b^2)/4a[/tex]

We can simplify the expression (4ac-b^2)/4a by recognizing that it is the discriminant of the quadratic equation [tex]ax^2 + bx + c = 0[/tex], which is [tex]b^2 - 4ac[/tex]. Therefore, we have:

[tex]ax^2 + bx + c = a(x + b/2a)^2 + (b^2 - 4ac)/4a[/tex]

We can rewrite this in the form [tex](x+p)^2 + q[/tex] by identifying p and q:

p = b/2a, and [tex]q = (b^2 - 4ac)/4a[/tex].

Therefore, the quadratic expression [tex]ax^2 + bx + c[/tex] can be expressed in the form [tex](x+p)^2 + q[/tex], where p = b/2a and [tex]q = (b^2 - 4ac)/4a[/tex].

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The mathematical expression that would make "h" the subject of the equation is h= k-1/3

To begin an equation is a mathematical expression that relates variables by using the = symbol. Due to this, the letters of variables involved can be changed to create equivalent equations.

The equation given is written is k= 3h - 1. To make h the subject of formula 3h= k - 1 Divide both sides by the coefficient of h which in this case is 33h/3= k-1/3h= k - 1/3.

Hence the value of h in the equation is k - 1/3

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