The floor of a rectangular room measures 5m by 4m and the ceiling is 3m from the floor. An ant is at the top of a corner of the room and crawls to the opposite bottom corner of the room. Find the shortest distance it can travel.

The solution of the system would be (3,6).

Lets solve the problem,

Other equation: y = 1/3x+5

Slope-Intercept Form: y = mx + b

Slope Formula:

y2-y1/x2-x1 = m

Have to find equation now:

m = (6 - 5)/(3 - 0)

m = 1/3

y = 1/3x + b

5 = 1/3(0) + b

b=5

y = 1/3x + 5

Lets put the substituition method here,

1/3x + 5 = -1/3x + 7

2/3x + 5 = 7

2/3x = 2

x = 3

y = 1/3(3) + 5

y = 1 + 5

y = 6

Therefore the coordinates are (3,6)

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The y-intercept of the function f(x) is (0, 10/3)

The function is given as:

[tex]f\left(x\right)=\frac{30}{1+2^{3-x}}[/tex]

As x approaches infinity, we have:

[tex]\lim_{x \to \infty} \frac{30}{1+2^{3-x}}[/tex]

This gives

[tex]\lim_{x \to \infty} \frac{30}{1+2^{3-\infty}}[/tex]

[tex]\lim_{x \to \infty} \frac{30}{1+2^{-\infty}}[/tex]

Evaluate

[tex]\lim_{x \to \infty} \frac{30}{1}[/tex]

[tex]\lim_{x \to \infty} 30[/tex]

As x approaches negative infinity, we have:

[tex]\lim_{x \to -\infty} \frac{30}{1+2^{3+x}}[/tex]

This gives

[tex]\lim_{x \to -\infty} \frac{30}{\infty}[/tex]

Evaluate

[tex]\lim_{x \to -\infty} 0[/tex]

Hence, the limits of the function are [tex]\lim_{x \to \infty} 30[/tex] and [tex]\lim_{x \to -\infty} 0[/tex]

Remove the denominator

f(x) = 30

Also, set the numerator to 0

f(x) = 0

Hence, the horizontal asymptotes are y = 30 and y = 0

Set x = 0

[tex]f\left(0\right)=\frac{30}{1+2^{3-0}}[/tex]

Evaluate

[tex]f\left(0\right)=\frac{30}{9}[/tex]

Divide

[tex]f\left(0\right)=\frac{10}3[/tex]

Hence, the y-intercept is (0, 10/3)

The numerator of f(x) is 30.

In (b), we have

The horizontal asymptotes are y = 30 and y = 0

This means that the horizontal asymptotes and the numerator are the same

See attachment for the graph of f(x)

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The 95 percent confidence interval for the mean is (266.76,426.24).

A confidence interval is how much uncertainty there is with any particular statistic. Confidence intervals are often used with a margin of error. It tells you how confident you can be that the results from a poll or survey reflect what you would expect to find if it were possible to survey the entire population. Confidence intervals are intrinsically connected to confidence levels.

The formula of Confidence Interval =  Mean ±   [tex]t\frac{Standard Deviation}{\sqrt{number of observations} }[/tex]

where t is a constant

Given:

Mean = 346.5

Standard Deviation = 170.378

t-critical value for 95% Confidence interval with degrees of freedom(df)=n-1= 19 is 2.093

∴ Substituting values in formula we get

E = [tex]2.093 X170.378/\sqrt{20}[/tex] = 2.0931.96 x 38.0976=79.74

95% Confidence interval : (346.5-79.74,346.5+79.74)

95% Confidence interval : (266.76,426.24)

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When calculating correlation and regression both sets of data must be Statistical.

According to the statement

we have to find the type of data when we calculate the correlation and regression both sets.

so, The difference between these two statistical measurements is that correlation measures the degree of a relationship between two variables (x and y), whereas regression is how one variable affects another.

And when we calculate both then data sets must be a statistical data. because correlation summarizing direct relationship between two variables  and regression predict or explain numeric response. So, without statistical data this is not possible to calculate correlation and regression both sets.

so, When calculating correlation and regression both sets of data must be Statistical.

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Given the 4th order linear ODE

[tex]u^{(4)} - u = 0[/tex]

we substitute

[tex]x_1 = u[/tex]

[tex]x_2 = {x_1}' = u'[/tex]

[tex]x_3 = {x_2}' = {x_1}'' = u''[/tex]

[tex]x_4 = {x_3}' = {x_2}'' = {x_1}''' = u'''[/tex]

Then the given equation transforms to

[tex]{x_4}' - x_1 = 0[/tex]

but we also need to relate this to the other derivative substitutions. This gives the system of differential equations

[tex]\begin{cases} {x_1}' = x_2 \\ {x_2}' = x_3 \\ {x_3}' = x_4 \\ {x_4}' = x_1 \end{cases}[/tex]

In matrix form,

[tex]\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix}' = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix}[/tex]

Answer:

4ab2(7a2b2 5a - 4b)

Step-by-step explanation:

28a3b4 20a2b2 − 16ab3

The GCF of 28, 20 and 16 is 4.

GCF of the variables is ab2

So the answer is

4ab2(7a2b2 5a - 4b)

Answer:

4ab2(7a2b2 5a - 4b)

Step-by-step explanation:

The given function shows imply that the graph g(x) is shifted two units right and three units up.

Option B is correct.

The transformation of a function occurs in a fancy way such that a function maps itself. f: x → x.

From the given information:

Let's take a look at the function f(x) that goes through 0 just when x = 0. Now we want to move it to the right by 2. It implies that it has to go through 0 when x = +2.

Now, we have to add 2 to x, then the function becomes f'(x) = f(x-2) will be shifted by 2 units to the right.

Similarly, the same scenario occurs when shifting up. Again imagine your function passes 0 when x = 0.  We have to add 3. Now, the function f’’(x) = f(x) + 3 will be shifted by 3 units up.

Combining the two transformations, we have:

Therefore, we can conclude that the function implies that the graph g(x) is shifted two units right and three units up.

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The graphs of the reciprocal functions are related to their corresponding original functions based on the information illustrated below.

It should be noted that the trigonometric functions are the real functions which relate an angle of a right-angled triangle to ratios of two side lengths.

There are six functions of an angle. They are sine (sin), cosine (cos), tangent (tan), cotangent, secant, and cosecant (csc).

It should be noted that all the y-coordinates of a reciprocal function will be the reciprocals of the y-coordinates of the original function. Then, the graph of a reciprocal function has a vertical asymptote at each zero of the original function. For example, y=arcsinx reflects against y=sinx.

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Answer:

B) 16

Step-by-step explanation:

12/3 = x/4

48 = 3x  Divide both sides by 3

16 = x

Using translation concepts, the inequality that represents the graph is given as follows:

f(x) > sqrt(1 - x)

A translation is represented by a change in the function graph, according to operations such as multiplication or sum/subtraction either in it’s definition or in it’s domain. Examples are shift left/right or bottom/up, vertical or horizontal stretching or compression, and reflections over the x-axis or the y-axis.

The parent square root function is given by:

y = sqrt(x)

It increases to right starting at x = 0, and in this problem it increases to left starting at x = 1, meaning that it was reflected over the y-axis and shifted right one unit, hence the function is:

sqrt(1 - x)

The part graphed is the values above the function, that is, the values that are greater, as the line is only dashed, hence the inequality is:

f(x) > sqrt(1 - x)

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Answer:

Step-by-step explanation:

First prime factorize 135,

135 = 5×3×3×3

Now prime factorize 128,

128=2×2×2×2×2×2×2 = 2^7

the highest prime factor of 135 is 5,

Let 5 be x

the highest power of the lowest prime factor of 128 which is the power 2 is 7

Let 7 be y

According to the question x^y = 5^7

Answer:

[tex]x^2+8x+8+\frac{-9}{x+4}[/tex]

Step-by-step explanation:

it would be simpler to do synthetic division by removing the variables and using the coefficients:

i hope this helped!

Length of each side is 176 feet ² .

each side of house = 22 feet long

Perimeter of octagon = 8 × sides

Length of each side of house = 8 × 22 ⇒ 176 feet²

Therefore, length of each side is 176 feet ² .

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The difference between the two given fractions is  [tex]\frac{x^{2} +4x-1}{x^{2} +2x}[/tex].

Fraction if a part of whole. It is represented in the form of numerator and denominator.

Given that,

[tex]\frac{x+5}{x+2} -\frac{x+1}{x^{2} +2x}[/tex]

=  [tex]\frac{x+5}{x+2} -\frac{x+1}{x(x+2)}[/tex]

Take the LCM of the denominators is [tex]x(x+2)[/tex].

=  [tex]\frac{x(x+5)-(x+1)}{x(x+2)}[/tex]

=  [tex]\frac{x^{2}+5x -x-1}{x^{2} +2x}[/tex]

=   [tex]\frac{x^{2}+4x-1}{x^{2} +2x}[/tex]

Thus, the difference between the two given fractions is  [tex]\frac{x^{2}+4x-1}{x^{2} +2x}[/tex] .

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The given question is not in correct form.

Answer:

The sum of all 2 digit multiples of 6 is 810

Step-by-step explanation:

12+18+24+30+36+42+48+54+60+66+72+78+84+90+96 = 810

Answer:

810

Step-by-step explanation:

2 digit multiples of 6 are 12,18,24...90,96

You can just add them on your calculator.

The expression which shows the earning of Mrs. Fresneda is 4x and the money that she will earn by selling 26 bars of soap is $144.

Given that price of soap is $4.00 per bar.

We have to find the expression which shows the earning of Mrs. Fresneda after selling soap bars and the money that she will earn by selling 26 bars of soap.

let the number of soap bars be x.

We know that the revenue is the product of price and quantity.

Money that Mrs.Fresneda earns is 4*x

=4x

Expression is 4x which shows that money that Mrs. Freneda will earn by selling soap bars.

We have to just put x=26 in the expression to calculate the money after selling 26 bars of soap.

Money=4*26

=$144

Hence the expression is 4x and the money that Mrs.Freneda will earnby selling 26 bars of soap is $144.

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Mark and Don have 18  and 92 marbles respectively.

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side. It demonstrates the equality of the relationship between the expressions printed on the left and right sides. We have LHS = RHS (left hand side = right hand side) in every mathematical equation. To determine the value of an unknown variable that represents an unknown quantity, equations can be solved.

Let the number of marbles Mark has be x.

According to the question,

Don has (5x+2) marbles.

Total number of marbles=110

Thus, the required equation is:

x+5x+2=110

6x=108

x=108/6

x=18

Mark has 18 marbles.

Don has (5*18+2)=92 marbles.

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Answer:

  f(x) = (x -6)² +14

Step-by-step explanation:

Completing the square involves writing part of the function as a perfect square trinomial.

The square of a binomial results in a perfect square trinomial:

  (x -h)² = x² -2hx +h²

The constant term (h²) in this trinomial is the square of half the coefficient of the linear term: h² = ((-2h)/2)².

One way to "complete the square" is to add and subtract the constant necessary to make a perfect square trinomial from the variable terms.

Here, we recognize the coefficient of the linear term is -12, so the necessary constant is (-12/2)² = 36. Adding and subtracting this, we have ...

  f(x) = x² -12x +36 +50 -36

Rearranging into the desired form, this is ...

  f(x) = (x -6)² +14

__

Additional comment

Another way to achieve the same effect is to split the given constant into two parts, one of which is the constant necessary to complete the square.

  f(x) = x² -12x +(36 +14)

  f(x) = (x² -12x +36) +14

  f(x) = (x -6)² +14

The point estimate of the proportion of students who prefer the new schedule is 0.22.

Definition of Point Estimate

Point estimation is a technique used in statistics to determine a single value that will serve as the "best guess" or "best estimate" of an unidentified population characteristic, known as the point estimate. More precisely, in order to produce a point estimate, we apply a point estimator to the data.

For the confidence interval (0.195, 0.245), point estimate is calculated as follows,

p = (0.195 + 0.245) / 2

p = 0.44 / 2

p = 0.22

Hence, the point estimate of the proportion of students who prefer the new schedule comes out to be 2.2.

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The smallest possible value of N is 23.

According to the statement

we have given that the sequence {1,2,3.....n}

and the sum is 241. we have to find the smallest value of n.

So, for this purpose we use summation formula of an arithmetic sequence

Sn = n /2 ( a1 +an )

Put the values in it then.

Note that the sum of the first 21 integers  is   21 * 22 /2  =  231...this isn't large enough as compare to given value.

And the sum of the first  22 integers  =   22 * 23 / 2  =   253

So    253 - 241  =  12 = omitted sum.....  

but  the sum of two consecutive integers must be odd

And......the sum of the first 23 integers is  23 * 24 / 2  = 276

So.......276 - 241   =  35    = omitted sum

So....the   consecutive integers omitted must be  17  and 18

So....... the smallest value of n  is   23

So, The smallest possible value of N is 23.

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The transformations applied to F(x) is a vertical shift of 2 and a phase shift of π/2.

When a function is transformed, the graph's curve either "moves to the left/right/up/down," "expands or compresses," or "reflects."

Function transformations are highly useful for graphing functions without having to start from scratch because they allow you to move, extend, compress, or reflect the curve.

The graph of the parent function is either "moved," "resized," or "reflected" by a function transformation. The three primary categories of function transformations are as follows:Translation,Dilation and Reflection.

The given function F(x)=cos x is transformed to G(x)=2-sin x, by applying a vertical shift of 2 units upwards and a phase shift of π/2.

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Answer:

See below.

Step-by-step explanation:

a. If h is the number of hours of labour required to print the cards, construct an equation for the cost of the cards, C.

C = $250 + ($40/hr)*h

b. You have budgeted $1000 for the printing job. How many hours of labour can you afford? Give your answer to the nearest minute.

$1000 = $250 + ($40/hr)*h

($40/hr)*h = $750

h = 18.75 hours

(18.75 hours)(60 min/hr) = 1125 minutes

c. The company estimates that it can print 1000 cards per hour of labour. How many cards will you get printed with your current budget?

(18.75 hr)*(1000 cards/hr) = 18,750 cards

d. An alternative to printing is photocopying. The company charges 15 cents per side for the first 10 000 cards and then 10 cents per side for the remaining cards. Which is the cheaper option for 18 750 single-sided cards and by how much?

Photocopy Cost for the first 1000 single-sided cards is:

(1000 cards)*($0.15) = $150

Photocopy Cost for the final 17,750 single-sided cards is:

(17,750)*($0.10) = $1,775

Totoal photocopy cost for 18,750 single-sided cards is $1,925.  [1775+150]

The cheapest option for 18,750 cards is printing:  $1,000

Printing is (1,925 - 1000) $925 cheaper than photocopying.

The package has a surface area of 4520cm². The area of the package is less than the area of the wrapping paper. So, dayson  can cover the package with the wrapping paper.

According to the question,

The first step is to determine the surface area of the package.

Total surface area of a cuboid = 2 (l w + w h + l h)

where:

• l = length

• w = width

• h = height

2[(50 x 20) + (50 x 18) + (20 x 18)] = 4520cm²

The second step is to convert 1 square meter to cm

1m² = 100 cm²

100 x 100 = 10,000

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Answer:

The gradient is -[tex]\frac{2}{3}[/tex] and the intercept is 2.

Step-by-step explanation:

First, transform the given equation into the slope-intercept form, y = mx + b. The variable m represents slope (gradient) and the variable b represents the intercept.

2x + 3y = 6

3y = -2x + 6

y = -[tex]\frac{2}{3}[/tex]x + 2

The gradient is -[tex]\frac{2}{3}[/tex] and the intercept is 2.

Answer:

[tex]\sf gradient =\dfrac{-2}{3}\\\\ y -intercept = 2[/tex]

Step-by-step explanation:

      [tex]\sf \boxed{\bf y = mx +b}[/tex]

Here m is the slope or gradient and b is the y-intercept.

Write the given equation in slope-intercept form.

  2x + 3y = 6

          3y = -2x + 6

            [tex]\sf y =\dfrac{-2}{3}x +\dfrac{6}{3}\\\\ y = \dfrac{-2}{3}x + 2[/tex]

[tex]\sf gradient =\dfrac{-2}{3}\\\\ y -intercept = 2[/tex]

The approximate measure of angle G is 41.4°. The correct option is C. 41.4°

From the question, we are to determine the measure of angle G

From the law of cosines, we can write that

[tex]cos\ G =\frac{e^{2}+f^{2}-g^{2} }{2ef}[/tex]

In the diagram,

e = 12

f = 10

g = 8

Putting the values into the equation

[tex]cos\ G =\frac{12^{2}+10^{2}-8^{2} }{2\times 12 \times 10}[/tex]

[tex]cos\ G =\frac{144+100-64 }{240}[/tex]

[tex]cos\ G =\frac{180}{240}[/tex]

cosG = 0.75

G = cos⁻¹(0.75)

G = 41.4°

Hence, the approximate measure of angle G is 41.4°. The correct option is C. 41.4°

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Hello,

[tex] \frac{ \frac{m - 4}{m + 4} }{m + 2} = \frac{ \frac{m - 4}{m + 4} }{ \frac{m + 2}{1} } = \frac{m - 4}{m + 4} \times \frac{1}{m + 2} = \frac{m - 4}{(m + 4)(m + 2)} [/tex]

Answer:

slope = 2

Step-by-step explanation:

calculate the slope m using the slope formula

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

with (x₁, y₁ ) = (- 1, 0) and (x₂, y₂ ) = (0,2) ← 2 points on the line

m = [tex]\frac{2-0}{0-(-1)}[/tex] = [tex]\frac{2}{0+1}[/tex] = [tex]\frac{2}{1}[/tex] = 2

Answer:

[tex]x=\frac{83}{36}[/tex]

Step-by-step explanation:

1) Convert [tex]2\frac{1}{4}[/tex] to  improper fraction. Use this rule: [tex]a\frac{b}{c} =\frac{ac+b}{c}[/tex].

[tex]x-\frac{2\times4+1}{4} = \frac{1}{18}[/tex]

2) Simplify [tex]2\times4[/tex] to 8.

[tex]x-\frac{8+1}{4} =\frac{1}{18}[/tex]

3) Simplify [tex]8+1[/tex] to 9.

[tex]x-\frac{9}{4} =\frac{1}{18}[/tex]

4) Add [tex]\frac{9}{4}[/tex] to both sides.

[tex]x=\frac{1}{18} +\frac{9}{4}[/tex]

5) Simplify [tex]\frac{1}{18} +\frac{9}{4}[/tex] to [tex]\frac{83}{36}[/tex].

[tex]x=\frac{83}{36}[/tex]

Decimal Form: 2.305556

Answer:

83/36

Step-by-step explanation:

x - 2 1/4 = 1/18

x = 1/18 + 9/4

L.C.M= 36,

x = 1x2/18x2 + 9x9/4x9

x = 2/36 + 81/36

x = 83/36

Decimal form = 2.305556