find the limit. lim t→0+ cos(t), cos(t), t3 ln(t)

Answer:

x = 3

y = -1

Step-by-step explanation:

             Method:  substitution

x - 3y = 6 --------------(I)

      x  = 6 + 3y -------------(II)

2x + 2y = 4 ----------(III)

Substitute x = 6 +3y in equation (II),

        2*(6+3y) + 2y = 4

Use distributive property to open the parenthesis,

     2*6 + 2*3y + 2y = 4

         12 + 6y + 2y = 4

Combine like terms.

            12 + 8y      = 4

Subtract 12 from both sides,

                        8y  = 4 - 12

                        8y = -8

Divide both sides by 8,

                          y = -8÷ 8

                            [tex]\boxed{\bf y = (-1)}\\[/tex]

Substitute y = -1 in equation (II) and find the value of x.

               x = 6 + 3*(-1)

                  = 6 - 3

               [tex]\sf \boxed{\bf x = 3}[/tex]

Answer:

(3, - 1 )

Step-by-step explanation:

x - 3y = 6 ( add 3y to both sides )

x = 3y + 6 → (1)

2x + 2y = 4 → (2)

substitute x = 3y + 6 into (2)

2(3y + 6) + 2y = 4

6y + 12 + 2y = 4

8y + 12 = 4 ( subtract 12 from both sides )

8y = - 8 ( divide both sides by 8 )

y = - 1

substitute y = - 1 into (1)

x = 3(- 1) + 6 = - 3 + 6 = 3

solution is (3, - 1 )

It is possible for a sample to follow a normal distribution, depending on the nature of the data being accumulated.

The regular distribution is a bell-shaped curve that is symmetric across the mean, and it's far typically used to model various natural phenomena and measurements in fields such as information, engineering, and social sciences.

If the data being gathered is continuous and the sample length is large sufficient, it is regularly assumed that the pattern follows a normal distribution.

This assumption is based at the critical limit Theorem, which states that the sampling distribution of the suggest of any independent, random variable will tend toward a ordinary distribution because the sample size increases.

But, it's far critical to notice that not all samples will observe a ordinary distribution, and in a few cases, different distributions may be greater suitable for the records being amassed.

It's far consequently essential to analyze the information and evaluate the assumptions being made before using the regular distribution as a model.

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Answer:

$450

Step-by-step explanation:

995/2 = 497.5

95/2 = 47.5

497.5 + 47.5 = 545

495 - 95 = 450

Answer:

The answer is 33°

Yes you are correct

Step-by-step explanation:

angles on a straight line equal 180°

57+90+x=180

x+147=180

x=180-147

x=33°

Hi, I can't tell if there's a little square drawn in red to indicate that there's a right angle in the middle. There's some squiggly lines and so it's kind of hard to tell.

I am going to assume that there is. Let me know if there isn't.

The sum of these 3 angles would be 180 degrees.

As an equation, 57 + 90 + x = 180

Solve for x.

147 + x = 180

x = 33 degrees.

So this would be correct ASSUMING that there's a little red square drawn indicating that there's a right angle.

The probability that either event will occur is 0.62

From the question, we have the following parameters that can be used in our computation:

Event A = 6 + 6 = 12

Event B = 20 + 6 = 26

Both A and B = 6

Other Events = 20

Using the above as a guide, we have the following:

Total = A + B + C + Others - Both

So, we have

Total = 12 + 26 - 6 + 20

Evaluate

Total = 52

So, we have

P(A) = 12/52

P(B) = 26/52

Both A and B = 6/52

For either events, we have

P(A or B) = (12 + 26 - 6)/52

Evaluate

P(A or B) = 0.62

Hence, the probability that either event will occur is 0.62

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The graph of the inequality y ≤ 1/3x + 1 is added as an attachment and it is represented with option (c)

From the question, we have the following parameters that can be used in our computation:

y ≤ 1/3x + 1

The above expression is an inequality that implies that

Next, we plot the graph

See attachment for the graph of the inequality

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The sum of the 24 terms of the arithmetic series is 900

How to solve:

To find the sum of an arithmetic series, we can use the formular:

Sn = (n/2)(2a + (n-1)d)

Where;

Sn is the series sum, n is the number of terms, a is the first term, and d is the common difference.

The first term (a) in this scenario is 3, and the common difference (d) is 3. With n equal to 24, we want to find the sum of the first 24 terms.

When the values are entered into the formula, we get:

The sum of the first 24 terms of the arithmetic series is 900 because;

S24 = (24/2)(2(3) + (24-1)(3))

= 12(6 + 23(3))

= 12(6 + 69)

= 12(75)

= 900.

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The multiplication of the expression (s - 3)³ is determined as s³ - 9s³ + 27s - 27.

The product of the expression given as (s - 3)³ is calculated  as follows;

The given expression;

(s - 3)³, this can also be written as;

(s - 3)³ =  (s - 3) (s - 3) x (s - 3)

We will simplify the expression as follows;

(s - 3)(s - 3) x (s - 3)

= (s - 3)(s - 3) x (s - 3)

= (s² - 3s -3s + 9 )(s - 3)

= ( s² - 6s + 9 ) (s - 3)

= s³ - 3s² - 6s² + 18s + 9s - 27

= s³ - 9s³ + 27s - 27

Thus, the given expression can be simplified using binomial theorem or using simple multiplication method as shown above.

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The classification , symmetry , maximum r-values and the zeroes of the function are solved

Given data ,

a)

Let the function be represented as A

Now , the value of A is

r² = 25 sin( 2θ )

The equation represents a cardioid

It is symmetric about the polar axis.

The maximum values of r occur at θ = (π/4) + kπ/2, and the maximum value of r is 5.

The equation has zeroes at θ = kπ/2, and the corresponding values of r are given by r = 0

b)

The equation represents a limaçon.

It is symmetric about the polar axis.

The maximum value of r is 7.

The equation has zeroes at θ = arccos(1/3) + 2kπ or θ = -arccos(1/3) + 2kπ, and the corresponding values of r can be found by substituting into the equation r = 4 - 3cos(θ).

c)

The equation represents an Archimedean spiral.

It does not possess any symmetry.

There is no maximum r value.

The equation has a zero at θ = 1/2.

Hence , the zeroes of the function are solved

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The height of domino #7 is approximately 6.89 cm tall.

The formula for the nth term of a geometric progression is:

[tex]a_n = a * r^(^n^-^1^)[/tex]

where:

a is the first term (the height of the smallest domino, 4.00 cm),

r is the common ratio (the growth rate, 1.12), and

n is the term number (for domino #7, n = 7 + 1 = 8, because the sequence starts with domino #0).

Let's plug these values into the formula:

[tex]a_8 = 4.00 cm * (1.12)^7[/tex]

(Note that we're using 7, not 8, because the first domino is #0, not #1.)

Now, compute the value:

[tex]a_8 = 4.00 cm * (1.12)^7[/tex]

=  6.89 cm

So, domino #7 is approximately 6.89 cm tall.

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The products of the matrices A and B are [tex]AB = \left[\begin{array}{cc}-50 &7\\44 &-10\end{array}\right][/tex] and [tex]BA = \left[\begin{array}{cc}-12&-24 \\-16&-48\end{array}\right][/tex]

From the question, we have the following parameters that can be used in our computation:

Matrix A = [tex]\left[\begin{array}{cc}-3&-8\\2&8\end{array}\right][/tex]

Matrix B = [tex]\left[\begin{array}{cc}6&3\\4&-2\end{array}\right][/tex]

Using the above as a guide, we have the following:

[tex]AB = \left[\begin{array}{cc}-3 * 6 + -8 * 4 &-3 * 3 -8 * -2\\2 * 6 + 8 * 4 &2 * 3 + 8 * -2\end{array}\right][/tex]

Evaluate

[tex]AB = \left[\begin{array}{cc}-50 &7\\44 &-10\end{array}\right][/tex]

Next, we have

[tex]BA = \left[\begin{array}{cc}6 * -3 + 3 * 2&6 * -8 + 3 * 8 \\4 * -3 + -2 * 2&4 * -8 + -2 * 8\end{array}\right][/tex]

Evaluate

[tex]BA = \left[\begin{array}{cc}-12&-24 \\-16&-48\end{array}\right][/tex]

Hence, the values are [tex]AB = \left[\begin{array}{cc}-50 &7\\44 &-10\end{array}\right][/tex] and [tex]BA = \left[\begin{array}{cc}-12&-24 \\-16&-48\end{array}\right][/tex]

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For the function y = 3sin[2(θ-90°)] + 2, the amplitude is 3, the period is π, the phase shift is -45° and the vertical shift is 2 units up.

The amplitude of a sine function represents the maximum displacement from the midline.

The coefficient in front of the sine function is 3, so the amplitude is 3.

The period of a sine function is the length of one complete cycle. It can be calculated using the formula T = 2π/b, where b is the coefficient of θ inside the sine function brackets.

Here, b = 2, so the period is T = 2π/2 = π.

The phase shift of a sine function indicates a horizontal shift to the left or right.

In this case, c = 90°, so the phase shift is θ = 0° - 90°/2 = -45°.

The vertical shift indicates a shift up or down on the y-axis.

The constant term outside the sine function is 2, so the vertical shift is 2 units up.

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Measure of angle 1 is,

Angler ACB = 36 degree

We have to given that;

In a parallelogram we get;

Angler CAD = 36 degree

Now, We know that;

Opposite sides are parallel to each other in a parallelogram.

Hence, We get;

Angler CAD = Angle ACB

Since, Angler CAD = 36 degree

Hence, We get;

Measure of angle 1 is,

Angler ACB = 36 degree

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150 degrees is the measure of the angle x from the figure

The given diagram is a circle and the angle x and 30 degrees can be seen to be on a straight line.

The sum of the measure of angle on a straight line is 180 degrees, hence:

x + 130 = 180

Subtract 130 degrees from both sides to have:

x + 30 - 30 = 180 - 30
x = 180 - 30
x = 150 degrees

Hence the measure of the angle x from the given equation is 150 degrees

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Answer:

16x+4y=2

Step by step explanation

The length of the midsegment of the trapezoid with bases 18 and 28 is gvien as follows:

23.

The trapezoid midsegment theorem, states that the length of the midsegment of the trapezoid is equals to the mean of the length of the bases of the trapezoid.

The bases of the trapezoid are given as follows:

18 and 28.

Hence the length of the midsegment of the trapezoid is given as follows:

(18 + 28)/2 = 23.

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The first term of a geometric sequence is 2 and the common ratio is 3.  The general term formula for the nth term of a geometric sequence is given by: an = a₁ × rⁿ⁻¹ Where, an is the nth term of the sequence, a₁ is the first term of the sequence, and r is the common ratio of the sequence. The 15th term of the sequence is 86,093,442.

To find the 15th term of the sequence, we need to use the above formula to find a₁ and r.The sequence is: 2, 6, 18, 54, ...Since the first term is 2, then a₁ = 2.The common ratio can be found by dividing any term by its preceding term. We will use the second and first term:6 / 2 = 318 / 6 = 3Again, the common ratio is 3. Now, we can use the formula to find the 15th term of the sequence: an = a₁ × rⁿ⁻¹a15 = 2 × 3¹⁵⁻¹a15 = 2 × 3¹⁴a15 = 2 × 43,046,721a15 = 86,093,442 The 15th term of the sequence is 86,093,442.

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The minimum number of monthly checks Regina needs to write for the first account to be a better option is 21 or more. Answer C. 21 is one of the suggested choices.

On the basis of the quantity of checks written, we can compare the overall costs of the two accounts.

Let's calculate the total cost for each account based on the given information:

First Account:

Monthly fee: $12

Per-check fee: $0.10

Total cost = Monthly fee + (Per-check fee * Number of checks)

Second Account:

Monthly fee: $9

Per-check fee: $0.25

Total cost = Monthly fee + (Per-check fee * Number of checks)

We must determine the point at which the first account's total cost is less than the second account's total cost. Let's construct the formula:

$12 + ($0.10 * Number of checks) < $9 + ($0.25 * Number of checks)

Simplifying the equation

$12 - $9 < ($0.25 - $0.10) * Number of checks

$3 < $0.15 * Number of checks

Dividing both sides by $0.15:

$3 / $0.15 < Number of checks

20 < Number of checks

Therefore, the minimum number of monthly checks Regina needs to write for the first account to be a better option is 21 or more. Answer C. 21 is one of the suggested choices.

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The surface area of the square pyramid is 224 square centimeters.

Since the base of the square pyramid is a square, all sides are equal in length.

Let's call the length of one side of the square base "s".

Then the surface area of the pyramid can be found using the formula:

Surface area = Area of base + Area of four triangles

The area of the base is just s², and the height of each triangle can be found using the Pythagorean theorem.

Let's call the height "h". Then:

h² = 14² - (s/2)² (using the longer side of the triangle as the hypotenuse)

h² = 196 - (s²/4)

Now, we need to find the surface area in terms of "s" and simplify:

Surface area = s² + 4(1/2)(s)(h)

Surface area = s² + 2sh

Surface area = s² + 2s√196 - s²/4))

Surface area = s²+ 2s√(784 - s²))/2

Surface area = s² + s√(784 - s²))

We are given that one side of the square base is 8cm, so:

Surface area = 8² + 8√784 - 8²

Surface area = 64 + 8√400

Surface area = 64 + 8(20)

Surface area = 224

Therefore, the surface area of the square pyramid is 224 square centimeters.

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The sine function with a midline of y=4, an amplitude of 2, a period of π, and a horizontal shift of π/4 to the right is y = 2sin(2(x - π/4)) + 4.

To write a sine function with the given properties, we can use the general form of a sine function: y = A * sin(B(x - C)) + D, where A represents the amplitude, B represents the frequency (or inverse of the period), C represents the horizontal shift, and D represents the vertical shift (or midline).

In this case:

The midline is y = 4, so D = 4.

The amplitude is 2, so A = 2.

The period is π, so the frequency is B = 2π/π = 2.

The horizontal shift is π/4 to the right, so C = -π/4.

Plugging these values into the general form, we get the sine function:

y = 2 * sin(2(x - (-π/4))) + 4

Simplifying further:

y = 2 * sin(2(x + π/4)) + 4

Therefore, the sine function with a midline of y = 4, an amplitude of 2, a period of π, and a horizontal shift of π/4 to the right is y = 2 * sin(2(x + π/4)) + 4.

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Answer:

The correct statements are:

A) There are no zeroes

D) The vertex is (1, 14)

H) The axis of symmetry is x = 1

J) The "a" of the parabola is positive

M) The parabola has a minimum

Therefore, the options B, C, E, F, G, I, K, N, O, P, Q, and R are all false.

Step-by-step explanation:

Step-by-step explanation:

Remember that, this is quadratic function graph.

Let's analyze fact the graph. We get :

There are two roots/zeroes

Two roots are x1 = -2 , x2 = 6

Vertex is maximum value of parabola (2,14)

Symmetry function is x = 2 (you can show that, x-axis from vertex)

"a" parabola indicated negative

The parabola with "a" negative, thus have maximum value (show that vertex is (2,14))

Conclusion :

The true statements are C,F,I,K,N,O (6 statements are shown)

Subject : Mathematics

Level : JHS

Chapter : Quadratic Functions

Answer:

(c):53

Step-by-step explanation:

Let's start with the equation:

3x+5y

Since we already have the values of x and y, we can just solve the equation.

3(11)+5(4)

33+20=53

The answer is (c)53

There are three area options for the number of bricks:

The number of bricks that can be painted out is equal to the paint area (A), in square centimeters, divided to the area of a brick (A'), in square centimeters. That is:

n = A / A'

A' = w · h

Where:

There are three possible answers:

Case 1: A = 93750 cm², w = 22.5 cm, h = 10 cm

A' = 22.5 · 10

A' = 225

n = 93750 / 225

n = 416.667

n = 416

Case 2: A = 93750 cm², w = 22.5 cm, h = 7.5 cm

A' = 22.5 · 7.5

A' = 168.75

n = 93750 / 168.75

n = 555.556

n = 555

Case 3: A = 93750 cm², w = 10 cm, h = 7.5 cm

A' = 10 · 7.5

A = 75

n = 93750 / 75

n = 1250

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The value of a scalar does not depend on the orientation.

The magnitude of a vector does not depend on the orientation of the axes.

(a) The value of a scalar does not depend on the orientation of the axes. Scalar is a quantity that has only magnitude and no direction, so it is independent of the coordinate system. Examples of scalars include temperature, mass, and time.

(b) Component of a vector depends on the orientation of the axes. If the coordinate axes are rotated or shifted, the components of a vector will change accordingly.

(c) A vector depends on the orientation of the axes. A vector has both magnitude and direction and changes when the orientation of the axes is changed.

(d) The magnitude of a vector does not depend on the orientation of the axes. The magnitude of a vector is defined as the length of the vector, which is independent of the coordinate system. For example, the magnitude of a velocity vector is the speed, which is the same regardless of the orientation of the axes.

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The surface areas of the solids are listed below:

Case I: A = 216 ft²

Case O: A = 204 mm²

Case H: A = 216 ft²

Case L: A = 284.6 cm²

Case E: A = 25.657 yd

Case N: A = 356 yd²

Case S: A = 361.88 ft²

Case T: A = 150 cm²

In this problem we find eight solids, whose surface areas, that is, the sum of the areas of all faces of the solid, must be found. Each face can be represented by the following area formula:

Rectangle

A = w · h

Triangle

A = 0.5 · w · h

Where:

Now we proceed to determine the surface areas:

Case I:

A = 6 · (6 ft)²

A = 216 ft²

Case O:

A = 4 · 0.5 · (6 mm) · (14 mm) + (6 mm)²

A = 204 mm²

Case H:

A = 2 · (8 ft) · (6 ft) + (8 ft) · (5 ft) + (6 ft) · (5 ft) + (10 ft) · (5 ft)

A = 216 ft²

Case L:

A = 2 · 0.5 · (7 cm) · (5 cm) + 2 · (6.1 cm) · (13 cm) + (7 cm) · (13 cm)

A = 284.6 cm²

Case E:

A = 2 · 0.5 · (2 yd)² + 2 · (2 yd) · (4 yd) + (√2 yd) · (4 yd)

A = 25.657 yd

Case N:

A = 2 · (3 yd) · (8 yd) + 2 · (8 yd) · (14 yd) + 2 · (14 yd) · (3 yd)

A = 356 yd²

Case S:

A = 2 · 0.5 · (8 ft) · (6 ft) + 2 · (14 ft) · (7.21 ft) + (14 ft) · (8 ft)

A = 361.88 ft²

Case T:

A = 6 · (5 cm)²

A = 150 cm²

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Answer:

The correct answer:

(B) SAS Similarity

Answer:

A. 4ab² + 2x

Step-by-step explanation:

If you want to get rid of those pesky parentheses, you have to use the superpower of distributive property. It lets you multiply everything inside the parentheses by the number outside. For example:

4(ab²+1/₂c + x) = 4ab² + 2c + 4x

Boom! No more parentheses!

To combine like terms, you have to add or subtract the numbers in front of the terms that have the same variable and exponent. For example:

2x - 4x = -2x

Bam! Only one x left!

Here are the steps to simplify the expression:

4(ab²+1/₂c + x) - 2(c+x)

= 4ab² + 2c + 4x - 2c - 2x (use superpower)

= 4ab² + 4x - 2x (combine like terms)

= 4ab² + 2x (simplify)

So, the answer is A. 4ab² + 2x.