Write one way to break apart the array. Then fine the product. 7x8

A postulate becomes a theorem through logical deduction and proof, starting with the postulates and leading to an indisputably true conclusion. Each step in a proof must follow logically from the ones before it, and the end result is a statement that holds true without premises, recognized universally in mathematics.

A postulate becomes a theorem through a process of logical deduction and proof. In mathematics, a postulate is an assumed statement used as a starting point for further reasoning and arguments. When we begin with our starting definitions and postulates, we can logically deduce new conclusions or theorems. If we can provide a proof of a statement A and show that it is definitely true based on the postulates and previously established theorems, then A becomes a theorem. Each line in a proof must follow from the rules cited, which are based on earlier lines of the proof or established logic. Once we derive the final conclusion, if every line of the proof is legitimate, then the whole proof is legitimate, and the sentence can be considered a theorem.

Moreover, a theorem is a statement that is true without any premises, meaning it derives from logical deductions of the initial assumptions. A theorem is proven via indisputable, step-by-step logical deductions. To illustrate, if an experiment or an established fact contradicts existing theories, it may lead to the development of a new theorem, as was the case with the postulates of Special Relativity. Ultimately, something is recognized as a theorem if it is a tautology—true in all possible models—thus making it a universally accepted truth in mathematics.

By applying logarithmic properties and solving a resulting quadratic equation, we find the solutions x = 3  and x = -7 for the given logarithmic equation.

To solve the logarithmic equation [tex]\( \log_4(x+5) + \log_4(x-1) = 2 \),[/tex] we can use the properties of logarithms to condense the equation into a single logarithm, then solve for x.

First, we apply the product rule of logarithms, which states that [tex]\( \log_a(b) + \log_a(c) = \log_a(bc) \)[/tex], to condense the two logarithms on the left side:

[tex]\[ \log_4((x+5)(x-1)) = 2 \][/tex]

Now, we have:

[tex]\[ (x+5)(x-1) = 4^2 \][/tex]

[tex]\[ (x+5)(x-1) = 16 \][/tex]

Next, we expand the left side of the equation:

[tex]\[ x^2 - x + 5x - 5 = 16 \][/tex]

[tex]\[ x^2 + 4x - 5 - 16 = 0 \][/tex]

[tex]\[ x^2 + 4x - 21 = 0 \][/tex]

Now, we can solve this quadratic equation for x. We can use the quadratic formula:

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

For [tex]\( x^2 + 4x - 21 = 0 \), \( a = 1 \), \( b = 4 \), and \( c = -21 \).[/tex]

Plugging in these values, we get:

[tex]\[ x = \frac{-4 \pm \sqrt{4^2 - 4(1)(-21)}}{2(1)} \][/tex]

[tex]\[ x = \frac{-4 \pm \sqrt{16 + 84}}{2} \][/tex]

[tex]\[ x = \frac{-4 \pm \sqrt{100}}{2} \][/tex]

[tex]\[ x = \frac{-4 \pm 10}{2} \][/tex]

[tex]\[ x = \frac{6}{2} \][/tex] or [tex]\( x = \frac{-14}{2}[/tex]

x = 3  or x = -7

So, the solutions to the equation are  x = 3  and x = -7 .

The question probable may be:

Given a log equation solve it:

[tex]\( \log_4(x+5) + \log_4(x-1) = 2 \),[/tex]

Answer:

15 :D

Step-by-step explanation:

Well, we know b = 3 so we can rewrite the expression to make it less confusing. So, we change b for 3, which is the same thing but number form:

2(3) + 9

Now we evaluate 2(3) which would be 2 x 3. 2 x 3 is 6, so we rewrite 2(3) as 6.

6 + 9

and finally, 6 + 9 is 15!

♡Hope this helps♡

The solution to 2b+9 when b=3 is 15.

Solutions are the values of an equation where the values are substituted in the variables of the equation and make the equality in the equation true.

We have,

2b + 9

Now,

The value of 2b + 9 when b = 3.

= 2b + 9

= 2 x 3 + 9

= 6 + 9

= 15

Thus,

The solution is 15.

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The correct answer is: 8

Explanation:

Given mixed number:

[tex] 7\frac{4}{5} [/tex]

In order to convert a mixed number into a decimal number, follow the steps mentioned below:

Multiple 7 with 5 and add the numerator to the answer you get by multiplying 7 with 5. After that, write that result in the numerator and divide it with 5:

Step-1:

Multiply 7 with 5.

7*5 = 35

Step-2:

Add numerator value, which is 4 in this case, in 35

35 + 4 = 39

Step-3:

Rewrite the fraction.

[tex] \frac{39}{5} [/tex]

Step-4:

Calculate.

[tex] \frac{39}{5} = 7.8 [/tex]

Now, as the number at the tenth place after the decimal point is greater than 5, we have to round it off to 8.0.

Hence the correct answer is 8.

Rounding 7 4/5 to the nearest whole number gives 8.

To round the fraction 7 4/5 to the nearest whole number, we examine the fractional part (4/5) of the mixed number.

The fractional part, 4/5, is greater than or equal to 1/2. In such cases, when the fractional part is greater than or equal to 1/2, we round the whole number up to the next higher whole number.

In this case, the whole number part is 7. Since the fractional part is greater than 1/2, we round the whole number up to the next higher whole number, which is 8.

Therefore, rounding 7 4/5 to the nearest whole number gives us 8.

In summary, when rounding a mixed number to the nearest whole number, we look at the fractional part. If it is greater than or equal to 1/2, we round the whole number up. If it is less than 1/2, we round the whole number down. In this case, since 4/5 is greater than 1/2, we round 7 up to 8.

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Answer:

RTP: 1 + cos2x = 2/1+tan^2x

LHS  =  1 + cos2x

        =  1 + 2cos^2x - 1

        =  2cos^2x

RHS = 2/1+tan^2x

        = 2/sec^2x

        = 2 x cos^2x

        = 2cos^2x

2cos^2x=2cos^2x

LHS = RHS

Step-by-step explanation:

Answer: substances that are made up of only one type of atom

Step-by-step explanation:

Element is a pure substance which is composed of atoms of similar elements. Example: Carbon (C)

Compound is a pure substance which is made from atoms of different elements combined together in a fixed ratio by mass.  Example: water [tex]H_2O[/tex]

The subatomic particles with a positive charge are called as protons and are located in the nucleus of the atom.

Thus elements are substances that are made up of only one type of atom.

Isometries are transformations that preserve lengths and angles. Reflection, Translation, and Rotation are isometries, while Dilation is not because it changes the size of the figure.

The student's question asks to identify which transformations are isometries. An isometry is a transformation in geometry that preserves lengths and angles, meaning that the figure's shape and size remain unchanged. The four options given are Reflection, Translation, Dilation, and Rotation.

Therefore, the transformations that are isometries are Reflection, Translation, and Rotation.

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Answer:  Option '4' is correct.

Step-by-step explanation:

Since we have given that

Total number of games =50

Firstly, he played 20 games.

Percentage of games they won = 60%

Number of games they won is given by

[tex]\dfrac{60}{100}\times 20\\\\=0.6\times 20\\\\=12[/tex]

Remaining number of games = 50-20=30

Percentage of remaining games they won = 80%

So, number of games they won is given by

[tex]\dfrac{80}{100}\times 30\\\\=0.8\times 30\\\\=24[/tex]

So, total number of games they won is given by

12+24 = 36

So, Percentage of games they won for the entire season will be

[tex]\dfrac{36}{50}\times 100\\\\=36\times 2\\\\=72\%[/tex]

Hence, Option '4' is correct.

To find y(7) given dy/dt = 4y and y(2) = 400, we can solve the differential equation, find the constant of integration, and substitute the value of t to find y(7).

To solve the differential equation dy/dt = 4y, we can separate the variables and integrate them. Starting with dy/y = 4dt, we get ln|y| = 4t + C, where C is the constant of integration.

Using the initial condition y(2) = 400, we can solve for C: ln|400| = 4(2) + C, C = ln|400| - 8. Now we can find y(7): ln|y(7)| = 4(7) + ln|400| - 8. Exponentiating both sides, we get |y(7)| = e^(28 + ln|400| - 8), and since y(7) is positive, y(7) = e^(28 + ln|400| - 8).

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To write a quadratic function in standard form with zeros 6 and -8, we use the fact that the zeros of a quadratic function are the values of x for which the function equals zero. The quadratic function in standard form with zeros 6 and -8 is x^2 + 2x - 48 = 0.

To write a quadratic function in standard form with zeros 6 and -8, we use the fact that the zeros of a quadratic function are the values of x for which the function equals zero. The standard form of a quadratic function is ax^2 + bx + c = 0. Since the zeros are 6 and -8, the factors of the quadratic function will be (x - 6) and (x + 8). Multiplying these factors gives us (x - 6)(x + 8) = 0. Expanding and simplifying this expression, we get x^2 + 2x - 48 = 0. Therefore, the quadratic function in standard form with zeros 6 and -8 is x^2 + 2x - 48 = 0.

The circumference of a child's swimming pool with a radius of 3 feet is approximately 18.84 feet. We derive this using the formula for the circumference of a circle, namely C = 2πr.

In the realm of Mathematics, the circumference of a circle can be calculated using the formula C = 2πr, where C represents the circumference, π is a constant approximately equal to 3.14, and r is the radius of the circle. In the case of a child's swimming pool with a radius of 3 feet, we substitute the radius (r) into the formula to get: C = 2*3.14*3= 18.84 feet. Therefore, the circumference of the child's swimming pool is approximately 18.84 feet.

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To find the lengths b and c of the cable supporting the ski lift, use the Pythagorean theorem. To find the measure of angle theta, use the tangent function.

To find the lengths b and c, we can use the Pythagorean theorem. Since the vertical rise of the cable is 2 feet for every 5 feet of horizontal length, we can create a right triangle where the vertical leg is 2x and the horizontal leg is 5x. The hypotenuse of this triangle, which represents the length of the cable, is 1320 feet. Using the Pythagorean theorem, we can solve for the value of x and then find the lengths b and c.

b = 5x = 5 * (1320 / √(25 + 4))

c = 2x = 2 * (1320 / √(25 + 4))

Next, to find the measure of angle theta, we can use the tangent function. The tangent of an angle is equal to the ratio of the length of the opposite side (2x) to the length of the adjacent side (5x). Therefore, tan(theta) = 2x / 5x = 2x / (5 * (1320 / √(25 + 4))). Taking the arctangent of both sides of the equation will give us the measure of theta.

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The lengths b and c are 3300 feet and 1320 feet respectively, and the angle theta is approximately 21.8°.

To solve this problem, let's denote:

- b is horizontal length from the lowest point to where the cable is fastened.

- c is vertical rise from the lowest point to where the cable is fastened.

- theta is angle that the cable makes with the horizontal.

From the problem statement:

- The cable rises 2 feet for each 5 feet of horizontal length. This gives us the slope of the cable:

[tex]\[ \frac{c}{b} = \frac{2}{5} \][/tex]

- The cable is fastened 1320 feet above the lowest point, which means:

[tex]\[ c = 1320 \text{ feet} \][/tex]

Now, solve for b:

[tex]\[ \frac{c}{b} = \frac{2}{5} \][/tex]

[tex]\[ \frac{1320}{b} = \frac{2}{5} \][/tex]

Cross-multiplying to solve for b:

[tex]\[ 1320 \times 5 = 2 \times b \][/tex]

[tex]\[ 6600 = 2b \][/tex]

[tex]\[ b = \frac{6600}{2} = 3300 \text{ feet} \][/tex]

Now, we have:

[tex]\[ b = 3300 \text{ feet} \][/tex]

[tex]\[ c = 1320 \text{ feet} \][/tex]

To find [tex]\( \theta \)[/tex], use the tangent function:

[tex]\[ \tan(\theta) = \frac{c}{b} = \frac{1320}{3300} = \frac{2}{5} \][/tex]

Therefore, the measure of angle [tex]\( \theta \)[/tex] is:

[tex]\[ \theta = \tan^{-1}\left(\frac{2}{5}\right) \][/tex]

Using a calculator to find [tex]\( \theta \):[/tex]

[tex]\[ \theta \approx \tan^{-1}(0.4) \][/tex]

[tex]\[ \theta \approx 21.8^\circ \][/tex]

The complete question is

The cable supporting a ski lift rises 2 feet for each 5 feet of horizontal length. The top of the cable is fastened 1320 feet above the cable’s lowest point. Find the lengths b and c, and find the measure of the angle theta.

A trend line shows the general pattern of the data, but does not try to connect all the data points.

The correct statement about trend lines is:

A. A trend line shows the general pattern of the data, but does not try to connect all the data points.

A trend line is used to represent the overall trend or pattern in a set of data. It is not necessary for all the data points to fall exactly on the trend line. The trend line provides a visual representation of the relationship between the variables being measured, such as the change over time.

When it comes to making a valid conclusion, D) No. Students currently taking an art class would be more likely to want to take another art class.

Why can the teachers not make a valid conclusion?

The sample in this case is limited to students who are already enrolled in the art class. Since the survey only includes students from the art class, it might not be representative of the entire student population at Albany Middle School.

Students who have chosen to enroll in the art class may have a specific interest or affinity for art, and their preferences may not be reflective of the broader student body.

If the goal is to determine elective preferences for the entire student population, it would be more appropriate to survey a random and diverse group of students rather than solely focusing on those already in the art class.

Options are:

A) Yes. Most students like to do art.

B) No. Students in middle school would probably like to take drama.

C) Yes. Students in an art class are a good group to get a sample of the general population.

D) No. Students currently taking an art class would be more likely to want to take another art class.

1.2 kilograms bag of pasta has 20 portions.

The division in mathematics is one kind of operation. In this process, we split the expressions or numbers into the same number of parts.

Given:

One portion of pasta has a mass of 60 grams.

And we have 1.2 kilograms of pasta.

So, number of portions,

= 1200/60

= 120/6

= 20

Therefore, 20 portions are there in a 1.2 kg bag of pasta.

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To find the rate at which the volume of the rind is growing, we calculate the volume of the entire watermelon and the volume of the watermelon not including the rind, at the end of the fifth week. Then, calculate the derivative of this volume difference with respect to time.

The topic at hand is related rates in calculus, specifically applied to a growing spherical watermelon. The rate of growth of the radius is given as 2 centimeters per week. The thickness of the rind is always one-tenth the radius, but we're interested in the volume of the rind. We use the formula for the volume of a sphere: V = (4/3)πr³, where r is the radius.

At the end of the fifth week, the radius is 2 cm/week x 5 weeks = 10 cm, therefore the outer radius of the watermelon is 10 cm and the inner radius is 9 cm (since the rind is one-tenth the radius). Therefore, the volume of the rind would be given by the volume of the watermelon minus the volume of the interior of the watermelon not including the rind. We can then calculate the derivative of the volume of the rind with respect to time to find the rate at which the volume is increasing at the end of the fifth week.

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Good evening,

___________________

Step-by-step explanation:

|3x – 5| > 10 ⇔ 3x – 5 > 10 or 3x – 5 < -10.

:)

Answer: 3x-5<-10 or 3x-5>10

Step-by-step explanation: