The cross-section of a motorway tunnel has the shape of a rectangle with a semicircle on top. How do the dimensions have to be chosen so that with a given cross-sectional area of 120 m
the circumference of the motorway tunnel will be the smallest?
Editing Note:
1. First draw a sketch.
2.) Set up an objective function with the variables x and y.
3.) Necessary and sufficient conditions for the minimum must be proven.

The chance that the voter will endorse the mayor's work on one of these three occasions is 0.189.

Given that,

Considering that 70% of voters support the mayor's efforts, 30% of voters oppose those efforts.

Formula for the binomial distribution

\(nCx. p^{x}.(1-p)^{n-x}\)

We find the number of ways only 1 person approves nCx of the mayor multiplied by the probability 1 person approves P^x and 2 people disapprove (1-p)^{n-x}

Only one person approves of in how many ways is

=  nCx. p^{x}.(1-p)^{n-x}

= 3C1. 7^1.(1-7)^(3-1)

By solving we get,

= 0.189

Hence,  the probability that on exactly one of these three occasions the voter approves of the mayor's work is 0.189

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Answer:

x = 80%

Step-by-step explanation:

Let the initial value be y

it rose by x%

That will be;

y + x/100 * y

= y + xy/100

Now, it fell by x%

That would give;

(y + xy/100)- (x/100(y + xy/100)) to give a total fallout of = y-0.64y = 0.36y

y + xy/100 - xy/100 - x^2y/10,000 = 0.36y

y - x^2y/10,000 = 0.36y

Cut out all y

1-x^2/10,000 = 0.36

Multiply through by 10,000

10,000 - x^2 = 3600

x^2 = 10,000-3600

x^2 = 6400

x = √6400

x = 80

Central tο Malthus' theοrem is the nοtiοn that pοpulatiοn grοws geοmetrically and the fοοd supply οnly increases arithmetically.

Central to Malthus' theorem is the notion that population growth tends to outpace the availability of resources, leading to potential scarcity and societal challenges. Malthus argued that human population has the tendency to increase exponentially, while the production of food and resources grows at a slower rate. This imbalance can result in population pressure, poverty, and other negative consequences.

Malthus' theorem, often referred to as the Malthusian theory, suggests that unless checked by factors such as disease, famine, war, or voluntary restraint, population growth will eventually exceed the capacity of the environment to sustain it. This theory has been influential in the field of demography and has sparked debates about population control, resource management, and sustainable development.

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Answer: -4h-4

Step-by-step explanation: the -4 stays the same, and then it’s just 2h -6h.

To find the 99% confidence interval for the mean starting salary of MBA graduates, we use the same formula as for the 95% confidence interval but with a larger critical z-score, resulting in a wider interval that provides greater confidence.

The given 95% confidence interval for the mean starting salary of MBA graduates is ($75, $95), which means that 95% of intervals obtained by repeatedly sampling MBA graduates will contain the true mean starting salary.

To find the 99% confidence interval, we use the same formula but with a different critical z-score. The critical z-score for a 99% confidence level is approximately 2.576, which is larger than the critical z-score for a 95% confidence level.

Substituting the given values into the formula, we get a 99% confidence interval estimate of ($85 ± $10.304 / √n) for the mean starting salary of MBA graduates. The interval width remains the same, but the larger critical z-score results in a wider interval that provides greater confidence.

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It should take John and Caroline approximately 0.1538 hours, or about 9.2 minutes, to wash the building windows when working together.

To solve this problem, we can use the formula:

Time taken when working together = (product of individual times) / (sum of individual times)

Let's first find the individual rates of work for John and Caroline:

John's rate of work = 1/2.5 = 0.4 windows per hour

Caroline's rate of work = 1/4 = 0.25 windows per hour

Now, we can substitute these values into the formula to find the time taken when working together:

Time taken = (0.4 x 0.25) / (0.4 + 0.25)

= 0.1 / 0.65

= 0.1538 hours

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Answer:

A

Step-by-step explanation:

\(\sqrt{50}\) is not a perfect square. This means that you can rule out answers B and D, because it won't be 'exactly' anything.

Let's think about the other answer choices. C says that the answer of \sqrt{50} will be between 24 and 25. This is easily checkable by doing \(24^{2}\), which equals 576. This easily eliminates option C.

Now, if the other answer choices weren't so obviously incorrect, let's see how you would solve this. What's the closest perfect square root under 50?

\(\sqrt{49}\), which equals 7.

What's the closest perfect square root above 50?

\(\sqrt{64}\), which equals 8.

This puts \(\sqrt{50}\) between 7 and 8.

Answer:

Option A

Step-by-step explanation:

Here we can find out the correct option by eliminating the wrong options . We know that √50 is not a perfect square.

This eliminates option d as it says that its value is exactly 7 .

\(\implies \red{\text{ Option D is wrong }}\)

Now look at Option C , it says that it lies between 24 and 25 . This is impossible since , 24² = 576 and 25² = 625 .

\(\implies \red{\text{ Option C is wrong }}\)

Again the option b is similar to option d , as it says its exact value is 25 which is not possible. Hence this option is also wrong .

\(\implies \red{\text{ Option B is wrong }}\)

Now we are ultimately left with option with shared that its value lies between 7 and 8 . We know that 7² = 49 and 8² = 64. And 50 lies in between them. Therefore Option A is the correct option .

\(\implies \green{\text{ Option A is correct. }}\)

When testing the differences between means, the research hypothesis suggests that population means are not equal.  (Option B)

Hypothesis refers to a concept or explanation for a trend that is based on known facts but has not yet been proved. A research hypothesis, also known as scientific hypothesis, is a statement about the expected result of a study. It is the proposed answer to the research question and generally includes an explanation (e.g., x affect y because …). It introduces a research question and proposes an expected result. It illustrates the relationship between two variables - an independent and dependent variable. Hence, when testing the difference between means, the hypothesis that suggests that population means are not equal is a research hypothesis.

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Answer:

7/15

Step-by-step explanation:

Step 1

We can't subtract two fractions with different denominators. So you need to get a common denominator. To do this, you'll multiply the denominators times each other... but the numerators have to change, too. They get multiplied by the other term's denominator.

So we multiply 4 by 3, and get 12.

Then we multiply 1 by 5, and get 5.

Next we give both terms new denominators -- 5 × 3 = 15.

So now our fractions look like this:

12

15

−  

5

15

Step 2

Since our denominators match, we can subtract the numerators.

12 − 5 = 7

So the answer is:

7

15

Step 3

Last of all, we need to simplify the fraction, if possible. Can it be reduced to a simpler fraction?

To find out, we try dividing it by 2...

Nope! So now we try the next greatest prime number, 3...

Nope! So now we try the next greatest prime number, 5...

Nope! So now we try the next greatest prime number, 7...

Nope! So now we try the next greatest prime number, 11...

No good. 11 is larger than 7. So we're done reducing.

Answer:

See explanation below for all parts

Step-by-step explanation:

Your current slope (m) is -1/2 (the orange colored slope) so the "opposite" slope would be 2/1 or 2(the blue colored slope)

The answer for number 3 is y=2x+b.

For number 4, plug in the point values for x and y and solve for b. The point given is (-4,-5).

-5=2(-4)+b Now simplify

-5=-8+b then add 8 to both sides to isolate b

3=b or b=3

For number 9, you now know that m is 2 and b is 3

y=2x+3

The simplification of the expression m²n³+ 18m³n² + 10m²n+2m²n is m²n ( n²+18mn+ 12)

Simplifying an expression is just another way to say solving a math problem. When you simplify an expression, you're basically trying to write it in the simplest way possible. At the end, there shouldn't be any more adding, subtracting, multiplying, or dividing left to do.

m²n³+ 18m³n² + 10m²n+2m²n = m²n³ + 18m³n² +12m²n

Therefore we factorize the common factors out, they are m²n

= m²n ( n²+18mn+ 12)

therefore the simplification of the expression m²n³+ 18m³n² + 10m²n+2m²n is m²n ( n²+18mn+ 12)

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Answer:

The answer is below

Step-by-step explanation:

A store carries four brands of DVD​ players, J,​ G, P and S. From past​ records, the manager found that the relative frequency of brand choice among customers varied. Using the given probability values for each of the four​ brands, find the probability that a random customer will choose brand J or brand P.

P(J)=0.22​, ​P(G)=0.18​, ​P(P)=0.35​, ​P(S)=0.25

Answer: Probability is the ration of possible outcomes to the total number of possible outcomes. The probability of mutually exclusive events i.e. events that cannot occur at the same time is the sum of their individual probabilities. If two events A and B are mutually exclusive events, then:

P(A or B) = P(A) + P(B)

Given that P(J)=0.22​, ​P(G)=0.18​, ​P(P)=0.35​, ​P(S)=0.25, the probability that a random customer will choose brand J or brand P is given by:

P(J or P) = P(J) + P(P) = 0.22 + 0.35 = 0.57

Answer:

a.) Marginal Product (MP) = 120

b.) Average Product = 126

c.) At x = 12, the output is maximum.

d.) After 5 levels of inputs diminishing returns set in.

Step-by-step explanation:

Given that,

Q = 72x + 15x² - x³

a.)

Marginal Product is equal to

\(\frac{dQ}{dx} = 72 + 30x - 3x^{2}\)

At x = 8

MP = 72 + 30(8) - 3(8)²

     = 72 + 240 - 192

     = 120

∴ we get

Marginal Product (MP) = 120

b.)

Average Product is equals to

= \(\frac{Q}{x}\)

= 72 + 15x - x²

At x = 6

Average Product = 72 + 15(6) - 6²

                             = 72 + 90 - 36

                             = 126

∴ we get

Average Product = 126

c.)

For Maximizing Q,

Put \(\frac{dQ}{dx} = 0\)

⇒72 + 30x - 3x² = 0

⇒24 + 10x - x² = 0

⇒x² - 10x - 24 = 0

⇒x² - 12x + 2x - 24 = 0

⇒x(x - 12) + 2(x - 12) = 0

⇒(x + 2)(x - 12) = 0

⇒x = -2, 12

As items can not be negative

∴ we get

At x = 12, the output is maximum.

d.)

Now,

For Diminishing Return

\(\frac{d(MP)}{dx} = \frac{d^{2}Q}{dx^{2} } < 0\)

⇒30 - 6x < 0

⇒-6x < -30

⇒6x > 30

⇒x > 5

∴ we get

For x > 5, the diminishing returns set in

i.e.

After 5 levels of inputs diminishing returns set in.

Answer:

The volume of gallons used by Silvio = \(7\frac{13}{32}\) gallons

Step-by-step explanation:

Total volume of milk = \(9\frac{7}{8}\) = \(\frac{79}{8}\)

amount used by Silvio = \(\frac{3}{4}\) of total volume

This means that if the total volume of milk were divided into four parts, Silvio takes 3 out of 4 parts, and this is converted to volume as follows:

∴ amount used by Silvio = \(\frac{3}{4}\) × \(\frac{79}{8}\) = \(\frac{237}{32}\) = \(7\frac{13}{32}\) gallons

Answer:

x = 74 degrees

Hope that helps

Answer:

74

Step-by-step explanation:

2x+32=180

2x=148

x=74

Answer:

4800 students

Step-by-step explanation:

120x40=4800students

4800 can sit in the class

Answer:

5.5

44 / 8 = 5.5

Answer:

Step-by-step explanation:

5

8 | 4 4

4 0

4

5 remander 4

Answer:

alternate exterior angles

Step-by-step explanation:

so are 2 & 8

Given:

The graph of a curve.

To find:

The domain and range for the given graph in interval notation.

Solution:

Domain: It is the set of input values.

Range: It is the set of output values.

Closed circle: It means that point is included in the solution set.

Open circle: It means that point is not included in the solution set.

(a) From the given graph it is clear that the function is defined between x=-5 to x=4. There is a closed circle at x=-5 and an open circle x=4, it means x=-5 is included and x=4 is not included in the domain.

\(Domain=-5\leq x<4\)

\(Domain=[-5,4)\)

(b) From the given graph it is clear that the values of the function lie between y=-3 to y=3. There is a closed circle at y=-3 and an open circle y=3, it means y=-3 is included and y=3 is not included in the range.

\(Range=-3\leq y<3\)

\(Range=[-3,3)\)

Therefore (a) \(Domain=[-5,4)\) and (b) \(Range=[-3,3)\).

The given differential equation y dx = 2(x y) dy is homogeneous, which can be solved by using the substitution y = ux. This leads to a separable differential equation in x and u, which can be integrated to obtain a general solution in terms of u. Finally, substituting back to y = ux gives the general solution to the original differential equation.

The given differential equation y dx = 2(x y) dy is homogeneous, which means that it is possible to solve it by using a substitution of the form

y = ux.

This yields the following differential equation in x and u:

u dx + x du = 2u du.

We can simplify this equation by rearranging the terms:

(x - 2u) du + u dx = 0. This equation is separable, which means that we can separate the variables and integrate both sides to solve for u.

To do this, we divide both sides by (x - 2u) u, which gives:

(1/u) du / (x - 2u) = -dx / u. Integrating both sides yields:

-ln|x - 2u| = ln|u| + C, where C is a constant of integration.

Exponentiating both sides and solving for u gives:

u = ±(Cx) / sqrt(4x^2 + C^2).

Substituting y = ux gives the general solution to the original differential equation: y = ±(Cx^2) / sqrt(4x^2 + C^2).

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Answer:

It will be at 33.3333 repeating so if you estimate the answer is y = 33

Step-by-step explanation:

Answer:

C

Step-by-step explanation:

yeah its C

The requried function with the smallest x-intercept value is C. f(x) = √(x+1), which has an x-intercept of -1.

In the equation intercept is the value of the linear function where either of the variables is zero.

Here,
The x-intercept is the point where the graph of a function intersects the x-axis. At this point, the value of y is 0. To find the x-intercept of each function, we set y equal to 0 and solve for x.

A. f(x) = 3x – 9

0 = 3x – 9

3x = 9

x = 3

B. f(x) = 1/x + 4

0 = 1/x + 4

-4 = 1/x

x = -1/4

C. f(x) = √(x+1)

0 = √(x+1)

x = -1

D. f(x) = x²

0 = x²

x = 0

Therefore, the function with the smallest x-intercept value is C. f(x) = √(x+1), which has an x-intercept of -1.

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For Quadrilateral ABCD is reflected over the x-axis, the coordinates of A' is at A'(2, 4)

For Quadrilateral ABCD is translated 2 units right and 1 unit down, the coordinates of A' is at A'(4, -5)

For Quadrilateral ABCD is dilated by a scale factor of 3, the coordinates of A' is at A'(6, -12)

For Quadrilateral ABCD is rotated 180° clockwise about the origin the coordinates of A' is at A'(-2, 4).

Transformation is the movement of a point from its initial location to a new position. The types of transformation include dilation, reflection, translation, and rotation. From the image attached, the coordinates of point A is at (2, -4).

If a point A(x, y) is reflected over the x-axis, the new point is at  A'(x, -y)

If a point A(x, y) is translated units right and b units down, the new location is at A'(x + a, y - b)

If a point A(x, y) is dilated by  factor of a, the new location is at A'(ax, ay)

If a point A(x, y) is  rotated 180° clockwise about the origin, the new location is at A'(-x, -y)

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The answers are:

\((a) \( \vec{a} \times \vec{b} = 7.0 \vec{i} + 42.0 \hat{j} - 43.0 \hat{k} \)(b) \( \vec{a} \cdot \vec{b} = 46.56 \)(c) \( (\vec{a}+\\)

\((a) To find the cross product of vectors \( \vec{a} \) and \( \vec{b} \), we can use the formula:\[ \vec{a} \times \vec{b} = (a_yb_z - a_zb_y) \vec{i} + (a_zb_x - a_xb_z) \hat{j} + (a_xb_y - a_yb_x) \hat{k} \]Substituting the values:\[ \vec{a} \times \vec{b} = (5.0 \cdot 1.4 - 8.6 \cdot 0) \vec{i} + (8.6 \cdot 5.0 - 4.6 \cdot 1.4) \hat{j} + (4.6 \cdot 0 - 5.0 \cdot 8.6) \hat{k} \]Simplifying the expression, we get:\[ \vec{a} \times \vec{b} = 7.0 \vec{i} + 42.0 \hat{j} - 43.0 \hat{k} \]\)

\((b) To find the dot product of vectors \( \vec{a} \) and \( \vec{b} \), we can use the formula:\[ \vec{a} \cdot \vec{b} = a_xb_x + a_yb_y + a_zb_z \]Substituting the values:\[ \vec{a} \cdot \vec{b} = (4.6 \cdot 8.6) + (5.0 \cdot 1.4) + (0 \cdot 0) \]Simplifying the expression, we get:\[ \vec{a} \cdot \vec{b} = 39.56 + 7.0 + 0 \]\[ \vec{a} \cdot \vec{b} = 46.56 \]\)

\((c) To find the dot product of \( (\vec{a}+\vec{b}) \) and \( (\vec{a}+\vec{b}) \), we can use the same formula as in part (b).Substituting the values:\[ (\vec{a}+\vec{b}) \cdot (\vec{a}+\vec{b}) = (4.6+8.6) \cdot (4.6+8.6) + (5.0+1.4) \cdot (5.0+1.4) + (0+0) \cdot (0+0) \]\)

\(Simplifying the expression, we get:\[ (\vec{a}+\vec{b}) \cdot (\vec{a}+\vec{b}) = 13.2 \cdot 13.2 + 6.4 \cdot 6.4 + 0 \]\[ (\vec{a}+\vec{b}) \cdot (\vec{a}+\vec{b}) = 174.24 + 40.96 + 0 \]\[ (\vec{a}+\vec{b}) \cdot (\vec{a}+\vec{b}) = 215.2 \]Therefore, the results are:(a) \( \vec{a} \times \vec{b} = 7.0 \vec{i} + 42.0 \hat{j} - 43.0 \hat{k} \)(b) \( \vec{a} \cdot \vec{b} = 46.56 \)(c) \( (\vec{a}+\\)

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The number of teams she can make is 10.

Division is the process of putting a number into equal groups using another number. The sign used to denote division is ÷. Division is one of the basic mathematical operations. Other basic mathematical operations include addition, subtraction, multiplication.

The number of teams that can be made can be determined by dividing the total number of friends by the total number of players in each team. The whole number that is derived from the division process is the number of teams teams that can be made given the total number of friends and the number of people that have to be in each group.

Number of teams she can make = total number of friends / number of people in each group

31 / 3 = 10 remainder 1

So, only 10 teams can be made. one person would not be on any team.

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Answer:

Step-by-step explanation:

Let x - be the number of sweets in small packs

y - be the number of sweets in big packs

Therefore, we have:

4x + 3y = 175 (1)

5x + 2y = 154 (2)

Now, we find the difference between (1) & (2) is:

y-x = 21. Thus, y = 21+x

Now we substitute the value of y = 21+x to any of the two statements, we have 4x + 3(21+x) = 175 => 4x + 63 + 3x = 175.

Hence, 7x = 175 - 63 = 112 or simply, x=16.

Now, finding the value of y:

5(16) + 2y = 154

80 + 2y = 154

2y = 154-80

2y = 74

y = 37.

Therefore, there are 16 sweets in each small pack and 37 sweets in each big pack.

The equation of the hyperbola is: (x^2/16) - (y^2/16) = 1

To begin, we need to use the definition of a hyperbola to find its equation. The standard form of the equation for a hyperbola centered at the origin is:

(x^2/a^2) - (y^2/b^2) = 1

Where a is the distance from the origin to the vertex along the x-axis, and b is the distance from the origin to the vertex along the y-axis.

We know that the foci of this hyperbola are at (4,0) and (-4,0). The distance between the foci is given by 2c, where c is the distance from the center of the hyperbola to a focus. So we have:

2c = 8
c = 4

We also know that the difference in the lengths of the focal radii from any point on the hyperbola is always ±6. This means that for any point (x,y) on the hyperbola, we have:

|sqrt((x-4)^2 + y^2) - sqrt((x+4)^2 + y^2)| = 6

Simplifying this equation, we get:

sqrt((x-4)^2 + y^2) - sqrt((x+4)^2 + y^2) = 6  (since the difference is always positive)
Squaring both sides, we get:

(x-4)^2 + y^2 - 2sqrt((x-4)^2 + y^2)*(x+4)^2 + y^2 = 36
Simplifying, we get:

(x-4)^2 + y^2 - (x+4)^2 - y^2 = 36
(x-4)^2 - (x+4)^2 = 36

Now we can simplify further using the difference of squares:

[(x-4)+(x+4)][(x-4)-(x+4)] = 36
2x*(-8) = 36
x = -9

Substituting x = -9 into our equation, we get:

(-9-4)^2 + y^2 - (-9+4)^2 - y^2 = 36
(-13)^2 - 5^2 = 36
144 = 36

This is a contradiction, so our original assumption that the hyperbola is centered at the origin must be incorrect. Instead, we know that the center of the hyperbola is at (0,0). This means that a = c = 4.

Substituting these values into the standard form of the equation for a hyperbola, we get:

(x^2/16) - (y^2/16) = 1

we can find the equation of the hyperbola. Since the foci are at (4, 0) and (-4, 0), the distance between them is 2a = 8, so a = 4. The difference in the lengths of the focal radii is 2b = ±6, so b = 3.

Now, we can use the formula for the equation of a hyperbola with horizontal major axis:

(x^2 / a^2) - (y^2 / b^2) = 1

Substitute the values of a and b:

(x^2 / 4^2) - (y^2 / 3^2) = 1

The equation of the hyperbola is:

(x^2 / 16) - (y^2 / 9) = 1

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The break-even point is 1100 products sold. This means that if the store can sell 1100 products, it will be able to cover its costs and make a profit.

What is average ?

The average is a measure of central tendency that is calculated by adding a set of values together and dividing the sum by the number of values. It is also known as the arithmetic mean. It is a way to represent a typical value of a dataset.

The break-even point is the point at which the cost of the products equals the revenue from the sales of the products. To find the break-even point, we need to set the cost function (C(x) = 550 + 3.00x) equal to the revenue function (R(x) = 5.50x) and solve for x.

C(x) = 550 + 3.00x = R(x) = 5.50x

3.00x = 5.50x - 550

0.50x = 550

x = 1100

So the break-even point is 1100 products sold. This means that if the store can sell 1100 products, it will be able to cover its costs and make a profit.

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Answer:

07698+96965#5_(-&)!:86539'6";

The solution to the given expression will be \(\dfrac{-8\pm\sqrt{180}}{6}\).

The polynomial having a degree of two or the maximum power of the variable in a polynomial will be 2 is defined as the quadratic equation and it will cut two intercepts on the graph at the x-axis.

The given equation will be solved as:-

= 3x² + 8x - 10  

Using the formula of factors:-

x = \(\dfrac{-b\pm\sqrt{b^-4ac}}{2a}\)

x = \(\dfrac{-8\pm\sqrt{64-(-120)}}{2\times 3}\)

x = \(\dfrac{-8\pm\sqrt{180}}{6}\).

Therefore the solution to the given expression will be \(\dfrac{-8\pm\sqrt{180}}{6}\).

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Answer:

(-1 , -5)

Hope this helped you! I would appreciate a Brainliest, if you wouldnt mind!