Consider the following improper rational function.
f(x) = x+5x+4/x²+3
Find the polynomial P(x) and proper rational function Q(x) such that f(x) = P(x) + Q(x).

The dotted line is the perpendicular bisector.

To minimize the total area of an equilateral triangle and square, the side length of the square should be 2.167 and the side length of the triangle should be 3.833.

To find the dimensions that minimize the total area, we can set up equations based on the given information. Let's denote the side length of the square as 's' and the side length of the equilateral triangle as 't'. The perimeter of the square is 4s, and the perimeter of the equilateral triangle is 3t. Given that the combined perimeter is 13, we have the equation 4s + 3t = 13.

To minimize the total area, we need to consider the formulas for the areas of the square and equilateral triangle. The area of the square is given by A_square = \(s^2\), and the area of the equilateral triangle is given by A_triangle = (\(\sqrt{(3)}\)/4) *\(t^2\).

To find the values that minimize the total area, we can substitute s = (13 - 3t)/4 into the equation for A_square and solve for t. By finding the derivative of the total area with respect to t and setting it equal to zero, we can find the value of t that minimizes the area.

Similarly, to find the dimensions that maximize the total area, we follow the same process but this time maximize the total area by finding the value of t that maximizes the area.

Performing the calculations, we find that to minimize the total area, the side length of the square is approximately 2.167 and the side length of the triangle is approximately 3.833. To maximize the total area, the side length of the square is approximately 4.333 and the side length of the triangle is approximately 1.667.

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a) We must solve the following compound inequality:

1) From the first inequality we have:

2) From the second inequality we have:

So we have that the compound inequality is equivalent to:

Which can be written as:

b) Graph of the compound inequality:

Answer

a. x < -6, x ≥ -1

b. Graph

Answer:

134or 2334

Step-by-step explanation:

The value of current  through the circuit is 1 mA.

To model the resistive touchscreen as a series combination of resistors, we need to divide the touchscreen into small strips of equal width and find the resistance of each strip using its length and resistivity.

The resistance of each strip can be calculated using the formula:

R = (ρ × L) ÷ A

where ρ is the resistivity, L is the length, and A is the cross-sectional area.

Since the touchscreen is divided into strips of equal width, the cross-sectional area of each strip is given by:

A = t × W

where t is the thickness and W is the width of the touchscreen.

Using the given numerical values, we can calculate the resistance of each strip:

For the first strip (x = 0 mm to x = 20 mm):

L =  = 20 mm

A = t × W

= 1 mm × 50 mm

= 50 mm²

= 50 × 10⁻⁶ m²

ρ =  = 0.5 Ωm

= (ρ × L) ÷ A

= (0.5 Ωm × 20 mm) ÷ (50 × 10⁻⁶ m²)

= 1000 Ω

= 1 kΩ

Similarly, we can calculate the resistance of each strip and find the total resistance of the touchscreen:

=  +  +  +

= 1 kΩ + 1.5 kΩ + 1 kΩ + 1.5 kΩ

= 5 kΩ

Using Ohm's law, we can find the current through the circuit:

=  ÷

= 5 V ÷ 5 kΩ

= 1 mA

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The apothem is the perpendicular distance from the center of a regular polygon to any one of its sides.

The apothem of a regular polygon is the perpendicular distance from the center of the polygon to any one of its sides because it serves as a measure of the inradius, or the radius of the circle inscribed within the polygon. In a regular polygon, all sides are congruent and all interior angles are equal, so the apothem is the same for all sides.

To see this, imagine drawing radii from the center of the polygon to each vertex, and then drawing the perpendicular bisector of each side. These perpendicular bisectors will all intersect at the same point, which is the center of the polygon. The apothem is the length of the line segment connecting this center point to any one of the sides of the polygon.

The apothem is useful in finding the area of a regular polygon. By using the formula for the area of a regular polygon in terms of the apothem and the number of sides, one can find the area of a regular polygon without having to find the exact shape of the polygon.

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Part A

The starting amount of water is 95 cubic meters or m^3 for short. This is the y intercept because it handles the starting amount.

Then the pool is filled at a rate of 4 m^3 per minute. This is the slope, i.e. the rate of change. Because this rate of change is constant, we have a linear equation.

We go from y = mx+b to y = 4x+95

Replace x with t to get the answer.

==============================================

Part B

Plug in t = 17 and compute

V(t) = 4t + 95

V(17) = 4*17 + 95

V(17) = 163

===============================================

Part C

We go in reverse compared to the previous part.

This time we don't know the t value, but we are given the V(t) value instead.

V(t) = 231

4t+95 = 231

4t = 231-95

4t = 136

t = 136/4

t = 34

The scale factor which is applied to the first rectangle to get the second rectangle is 2.5 .

The side length of the first rectangle is 3 units ;

the side length of the second rectangle is 7.5 units ;

To find the Scale Factor that was applied to the rectangle of side length 3 units to get the rectangle of side length 7.5 units,

We divide the side length of first rectangle by the side length of the second rectangle.

So, the scale factor is ⇒ 7.5/3 = 2.5 .

Therefore, a scale factor of 2.5 was applied to the rectangle of side length 3 units to get the rectangle of side length 7.5 units.

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The given question is incomplete , the complete question is

What scale factor was applied to the first rectangle of side length 3 units  to get the second rectangle of side length 7.5 units?

Answer:

10

Step-by-step explanation:

Because 5 + 5 is 10, so 1/2 of 10 + 1/2 of 10 = 10

Answer:y intercept = 5

Step-by-step explanation:f(x)=5•(1/6)^xThe y intercept is when x =0Let x =0f(0)=5•(1/6)^0      = 5* 1     = 5The y intercept is 5If the question is f(x)=5•(1/6)xalthough I have never seen the question written this wayThe y intercept is when x =0Let x =0f(0)=5•(1/6)0      = 5* 0     = 0The y intercept is 0

The total number of ways are 25,905,981,685,760 to form 8 groups of 3 if you insist that no two students in the same group differ by more than 3 hours

There are 24 time zones, and we need to form 8 groups of 3 students each. We need to ensure that no two students in the same group differ by more than 3 hours.

First, let's consider the first group of 3 students. We can choose the first student from any of the 24 time zones. For the second student, we can choose from the 3 time zones that are within 3 hours of the first student's time zone.

For the third student, we can choose from the 2 time zones that are within 3 hours of the first student's time zone and not already chosen.

So the number of ways to form the first group is 24 * 3 * 2 = 144.

Now, let's consider the second group of 3 students. We can choose the first student from any of the remaining 21 time zones. For the second student, we can choose from the 3 time zones that are within 3 hours of the first student's time zone and not already chosen.

For the third student, we can choose from the 2 time zones that are within 3 hours of the first student's time zone and not already chosen.

So the number of ways to form the second group is 21 * 3 * 2 = 126.

We can continue this process for the remaining 6 groups. The number of ways to form the third group is 18 * 3 * 2 = 108, the number of ways to form the fourth group is 15 * 3 * 2 = 90, and so on.

So the total number of ways to form 8 groups of 3 students each is 144 * 126 * 108 * 90 * 72 * 54 * 36 * 18 = 25,905,981,685,760.

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The bond energies are;

1) -96 kJ/mol

2) -930kJ/mol

3) -1722 kJ/mol

4) 2196 kJ/mol

Bond energy values can be determined experimentally using various techniques, including spectroscopy and calorimetry.

For reaction 1;

2[945 + 201] - [(2(945) + 498]

=2292 - 2388

= -96 kJ/mol

For reaction 2;

[8(806) + 10(464)] - [4(346) + 10(416) + 13(498)]

(6448 + 4640) - (1384 + 4160 + 6474)

11088 - 12018

= -930kJ/mol

For reaction 3;

[20(806) + 22(464)] - [10(346) + 22(416) + 31(498)]

(16120 + 10208) - (3460 + 9152 + 15438)

26328 - 28050

= -1722 kJ/mol

For reaction 4;

4(1072) - 4(523)

4288 - 2092

= 2196 kJ/mol

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Hey Mate!!

Your answer will be below.

Step-by-step explanation:

The answer is C. Because, There's vertical asymptotes at 1 and -1. Vertical asymptotes are present where the x on the denominator is equal to 0. So, Both the values are 1 and -1 set are equal to the 0 in the denominator, So, They're both horizontal asymptotes. Good Luck!

To prove the formula\(\(\boldsymbol{S}(\boldsymbol{n}) = \sum_{\boldsymbol{k}=\mathbf{1}}^{n} \boldsymbol{k}^{\mathbf{3}} = \left(\frac{n(n+1)}{2}\right)^{2}\)\)by induction, we will first establish the base case and then proceed with the inductive step.

Base case (n = 1): When \(n = 1\), the formula becomes\(\(\boldsymbol{S}(1) = 1^{3} = \left(\frac{1(1+1)}{2}\right)^{2} = 1\),\) which holds true.

Inductive step: Assume that the formula holds true for some arbitrary positive integer \(k\), i.e.,\(\(\boldsymbol{S}(k) = \sum_{\boldsymbol{i}=\mathbf{1}}^{k} \boldsymbol{i}^{\mathbf{3}} = \left(\frac{k(k+1)}{2}\right)^{2}\).\)

We need to show that the formula also holds true for \(n = k+1\), i.e., \\((\boldsymbol{S}(k+1) = \sum_{\boldsymbol{i}=\mathbf{1}}^{k+1} \boldsymbol{i}^{\mathbf{3}} = \left(\frac{(k+1)(k+2)}{2}\right)^{2}\).\)

Expanding the sum on the left side, we have\(\(\boldsymbol{S}(k+1) = \boldsymbol{S}(k) + (k+1)^3\). Using the induction hypothesis, we substitute \(\boldsymbol{S}(k) = \left(\frac{k(k+1)}{2}\right)^{2}\)\).

By simplifying, we get \(\(\boldsymbol{S}(k+1) = \left(\frac{k(k+1)}{2}\right)^{2} + (k+1)^3\). Rearranging this expression, we obtain \(\boldsymbol{S}(k+1) = \left(\frac{(k+1)(k^2+4k+4)}{2}\right)^{2}\).\)

Finally, we can simplify the right side to \(\(\left(\frac{(k+1)(k+2)}{2}\right)^{2}\)\), which matches the desired form.

Since the base case is true, and we have shown that if the formula holds for \(k\), it also holds for \(k+1\), we can conclude that the formula \\((\boldsymbol{S}(\boldsymbol{n}) = \sum_{\boldsymbol{k}=\mathbf{1}}^{n} \boldsymbol{k}^{\mathbf{3}} = \left(\frac{n(n+1)}{2}\right)^{2}\)\) holds for all positive integers \(n\) by the principle of mathematical induction.'

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Using the product rule, we get:
dy/dx = (d(e^(5x))/dx) * 4x + e^(5x) * (d(4x)/dx)
dy/dx = (5e^(5x)) * 4x + e^(5x) * 4

Now, we are given x = 0 and δx = dx = 0.05. We will first find dy:
dy = dy/dx * dx
dy = (5e^(5*0)) * 4*0 + e^(5*0) * 4 * 0.05
dy = (5*1) * 0 + 1 * 4 * 0.05
dy = 0 + 0.2
dy = 0.2

For small δx, δy ≈ dy, so:
δy ≈ 0.2
In summary, dy = 0.2 and δy ≈ 0.2 for the given function and values of x and δx.

To compute the values of dy and δy for the function y=e5x 4x given x=0 and δx=dx=0.05, we first need to find the derivative of the function.
y = e^(5x) * 4x

To find dy, we can take the derivative of the function with respect to x:
dy/dx = 20xe^(5x) + 4e^(5x)

Now we can substitute the given value of x and δx:
dy/dx = 20(0)e^(5(0)) + 4e^(5(0)) = 4
So dy = 4 * 0.05 = 0.2

To find δy, we can use the formula:
δy = |dy/dx| * δx
δy = |4| * 0.05 = 0.2

Therefore, the values of dy and δy for the given function and values are dy = 0.2 and δy = 0.2. To compute the values of dy and δy for the function y = e^(5x) * 4x, we first need to find the derivative of the function with respect to x.

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To prove that the sum of two interior angles of a triangle is equal to the exterior angle, we will need to use some basic principles of geometry.

First, we will need to establish some definitions:

An interior angle is an angle formed by two adjacent sides inside a polygon.

An exterior angle is an angle formed by one side of a polygon and the extension of an adjacent side.

Now, consider the following diagram of a triangle ABC:

Let angle A be an exterior angle of the triangle, formed by side AB and the extension of side AC. We will prove that the sum of angles B and C (the interior angles adjacent to angle A) is equal to angle A.

To do this, we will use the fact that the sum of the angles in any triangle is always 180 degrees. Therefore, we have:

angle B + angle C + angle A = 180 degrees

Next, we will use the fact that angles A and B form a straight line, and angles A and C form another straight line. Therefore, we have:

angle A + angle B = 180 degrees (1)

angle A + angle C = 180 degrees (2)

Now, we can subtract equation (1) from equation (2) to eliminate angle A:

angle A + angle C - (angle A + angle B) = 0

angle C - angle B = 0

Therefore, we have:

angle C = angle B

This means that the sum of angles B and C is equal to 2B, or:

angle B + angle C = 2B

Substituting this into our original equation, we get:

angle B + angle C + angle A = 180 degrees

2B + angle A = 180 degrees

Solving for angle A, we get:

angle A = 180 degrees - 2B

This is exactly the definition of the exterior angle formed by side AB and the extension of side AC. Therefore, we have proved that the sum of two interior angles of a triangle is equal to the exterior angle.

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Answer:

Move all terms that don't contain  x to the right side and solve.

x = 3

Step-by-step explanation:

Answer:

x = 3

Step-by-step explanation:

3x + 2 = 11

Subtract by 2 on both sides

3x = 9

Divide by 3 on both sides

x=3

hope it helped

3(y-1)=2x+2

3(y-1)/3=2x+2/3

y-1=2x/3+2/3

y=2x/3+2/3+1

y=2x/3+5/3

m=2/3

To calculate the cost of shortage (Cs), we need to find the area under the normal distribution curve to the left of the optimal service level. The cost of shortage is the difference between the selling price and the Monday price, multiplied by the probability of shortage.

To calculate the cost of excess (Ce), we find the area under the normal distribution curve to the right of the optimal service level. The cost of excess is the difference between the cost of making the cake and the selling price, multiplied by the probability of excess. The optimal service level is determined by minimizing the total cost, which is the sum of the cost of shortage and the cost of excess. We can find the corresponding z value for the optimal service level using the standard normal distribution table. Once we have the optimal service level (z value), we can find the corresponding demand value. Since demand is normally distributed, we can calculate the optimal number of birthday cakes to make for the weekend by subtracting the expected demand from the optimal service level demand. However, without specific information on the desired service level or target level of shortage/excess, it is not possible to provide numerical answers to the questions. The optimal service level, corresponding z value, and the optimal number of cakes will depend on the specific parameters and objectives of the supermarket bakery.

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The solution set of the given system of linear equations can be expressed as a line in R^3 vector as r = (6, -2) + x_3(3, -5).

To express the solution set of the given system of linear equations as a line in R^3, we first need to express the solution set as a vector equation. We can do this by expressing the equations in terms of x_3 as follows:

x_1 = 6 + 3x_3,

x_2 = -2 - 5x_3

Therefore, the solution set of the given system of linear equations can be expressed as a line in R^3 as follows:

r = (6, -2) + x_3(3, -5).

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using exponents, On simplifying the equation, we get =4m+8.

The PEMDAS order of operations must be followed when you want to simplify a mathematical equation without using parenthesis (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction). There are no parenthesis in the expression, so you may start looking for exponents. If it does, first make that simpler.

Work simplification is to develop better work processes that boost output while cutting waste and costs.

Simplifying an expression is the same as solving a mathematical issue. When you simplify an equation, you essentially try to write it as simply as you can. There shouldn't be any more multiplication, dividing, adding, or deleting to be done when the process is finished.

Given equation,

m-(8-3m)

=m-8+3m

=4m+8

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Answer:

Step-by-step explanation:

" ... at most 5 " ⇒ 5 is max value

\(\frac{x}{3}\) ≤ 5

Answer:

88 m²

Step-by-step explanation:

Add the two bottom lengths to get

11 m

Multiply it with the height which is 8 m

11 m·8 m

88 m²

Answer:

4th answer is correct

Step-by-step explanation:

First, let us make x the subject.

\(\sf \frac{x+4}{4} =\frac{y+7}{7}\)

Use cross multiplication.

\(\sf 7(x+4)=4(y+7)\)

Solve the brackets.

\(\sf 7x+28=4y+28\)

Subtract 28 from both sides.

\(\sf 7x=4y+28-28\\\\\sf7x=4y\)

Divide both sides by 7.

\(\sf x=\frac{4y}{7}\)

Now let us find the value of x/4.

To find that, replace x with (4y/7).

Let us find it now.

\(\sf \frac{x}{4} =\frac{\frac{4y}{7} }{4} \\\\\sf \frac{x}{4} =\frac{4y}{7}*\frac{1}{4} \\\\\sf \frac{x}{4} =\frac{4y}{28}\\\\\sf \frac{x}{4} =\frac{y}{7}\)