Factor the expressions:​

Answer:

18 cm²

Step-by-step explanation:

The given triangle is a right angled triangle.

By tilting the triangle & placing the 90° to the base side, we can see that,

Base (b) = 6 cm

Height (h) = 66cm

=》Area of the triangle

= ½bh

= ½ × 6 × 6

= ½ × 36

= 18 cm²

■ Refer to the image.

Hope it helps ⚜

leg (a)= 6

leg (b)= 6

the area of the triangle.

\(area = \frac{ab}{2} \)

\(a = \frac{6 \times 6}{2} \)

\(a = 18 \: {cm}^{2} \)

hence, the area of the triangle is 18 square centimeters.

The minimum surface area of a rectangular box with a square base is 300 square inches.

Surface area is the amount of space on the outside of a three-dimensional form. The surface area of a solid item is the total area that the object's surface occupies.

The mathematical definition of surface area in the presence of curved surfaces is significantly more involved than the definition of arc length for one-dimensional curves or surface area for polyhedra , where the surface area is the sum of the areas of its faces. Smooth surfaces, such as spheres, are ascribed surface area through their representation as parametric surfaces.

The volume of a rectangular box with a square base, an open-top is 500 cubic inch

So, V=lbh

500 inch³=lbh

Also, given that base is a square, so l=b

h= 500 inch³/lb

h=500 inch³/l²

Now, Surface area(A)= Area of base+ Area of 4 walls

=l²+4lh

=l²+4l(500/l²

=l²+(2000/l)

To minimize the surface area, differentiate it with respect to l.

A'(l)=d/dl(l²+(2000/l))

=2l-(2000/l²)

=(2l³-2000)/l²

Substitute A'(l)=0 to obtain the critical point

(2l³-2000)/l²=0

2l³-2000=0

2l³=2000

l³=1000

l=10

Again differentiating A'(l) to confirm the minimum value at l=10

A"(l)=d/dl((2l³-2000)/l²)

A"(l)=2+(4000/10³)

A"(10)=2+(4000/10³)

A"(10)=6>0

Thus, the required surface area is,

Surface area=l²+(2000/l)

=10²+(2000/l)

=300

Therefore,  the minimum surface area of a rectangular box is 300 square inches.

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The Answer is 1440 There are two possibilities for the person who demands to be either first or last: they can be first in line, or they can be last in line.

The problem is asking for the number of ways 7 engineering students can line up for presentations at a hiring fair, where one person demands to be either the first or last in line.

If they are first in line, then there are 6 students left to line up. The number of ways to line up 6 students is 6!, which is 720.

If they are last in line, then there are also 6 students left to line up, and the number of ways to line up 6 students is 6! or 720.

So, the total number of ways to line up the 7 students is 720 + 720 = 1440.

Therefore, the answer to the question is 1440.

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Answer:

(3,0)

Step-by-step explanation:

put x-3=0 and solve for it whic hwould be x=3

Answer:

11 and 12

Step-by-step explanation:

11 and 12 is the square root of 256 and 144

in the question so

We can conclude that log10(7) is irrational using a proof by contradiction with the steps given below.

To prove that log10(7) is irrational using a proof by contradiction, we start by assuming that log10(7) is rational, which means that it can be expressed as a ratio of two integers p and q:

log10(7) = p / q

where p and q have no common factors.

We can then use the definition of logarithms to rewrite this equation in exponential form:

10^(p/q) = 7

Next, we raise both sides of the equation to the power of q to eliminate the denominator:

10^p = 7^q

Now, we can assume the opposite, which is that log10(7) is rational and then find a contradiction. If log10(7) is rational, then 10^p and 7^q are also integers. However, 7 and 10 are relatively prime, which means that they do not share any common factors.

This implies that the prime factorization of 7^q cannot include any factors of 2 or 5, which are the prime factors of 10. But since 10^p is an integer, it must have a prime factorization that includes at least one factor of 2 or 5.

Therefore, we have arrived at a contradiction, which shows that our assumption that log10(7) is rational is false. Hence, we can conclude that log10(7) is irrational.

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Answer:

You need to deposit a minimum of $15.50 to avoid being charged a service fee.

Explanation:

Let x represent the amount needed to be deposited to avoid being charged a service fee.

Given;

Initial balance = 516.46 dollars

Amount on cheque to be withdrawn = 31.96 dollars

Minimum amount to avoid being charged = 500 dollars

So we have;

Therefore, you need to deposit a minimum of $15.50 to avoid being charged a service fee.

Answer:

slop: 5/2 (rise 5 run 2)

this means that 5mm of rainfall falls every 2 hours

(I hope this help, your question is a little confusing because it doesn't give the drop down menus.)

Step-by-step explanation:

Answer:

The slope of the line is given by:

\((y_2 - y_1) / (x_2-x_1)\)

We will treat \((2, 5)\) as point 1, and \((4, 10)\) as point 2.

So we have:

\(Slope=(10-5)/(4-2)=5/2=2.5\)

What this means is that for every hour, we have \(2.5mm\) of rain. Or in \(2\) hours, we will have \(5mm\) of rain. And in three hours we will have \(7.5mm\) of rain and so on.

Answer: A.82

have: x + 105 + x + 89 = 180

⇔ 2x + 194 = 180

⇔ 2x = 180 - 194 = -14

⇔ x = -14/2 = -7

=>  the measure of the angle is -7 + 89 = 82

Step-by-step explanation:

All of the conditions for inference are met, so it is appropriate to construct a 99% confidence interval for the true proportion of all magazine subscribers who do not read the magazine they subscribe to.

Random sample: The sample of 100 magazine subscribers was selected randomly, so this condition is met.

10% condition: The sample size is 100, which is less than 10% of the population of all magazine subscribers, so this condition is met.

Large counts condition: To check the large counts condition, we need to calculate the expected number of subscribers who do not read the magazine they subscribe to, which is n × p = 100 × 0.38 = 38. Since this number is greater than 5, the large counts condition is met.

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Answer:

Random condition:

✔ met

10% condition:

✔ met

Large counts condition:

✔ met

Are the conditions for inference met?

✔ yes

When you develop an argument with a major premise, a minor premise, and a conclusion, you are using deductive reasoning. When constructing an argument using deductive reasoning, three components are involved: a major premise, a minor premise, and a conclusion.

Deductive reasoning is a logical process where the conclusion is derived from the major and minor premises. The major premise is a general statement or principle that establishes a broad context or rule.

The minor premise is a specific statement or evidence that relates to the major premise. Finally, the conclusion is the logical inference or outcome that follows from the combination of the major and minor premises.

Deductive reasoning allows for the logical progression from general principles to specific conclusions, making it a valuable tool in fields such as mathematics, logic, and philosophy.

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We have to divide the shape into two

It will give us a Rectangle and a Trapezium

Answer: y=5x+100

Step-by-step explanation:

the y intercept is the 100$ invested into the account, and the rate of change is 5

Answer:

\(y=c_1\cos(x)+c_2\+\sin(x)+\sin^2(x)-\frac{1}{3}\sin^4(x)+\frac{1}{3}\cos^4(x)}}}\)

Step-by-step explanation:

Given the second-order differential equation, \(y'' + y = cos2(x)\), solve it using variation of parameters.

(1) - Solve the DE as if it were homogenous and find the homogeneous solution\(y'' + y = cos2(x) \Longrightarrow y'' + y =0\\\\\text{The characteristic equation} \Rightarrow m^2+1=0\\\\m^2+1=0\\\\ \Longrightarrow m^2=-1\\\\\ \Longrightarrow m=\sqrt{-1} \\\\\Longrightarrow \boxed{m=\pm i} \\ \\\text{Solution is complex will be in the form} \ \boxed{y=c_1e^{\alpha t}\cos(\beta t)+c_2e^{\alpha t}\sin(\beta t)} \ \text{where} \ m=\alpha \pm \beta i\)

\(\therefore \text{homogeneous solution} \rightarrow \boxed{y_h=c_1\cos(x)+c_2\sin(x)}\)

(2) - Find the Wronskian determinant

\(|W|=\left|\begin{array}{ccc}y_1&y_2\\y'_1&y'_2\end{array}\right| \\\\\Longrightarrow |W|=\left|\begin{array}{ccc}\cos(x)&\sin(x)\\-sin(t)&cos(x)\end{array}\right|\\\\\Longrightarrow \cos^2(x)+\sin^2(x)\\\\\Longrightarrow \boxed{|W|=1}\)

(3) - Find W_1 and W_2

\(\boxed{W_1=\left|\begin{array}{ccc}0&y_2\\g(x)&y'_2\end{array}\right| and \ W_2=\left|\begin{array}{ccc}y_2&0\\y'_2&g(x)\end{array}\right|}\)

\(W_1=\left|\begin{array}{ccc}0&\sin(x)\\\cos^2(x)&\cos(x)\end{array}\right|\\\\\Longrightarrow \boxed{W_1= -\sin(x)\cos^2(x)}\\\\W_2=\left|\begin{array}{ccc}\cos(x)&0\\ -\sin(x)&\cos^2(x)\end{array}\right|\\\\\Longrightarrow \boxed{W_2= \cos^3(x)}\)

(4) - Find u_1 and u_2

\(\boxed{u_1=\int\frac{W_1}{|W|} \ and \ u_2=\int\frac{W_2}{|W|} }\)\

u_1:

\(\int(\frac{-\sin(x)\cos^2(x)}{1}) dx\\\\\Longrightarrow-\int(\sin(x)\cos^2(x)) dx\\\\\text{Let} \ u=\cos(x) \rightarrow du=-sin(x)dx\\\\\Longrightarrow\int u^2 du\\\\\Longrightarrow \frac{1}{3}u^3\\ \\\Longrightarrow \boxed{u_1=\frac{1}{3}\cos^3(x)}\)

u_2:

\(\int\frac{\cos^3(x)}{1}dx\\ \\\Longrightarrow \int \cos^3(x)dx\\\\ \Longrightarrow \int (\cos^2(x)\cos(x))dx \ \ \boxed{\text{Trig identity:} \cos^2(x)=1-\sin^2(x)}\\\\\Longrightarrow \int[(1-\sin^2(x)})\cos(x)]dx\\\\\Longrightarrow \int \cos(x)dx-\int (\sin^2(x)\cos(x))dx\\\\\Longrightarrow \sin(x)-\int (\sin^2(x)\cos(x))dx\\\\\text{Let} \ u=\sin(x) \rightarrow du=cos(x)dx\\\\\Longrightarrow \sin(x)-\int u^2du\\\\\Longrightarrow \sin(x)-\frac{1}{3} u^3\)\

\(\Longrightarrow \boxed{u_2=\sin(x)-\frac{1}{3} \sin^3(x)}\)

(5) - Generate the particular solution

\(\text{Particular solution} \rightarrow y_p=u_1y_1+u_2y_2\)

\(\Longrightarrow y_p=(\frac{1}{3}\cos(x))(\cos(x))+(\sin(x)-\frac{1}{3} \sin^3(x))(\sin(x))\\\\ \Longrightarrow y_p=\frac{1}{3}\cos^4(x)+\sin^2(x)-\frac{1}{3}\sin^4(x)\\\\\Longrightarrow \boxed{y_p=\sin^2(x)-\frac{1}{3}\sin^4(x)+\frac{1}{3}\cos^4(x)}\)

(6) - Form the general solution

\(\text{General solution} \rightarrow y_{gen.}=y_h+y_p\)

\(\boxed{\boxed{y=c_1\cos(x)+c_2\+\sin(x)+\sin^2(x)-\frac{1}{3}\sin^4(x)+\frac{1}{3}\cos^4(x)}}}\)

Thus, the solution to the given DE is found where c_1 and c_2 are arbitrary constants that can be solved for given an initial condition. You can simplify the solution more if need be.

The maximum value of the objective function will be:50(100) + 50(20) = 5000 + 1000 = 6000

The given linear programming is:Max 50A + 50Bs.t.10A ≤ 100010B ≤ 80020A + 40B ≤ 4000A, B ≥ 0 To solve the linear programming using the solver follow these steps:Step 1: Open the Excel spreadsheet and go to Data, choose the solver and enable it.

Step 2: A Solver Parameters dialog box will appear. Input the necessary details as follows:Set Objective: 50A + 50BTo: Max Subject to constraints:

10A ≤ 100010B ≤ 80020A + 40B ≤ 4000

Select variables to change: A, BStep 3: Click Solve and the optimal value of A and B will be found.In this case, the optimal value of A is 100 and that of B is 20.

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Answer:

66

Step-by-step explanation:

Answer:

66? ................... I think

The density function of area is is ½({(y/\pi)}^{1/2}(e^{{\lambda(y/\pi)}^{1/2}}+ e^{{-\lambda(y/\pi)}^{1/2}}).

A random variable can be defined as a variable whose value is unknown or as a function that assigns numerical values to each of the outcomes of an experiment. It can also be defined as a rule that assigns a numerical value to each outcome in a sample space.

The density function (1.4-5)P(x) = an exp(-ax), if x0,0, if x>0, where an is any positive real number, defines the exponential random variable.

radius R has df f(r) ={\lambdae}^{-\lambdar}

area Y= *r2

r = \pm (Y/(\pi))1/2

hence P(Y<y) =F(Y) =P( *r2<y) =P(r<+/- (Y/( ))1/2 ) = \int_{{-y/\pi}^{1/2}}^{{(y/\pi)}^{1/2}}{\lambdae}^{-\lambdar} dr

= e^{{\lambda(y/\pi)}^{1/2}}- e^{{-\lambda(y/\pi)}^{1/2}}

Therefore P(df of y) = f(y) = d/dyf(y) =(d/dy) (e^{{\lambda(y/\pi)}^{1/2}}- e^{{-\lambda(y/\pi)}^{1/2}})

= ½({(y/\pi)}^{1/2}(e^{{\lambda(y/\pi)}^{1/2}}+ e^{{-\lambda(y/\pi)}^{1/2}})

Therefore the density function of area is ½({(y/\pi)}^{1/2}(e^{{\lambda(y/\pi)}^{1/2}}+ e^{{-\lambda(y/\pi)}^{1/2}})

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Answer:

Step-by-step explanation:

110+z=180  180-110=70  z=70

y=110  x=70

The function for the arithmetic sequence (fn) = 8n

The domain is {1,2,3,…10}

An arithmetic sequence is a series of numbers in which each term (save the first) is obtained by adding a constant number to the preceding term. For example, 2, 4, 6, 8,... is an arithmetic sequence because each term is created by adding 2 (a constant integer) to the previous term.

Given that

The arithmetic sequence is 8, 16, 24, 32, 40, 48, 56

Common difference = 16-8 = 8

The sequence is increasing so the common difference is positive 8

fn = 8 + (n-1)d

fn = 8 + (n-1)8

fn = 8 + 8n – 8

fn = 8n

So the function for the arithmetic sequence (fn) = 8n

b. The points to the graph are represented by (nfn). So the points are (1,8);(2,16);(3,24);(4,32);(5,40);(6,48);(7,56)

The domain of the function is the number 10

So, the domain is {1,2,3,…10}

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The result of the each of the following set is

A ∪ B = {1, 3, 5, 6, 7, 9}

A ∩ B = {3, 9}

A ∪ C = {1, 2, 3, 4, 5, 6, 7, 8, 9}

A ∩ C = {∅}

A - B = {1, 5, 7}

B - A = {6}

B U C = {2, 3, 5, 6, 8, 9 }

B ∩ C = {6}

The given values are

A = {1, 3, 5, 7, 9}

B = {3, 6, 9}

C = {2, 4, 6, 8}

Then find the each given terms in set roaster notation

Union of the set, intersection of the set  and the difference of the set are the basic operations of set

A ∪ B = {1, 3, 5, 6, 7, 9}

A ∩ B = {3, 9}

A ∪ C = {1, 2, 3, 4, 5, 6, 7, 8, 9}

A ∩ C = {∅}

A - B = {1, 5, 7}

B - A = {6}

B U C = {2, 3, 5, 6, 8, 9 }

B ∩ C = {6}

Therefore, all the given terms has been found

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Answer:

Number of people with polio.

Ice cream sales.

Seems like a weird thing to be correlated but I saw a graph with this example and people used to think that ice cream was the reason for polio. It's obvious there's no causation!

hope this helps!!

Answer:

$735

Step-by-step explanation:

Multiply 24 and 30: 24*30=720

Add 15: 720+15

=735

Answer:

the correct answer is 735

Step-by-step explanation:

Answer:

B. 5 units

Step-by-step explanation:

Transformation is the movement of a point from its initial location to a new location. Types of transformation are reflection, dilation, rotation or translation.

If a point X(x,y) is rotated about the origin 180 degrees clockwise the new point is at X'(-x, -y).

If the square with vertices at A(0, 0), B(0, 5), C(5, 5) and D(5, 0) is rotated about the origin 180 degrees clockwise the new points are at A'(0, 0), B'(0, -5), C'(-5, -5), D'(-5, 0)

A square has four equal sides. The distance between two points \(X(x_1,y_1)\ and\ Y(x_2,y_2)\) is:

\(|XY|=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)

Hence:

\(|A'B'|=\sqrt{(0-0)^2+(-5-0)^2} =5\)

The mass of the marble is 2.7 x 1 x 0.6 x 0.3 x 1000 = 162.6 kg. The density of the marble is 2.7 g/cm3. To calculate the mass of the marble, multiply its volume by its density and then multiply that result by 1000.

Step 1: Determine the volume of the marble using the given dimensions.

Volume = 1 m x 0.6 m x 0.3 m = 0.18 m3

Step 2: Multiply the volume of the marble by its density to get the mass.

Mass = 0.18 m3 x 2.7 g/cm3 x 1000 = 162.6 kg

The mass of the marble can be determined by first calculating its volume. The given dimensions of the cuboid are 1 m by 0.6 m by 0.3 m. The volume of the marble is therefore 0.18 m3. The density of the marble is 2.7 g/cm3. To calculate the mass of the marble, multiply its volume by its density and then multiply that result by 1000. This gives a result of 162.6 kg. Therefore, the mass of the marble is 162.6 kg.

the complete question is : A culpture need to lift a piece of marble. It is a cuboid with dimenions 1 m by 0. 6 m by 0. 3 m. Marble has a denity of 2. 7g/cm cubed. What is the mass of the marble in kg?

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The required point is 0.25 miles away from the nearest point on the shore.

The differentiation of a function is defined as rate of change of its value at a point. It can be written as f'(x) =  lim h--> 0 (f(x + h) - f(x)) /(x + h - x).

here, we have,

It can be applied to minimize or maximize a function.

The given problem can be shown in the form of a diagram as follows

Suppose the underwater cable meet the shore at a distance x from the nearest point.

Then, the distance of cable in water is x² + 3.5² = x² + 12.25

And, the distance of cable in the ground is 10 - x.

As per the question, the cost of the laying cable can be represented as follows,

C(x) = 1200(10 - x) + 2400(x² + 12.25)

In order to minimize the cost the above function can be differentiated and equated to zero as,

C'(x) = -1200 + 4800x = 0

=> x = 1200/4800

      = 1/4

      = 0.25

Hence, in order to minimize the cost of the entire project the underwater cable should meet the shore at 0.25 miles away from the nearest point.

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Answer:

There are 192 employees in the company.

Step-by-step explanation:

Let x be the total number of employees

Use ratio and proportion

\(\frac{120}{x} =\frac{5}{8} \\5x = 120(8)\\5x = 960\\\frac{5x}{5} = \frac{960}{5} \\x = 192\)

x = 192

Wouldnt that be zero cuz they used more then they started of with?