(-4)³·7⁰
_____ this a fraction!
(-1³)

The statements true is :

The standard equation of a circle is expressed as:

x²+y²+2gx+2fy+C = 0

Centre is (-g, -f)

radius = √g²+f²-C

Given a circle whose equation is x²+y² - 2x- 8 = 0

Get the center of the circle

2gx = -2x

2g = -2

g = -1

Similarly, 2fy = 0

f = 0

Centre = (-(-1), 0) = (1, 0)

This shows that the center of the circle lies on the x-axis

r = radius = √g²+f²-C

radius = √1²+0²-(-8)

radius =√9 = 3 units

The radius of the circle is 3 units.

For the circle x² + y² = 9, the radius is expressed as:

r² = 9

r = 3 units

Hence the radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.

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The dimensions of the box with a volume of 4096 cm³ that has a minimal surface area are 16cm×16cm×16 cm.

A solid object's surface area is a measurement of the overall space that the object's surface takes up. It is also known as the overall area of a three-dimensional shape's surface. The formula to calculate surface area is S = 2(xy+yz+zx) where x is length, y is width and z is the height of the object.  

Given the volume is 4096 cm³.

Then, V = xyz = 4096. Deriving the value of z from this, we get z = 4096/xy. Substitute value of z in S = 2(xy+yz+zx), we get,

[tex]\begin{aligned}S&=2\left(xy+y\times\frac{4096}{xy}+x\times\frac{4096}{xy}\right)\\&=2\left(xy+\frac{4096}{x}+\frac{4096}{y}\right)\\&=2xy+\frac{8192}{x}+\frac{8192}{y}\\&=2xy+8192\left(\frac{1}{x}+\frac{1}{y}\right)\end{aligned}[/tex]

Differentiating S with respect to x, we get,

[tex]\frac{dS}{dx}=2y-\frac{8192}{x^2}[/tex]

Equating this to zero and multiplying with x² we get,

[tex]\begin{aligned}2y-\frac{8192}{x^2}&=0\\2x^2y-8192&=0\\x^2y&=\frac{8192}{2}\\x^2y&=4096\end{aligned}[/tex]

Differentiating S with respect to y, we get,

[tex]\frac{dS}{dy}=2x-\frac{8192}{y^2}[/tex]

Equating this to zero and multiplying with y² we get,

[tex]\begin{aligned}2x-\frac{8192}{y^2}&=0\\2xy^2-8192&=0\\xy^2&=\frac{8192}{2}\\xy^2&=4096\end{aligned}[/tex]

Dividing x²y by xy², we get,

[tex]\begin{aligned}\frac{x^2y}{xy^2}&=\frac{4096}{4096}\\\frac{x}{y}&=1\\x&=y\end{aligned}[/tex]

Substitute x = y in x²y = 4096, we get,

[tex]\begin{aligned}y^3&=4096\\y&=16=x\end{aligned}[/tex]

Then, the z value is

[tex]\begin{aligned}z &= \frac{4096}{16\times 16}\\&=16\end{aligned}[/tex]

The required answer is 16cm×16cm×16 cm.

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The domain and the solution of the equation log₄²(1+x²)=0.25  are x = ±1

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

log₄²(1+x²)=0.25

In this case, the domain of the expression and the solution are the same

So, we start by taking the square root of both sides

log₄(1+x²) = 0.5

Next, we apply the change of base law of logarithm

So, we have

1 + x² = 4⁰°⁵

Evaluate the exponent

1 + x² = 2

Evaluate the like terms

x² = 1

Take the square root of both sides

x = ±1

Hence, the domain and the solution are x = ±1

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The pressure of 28 inches of mercury occurs about 6 miles from the eye of the hurricane. We get this from the given algebraic expression.

An expression is formed by variables, constants, and algebraic operations. Since the operation among them is an algebraic or arithmetic operation, it is said to be an algebraic expression.

It is given that the algebraic expression that relates the barometric pressure and the eye of the hurricane as

f(x) = 0.48 ln(x+2) + 27

Here x is the distance in miles from the eye of the hurricane.

f(x) is the pressure of the mercury in a barometer in inches

So, the required distance from the eye of the hurricane when the pressure of 28 inches of mercury in the meter is

(Here f(x) = 28)

f(x) = 0.48 ln(x+2) + 27

⇒ 28 = 0.48 ln(x+2) + 27

⇒ 0.48 ln(x+2) = 28 - 27

⇒ ln (x+2) = 1/0.48

⇒ ln(x+2) = 2.0833

Applying exponential base "e" on both sides, we get

(x+2) = [tex]e^{2.0833}[/tex]

⇒ x + 2 = 8.0309

⇒ x = 8.0309 - 2 = 6.0309

When the result is rounded to the nearest whole number, we get x = 6 miles.

Thus, for the pressure of 28 inches of mercury, the eye of the hurricane is 6 miles far from the barometer.

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Answer:

The probability of rolling an odd sum and a sum less than or equal to four is 1/12

Step-by-step explanation:

The equation -2/3x + y = -2 is in the slope-intercept form, which is a common form for linear equations.

What is the slope of the equation?

The slope of a line is a measure of its steepness. Mathematically, the slope is calculated as "rise over run" (change in y divided by change in x).

we can use the two-point form to find the equation of the line:

y - y1 = m(x - x1)

Substituting the coordinates of the given point and the slope, we get:

y - (-2) = (-2/3)(x - 9)

Combining like terms and multiplying through the parentheses, we get:

y + 2 = -2/3x + 6

This equation is already in the form Ax + By = C, so we can simply use it as is:

-2/3x + y + 2 = 0

This equation can also be written as -2x + 3y = -6 by multiplying all the terms by 3 to get rid of the fraction.

The equation -2/3x + y = -2 is in the slope-intercept form, which is a common form for linear equations. In this form, the coefficient of x (-2/3) is the slope of the line, and the constant term (-2) is the y-intercept, which is the point where the line intersects the y-axis.

Hence, The equation -2/3x + y = -2 is in the slope-intercept form, which is a common form for linear equations.

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Answer:

0.4, 14%, 70%

Step-by-step explanation:

0.4= 0.4%

14= 14%

70= 70%

According to the combination technique,

A) 3 from A and 1 from B = 16320

B) 2 from A and 2 from B = 29070

C) All from A = 3060

D) 4 people regardless of department = 73815

E) at least 3 from department A = 19380

In math combination technique is used to determines the number of possible arrangements in a collection of items where the order of the selection does not matter.

Here we have given that 4-person grievance committee is to be selected out of 2 how many ways can the following committees be selected

And we need to find the following

Here we know that  4-person grievance committee is to be selected out of 2 departments, A and B, with 18 and 20 people.

In this one we have to find the number ways can the following committees be selected.

A) While the first condition is to 3 from A and 1 from B

Here the term "and" means to multiply

Then it can be written as

=> (18 people choose 3) x (20 people choose 1)

=> (18C3)(20C1) = (816)(20) = 16320

B) And then the next condition is 2 from A and 2 from B

In this one is written as,

=> (18 people choose 2) x (20 people choose 2)

=> (18C2)(20C2) = (153)(190) = 29070

C) All from A

which is equal to the following,

=> 18 people choose 4 = 18C4 = 3060

D) Next, we have to do the calculation for 4 people regardless of department

=> 18+20 or 38 people choose 4 = 38C4 = 73815

E) Finally, at least 3 from department A.

'In other words, it can be written as choose 3 from department A and choose 1 from department B or  Choose 4 all from department A

Where "And" means to multiply and "or" means to add.

Therefore, the condition is written as,

=> [(18 people choose 3) times (20 people choose 1)] plus  (18 people choose 4)

=> (18C3)(20C1) + (18C4) = (816)(20) + 3060 = 19380

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(a) February 4 and November 3 of the year have about 9 h of daylight.

(b) 271 days with more than 9 hours of daylight.

What is a function?

A function, according to a technical definition, is a relationship between a set of inputs and a set of potential outputs, where each input is connected to precisely one output.

Here, we have

Given

(a) L(t) = 11+ 2.86 sin(2π/365 (t-80))

L(t) = 9

9 = 11+ 2.86 sin(2π/365 (t-80))

-2 = 2.86 sin(2π/365 (t-80))

-2/2.86 = sin(2π/365 (t-80))

-2/2.86 = -0.6993

Sin function is negative on [π, 3π/2] and [3π/2, 2π]

sin⁻¹(-0.6993) = 2π/365 (t-80)

2π/365 (t-80) = 3.9160 or 5.5087

t = 80 + 365/2π(3.9160 or 5.5087)

t = 35 or 307

After January 31 days and February 4 days

For t = 35 days =  February 4

For t =307days = November 3

Hence, February 4 and November 3 of the year have about 9 h of daylight.

(b) There are 35 + (365-307) = 35 + 58 = 94 days with 9 hours or less

Hence, 365 - 94 = 271 days with more than 9 hours of daylight.

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Using division and multiplication, the values of the operation are 5/4, 203/6, 1/3 and 32/21 respectively

Division of numbers is the process of breaking down a number into definite proportions of value. In dividing a number, we are reducing it's size by a specific factor. The number to be divided is the dividend. The number by which we divide is the divisor.

In the question given, we have several division operations to carry out.

A) 5/11 ÷ 4/11

Since this are fractions, when we take the inverse of one of the values, we change the sign to multiplication.

5/11 * 11 / 4

This becomes 5/4

B) 29 ÷ 6 / 7

We can also do the same thing we did above by simply taking the inverse of the divisor and then change the sign to multiplication.

29 × 7 / 6 = 203/6

C) 5/9 ÷ 1 2/3

We have to first convert the mixed fraction into an improper fraction and the take the inverse of it and multiply.

5/9 ÷ 5/3

5/9 × 3/5 = 3/ 9 = 1/3

D) 5 3/5 ÷ 3 1/2

We also change the mixed fraction into improper fraction and then take inverse before multiplying.

16/3 ÷ 7/2

16/3 × 2/ 7 = 32/21

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Answer:

m² + 7m - 9

Step-by-step explanation:

[tex]\frac{m^3+16m^2+54m-81}{m+9}[/tex]

setting up the synthetic division

- 9 | 1   16   54   - 81

       ↓  - 9   - 63   81

      -------------------------

        1   7    - 9        0 ← remainder

quotient is m² + 7m - 9

since the remainder is zero then (m + 9) is a factor of the polynomial

The "method is least square" is a least-squares fitting of a simple regression model minimizes the sum of the squares of the perpendicular distances from the data points to the regression line.

The term linear regression refers the predictive method to predict or estimate the values of a dependent quantitative variable denoted by y with the help of other independent variables denoted by x1,x2,x3 by using one of many methods like that of ordinary least squares.

Here we need to find what is the method for a least-squares fitting of a simple regression model minimizes the sum of the squares of the perpendicular distances from the data points to the regression line.

As per the definition of the linear regression, we have know that a least-squares fitting of a simple regression model  which minimizes the sum of the squares of the perpendicular distances from the data points to the regression line.

Here we have to take the square for the number to fit into the given interval.

So, this is known as method for least square and it does not use the sum of perpendicular distances between the points and the ‘estimated regression equation’ does not used to minimize.

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Since the normal distribution is symmetric, the response is 50%.

The Latin term "per centum," which signifies "by the hundredth," was the source of the English word "percentage." Segments with a denominator of 100 are considered percentages. In other words, it is a relationship where the worth of the entire is always considered 100.

As per the given information in the question,

According to the Empirical Rule, if a random variable is regularly distributed:

One standard deviation from the mean is found for 68% of the measurements.

The mean is within two standard deviations for 95% of the measurements.

99.7% of the measurements are found within three standard deviations of the mean.

Additionally, because of the symmetry of the normal distribution, 50% of the measurements fall below the mean, and 50% fall above.

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Statistic value t=2.27

p-value: < 0.00001

To obtain information on the corrosion-resistance properties of a certain type of steel conduit a random sample of 45 specimens was taken and buried for two years.

The study variable is:

X: Max. penetration of a steel conduit.

The data of the sample

n= 45

sample mean X[bar]= 52.9

sample standard deviation S= 4.2

The conduits are manufactured to have a true average penetration of at most 50 mills, symbolically: μ ≤ 50

The hypothesis is:

H₀: μ ≤ 50

H₁: μ > 50

α: 0.05

To choose the corresponding statistic to use to study the population mean, the variable must have a normal distribution. There is no available information to check this, so I'll just assume that the variable has a normal distribution and, since the population variance is unknown and the sample is small, the statistic to use is a Student t.

Under the null hypothesis, the critical region and the p-value are one-tailed.

Critical value: [tex]t_{n- 1;1-\alpha} =t_{44;0.95} =1.68[/tex]

Rejection rule:

Reject the null hypothesis when t ≥ 1.68

t= [tex]\frac{52.9-50}{\frac{4.2}{\sqrt{45} } }[/tex]

t= 2.9-0.626 = 2.27

If the calculated value is greater than the critical value, the decision is to reject the null hypothesis.

p-value:

P(t ≥ 2.27) = 1 - P(t < 2.27) = < 0.00001.

The p-value is less than α so the decision is to reject the null hypothesis.

Since the null hypothesis was rejected, then the population average of the penetration of the conduits specimens is greater than 50 mils. It is not recommendable to use these conduits.

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The given sequence  48, 12, 3, .... written in the explicit formula is as follows:

aₙ = 48·(1/4)^(n-1)

The formal equations for L-functions in mathematics are Riemann's zeta function and links between sums over an L-complex function's number zeroes and sums over prime powers.

An arithmetic sequence has the explicit formula aₙ = a + (n - 1)d, and each term of the sequence may be computed without knowing the values of the previous terms.

The nth term of arithmetic, geometric, or harmonic series is usually the explicit formula.

So, the sequence we have is as follows:

48, 12, 3, ....

The common ratio of the terms is 1/4, and the first term is 48.

The universal equation for a geometric sequence's n-th term is:
aₙ = a1·r^(n-1)

This is true when r = 1/4 and a1 = 48:

aₙ = 48·(1/4)^(n-1)

Therefore, the given sequence  48, 12, 3, .... written in the explicit formula is as follows:

aₙ = 48·(1/4)^(n-1)

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Answer:

Definition of an isosceles triangle

Therefore ,the linear function that best describes the scatterplot relating the number of books in the library is y = 10 + 8x

Two separate but related concepts are referred to as linear functions in mathematics: A polynomial function of degree 0 or 1 is referred to as a linear function in calculus and related fields if its graph is a straight line.

Here,

The points given on the scatterplot are (0,10),(4,42),(8,74) & (12,106)

By putting in the points in the given linear functions

y = 8 − 10x

y = 8 + 10x

y = 10 − 8x

y = 10 + 8x

Thus ,the  linear function which best satisfy the points given on the scatterplot is  y = 10 + 8x

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Answer:

y = 10 + 8x

Step-by-step explanation:

we can find the y-intercept first, by observing the point at which x=0, (0,y)

when x = 0, y = 10. The y-intercept is 10.

Let's find the slope. Given the 2 points (0, 10) and (4, 42), we can use the slope calculating formula:

[tex]m(slope)=\frac{y_{2}-y_{1} }{x_{2} -x_{1} }[/tex]

we have:

m = (42-10)/(4-0)

m = 32/4

m=8.

we have y = 8x + 10 (slope-intercept form), which is equal to y = 10 + 8x (unofficial confusing form.)

Hope this helped!

Answer:

60

Step-by-step explanation:

Points A, B, and C are collinear then the possibility of AB will be equal to 4.7. Hence, option A is correct.

Points that are parallel to or within a single line are referred to as collinear points. In Euclidean space, two or more points have been shown to be collinear if they are situated on a line that is either close to or far from another.

As per the given information in the question,

The case is that when B is in between A and C, which means A, B, and C are collinear.

So,

AB + BC = AC

AB + 8.5 = 13.2

AB = 13.2 - 8.5

AB = 4.7

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The required value of x  for y = 25 is 32.4

What is a linear function?

A polynomial function of degree 0 or 1 that has a straight line as its graph is said to be linear.

A linear function is given by:

y = mx + b

where:

The slope, or m, indicates how much y changes when x changes by one, or the rate of change.

b is the y-intercept, which is also known as the function's starting point and is the value of y at x = 0.

From the given graph, the line passes through points (32,25) and (48,50), then the slope of the line is given by:

m = (50-25)/(48-32) = 25/16 = 1.6

Then, y = 1.6x + b

For points (48,50) then,

or, 50 = 1.6*48 + b

or, 50 = 76.8 + b

or, b = 50 - 76.8 = -26.8

Then, the equation will be

y = 1.6x - 26.8

So, for y = 25 then x becomes

25 = 1.6x - 26.8

or, 1.6x = 25 + 26.8

or, 1.6x = 51.8

or, x = 32.4

Hence, the required value of x is 32.4.

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Triangle ABC is equilateral and hence proved.

According to the question,

Let ABC be a triangle

And the given points A1, B1, and C1 are on its sides BC, AC, and AB respectively

Since according to the first equality says that A1, B1, and C1 are not the midpoints and just normal points on the lines.

According to what equality says that ΔABC = ΔA1B1C1 because the sides are in proportion.

Let us consider 3 more triangles = ΔAB1C1, ΔBC1A1, and ΔCA1B1

Now, these above-mentioned triangles will also be similar due to the proportionality insides.

Which results in ⇒ ∠A = ∠B = ∠C due to the phenomenon of corresponding angles of triangles.  

Therefore, Triangle ABC is equilateral and hence proved.

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Answer: [tex]x=4\sqrt{2}[/tex]

Step-by-step explanation:

Using the properties of a 45-45-90 triangle, the hypotenuse is [tex]\sqrt{2}[/tex] times the length of a leg.

Therefore, [tex]x=4\sqrt{2}[/tex].

Answer:

its A

Step-by-step explanation:

cause 2.5 x 3 is 7.5 and 3500+7.5% is 3762.50

Answer:

24.4 km

Step-by-step explanation:

[tex]\sqrt{14^2 + 20^2} = \sqrt{196 + 400} = \sqrt{596}[/tex] = 24.413111 = 24.4 km

Using the concept of Pythagorean theorem, the distance between Closside and Farlisle is 24.4km

The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Mathematically, the Pythagorean theorem can be expressed as:

c² = a² + b²

In this problem, we can substitute the values and solve the distance;

c² = 14² + 20²

c² = 596

c = 2√149

c = 24.4km

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Answer: 11/12

Step-by-step explanation:

it is a rise over run so the slope would be 11/12

The total cost of all the remaining tickets for the show will be equal to $1564.

The four basic mathematical operations are the addition, subtraction, multiplication, and division of two or even more integers. Among them is the examination of integers, particularly the order of actions, which is crucial for all other mathematical topics, including algebra, data organization, and geometry.

As per the given data in the question,

Total tickets left = 46

Price of each ticket = $34

Then, the price of 46 tickets will be,

46 × 34

= $1564

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As a result of it being so accessible, we became incredibly lax over the years.

The Complete Question.

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The amount in the account after 2½ years is $3231.85 and the interest is $231.85

The amount in the account after 4 years is $3379.47and the interest is $379.47

The amount after 2½ years

The given variables can be represented as:

Deposit amount P = $3,000

Rate of interest, R = 3%

Time, t = 2½ years

The amount from compound interest formula can be calculated as:

A = P(1 + r/n)^nt

Also from the question, we have

n = 2 i.e. compounded semi-annually.

Solving further, we replace the variables with their values in the above equation

A = 3000 * (1 + (3%/2))^(2*2.5)

Evaluate

A = 3231.85

The interest is

I = 3231.85 - 3000

I = 231.85

The amount after 4 years

Here, we have

Time, t = 4 years

The amount from compound interest formula can be calculated as:

A = P(1 + r/n)^nt

Solving further, we replace the variables with their values in the above equation

A = 3000 * (1 + (3%/2))^(2*4)

Evaluate

A = 3379.47

The interest is

I = 3379.47 - 3000

I = 379.47

So, the interest is 379.47

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The equation for the total cost is y = 22.5x, and the total cost of four more students joining the group is $292.50.

What is a linear equation?

It is defined as the relation between two variables, if we plot the graph of the linear equation, we will get a straight line.

If in the linear equation, one variable is present, then the equation is known as the linear equation in one variable.

Given:

Mrs. Jameson paid $202.50 for a group of 9 students to visit an amusement park.

Each student cost = 202.50/9 = $22.5

Total cost equation (y):

y = 22.5x

Here x is the number of students.

Plug x = 4 in the equation y = 22.5x

⇒ y = 22.5 (4)

y = $90

Total cost = 90 + 202.50 = $292.50

Hence, the equation for the total cost is y = 22.5x, and the total cost of four more students joining the group is $292.50.

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Answer:

$3893.97

Step-by-step explanation:

5000 = x (1 + 0.05/365)^365 * 5

5000 = x ( 1 + 0.000137)^1825

5000 =  x (1.000137)^1825

5000 = 1.284036 x

1.284036 / 1.284036 x = 5000 / 1.284036

x = 3 893.97182

x = 3 893.97

Answer:

$3,894.07

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{8.5 cm}\underline{Compound Interest Formula}\\\\$ A=P\left(1+\frac{r}{n}\right)^{nt}$\\\\where:\\\\ \phantom{ww}$\bullet$ $A =$ final amount \\ \phantom{ww}$\bullet$ $P =$ principal amount \\ \phantom{ww}$\bullet$ $r =$ interest rate (in decimal form) \\ \phantom{ww}$\bullet$ $n =$ number of times interest is applied per year \\ \phantom{ww}$\bullet$ $t =$ time (in years) \\ \end{minipage}}[/tex]

Given:

Substitute the given values into the formula and solve for P:

[tex]\implies 5000=P\left(1+\dfrac{0.05}{365}\right)^{365 \times 5}[/tex]

[tex]\implies 5000=P\left(1.000136986...\right)^{1825}[/tex]

[tex]\implies P=\dfrac{5000}{\left(1.000136986...\right)^{1825}}[/tex]

[tex]\implies P=3894.070588...[/tex]

Therefore, the amount you would need to deposit in an account now in order to have $5,000 in the account in 5 years time is $3,894.07.