The length of a rectangle is six times its width. The area of the
rectangle is 294 square centimeters. Find the dimensions of the
rectangle.

Answer: x = {-2, 2}

Step-by-step explanation:

Tangent means it is touching. Find the intersection of the two equations.

Solve the linear equation for y, then set the two equations equal to each other.

[tex]4y=x+8\qquad \rightarrow \qquad y=\dfrac{x+8}{4}[/tex]

[tex]\dfrac{x-1}{x}=\dfrac{x+8}{4}\\\\\\\text{Cross multiply and solve for x:}\\4(x-1)=x(x+8)\\4x-4=x^2+8x\\.\qquad 0=x^2+4x+4\\.\qquad 0=(x+2)^2\\.\qquad 0=x+2\\.\qquad x=-2[/tex]

To find the next point that is parallel to the linear equation and tangent to the curve, we need to use the linear equation with slope (m) = [tex]\dfrac{1}{4}[/tex] and unknown b.

Let's try b = 0, then the equation of the linear equation is: [tex]y=\dfrac{1}{4}x[/tex]

Set the equations equal to each other and solve for x:

[tex]\dfrac{x-1}{x}=\dfrac{x}{4}\\\\\\4(x-1)=x^2\\4x-4=x^2\\.\qquad 0=x^2-4x+4\\.\qquad 0=(x-2)^2\\.\qquad 0=x-2\\.\qquad x=2[/tex]

This works!!!  If it didn't work, we would have tried other values for b until we arrived at a solution.

Answer:

Step-by-step explanation:

Substitute n = 1, n = 2, n = 3 and n = 4 to the equation f(n) = n² - 1:

f(1) = 1² - 1 = 1 - 1 = 0

f(2) = 2² - 1 = 4 - 1 = 3

f(3) = 3² - 1 = 9 - 1 = 8

f(4) = 4² - 1 = 16 - 1 = 15

Answer:

a)  [tex]\frac{2}{3} \,\frac{lb}{bread}[/tex]

b)  [tex]1\frac{1}{4} \,\frac{in}{domino}[/tex]

Step-by-step explanation:

Part a:

every 4 lbs of flour, she makes 6 loaves of bread. this as a rate in simplest fraction form is:

[tex]\frac{4}{6} \,\frac{lb}{bread} = \frac{2}{3} \,\frac{lb}{bread}[/tex]

Part b:

every 10 inches , 8 dominoes can be placed. then the rate can be written as:

[tex]\frac{10}{8} \,\frac{in}{domino} = \frac{5}{4} \,\frac{in}{domino} =1\frac{1}{4} \,\frac{in}{domino}[/tex]

Answer:

28

Step-by-step explanation:

140 calories over 5 ounce

= 28

The unit rate of calories per ounce will be 28 calories/ ounce

What is proportion ?

A proportion is an equation based on the equality of two ratios.

It is given that 5-ounce container of Greek yogurt contains 140 calories and it is to calculate for one ounce calories contain in Greek yogurt :

[tex]\begin{aligned}5 \text{\:ounce}&\rightarrow 140 \text{\:calories}\\1 \text{\:ounce}&\rightarrow \frac{140}{5}\text{\:calories} \\&\rightarrow 28 \text{\:calories}\end{aligned}[/tex]

Therefore, the unit rate of calories per ounce will be 28 calories/ ounce

Read more about ratio at:

https://brainly.com/question/17869111

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Answer:

The answer is (1,-5). (i.e x=1 and y=-5).

Hope it helps..

Answer:

(1,-5)

Step-by-step explanation:

Answer:

there were 63,000L of water initially in the pool

Step-by-step explanation:

do 900L times the first 20 minutes to find out how much water was drained during the first 20 minutes and you get 18,000L drained

then, add 18,000L plus the rest of the 45,000L of water drained from the pool to get 63,000L of water intially in the pool

Answer:

B: The mean will increase

Step-by-step explanation: The outlier is 46, which is way below all the other numbers, which is the definition of an outlier. If we remove a really low number from the set, then the mean(average) will increase.

Answer:

see explanation

Step-by-step explanation:

The n th term of an AP is

[tex]a_{n}[/tex] = a₁ + (n - 1)d

where a₁ is the first term and d the common difference

Given

6(a₁ + 5d) = 13(a₁ + 12d) ← distribute parenthesis on both sides

6a₁ + 30d = 13a₁ + 156d ( subtract 13a₁ from both sides )

- 7a₁ + 30d = 156d ( subtract 30d from both sides )

- 7a₁ = 126d ( divide both sides by - 7 )

a₁ = - 18d

Now

a₁₉ = a₁ + 18d = - 18d + 18d = 0 ← as required

Answer:

The length and width of the floor of the shed are 8 feet and 5.5 feet, respectively.

Step-by-step explanation:

Given that the shape of the shed is a rectangle, the expression for the area is:

[tex]A = w \cdot l[/tex]

Where [tex]w[/tex] and [tex]l[/tex] are the width and length of the shed, measured in feet. In addition, the statement shows that [tex]l = 2\cdot w - 3\,ft[/tex]. Then, the equation of area is expanded by replacing length:

[tex]A = w\cdot (2\cdot w - 3)[/tex]

[tex]A = 2\cdot w^{2} - 3\cdot w[/tex]

If [tex]A = 44\,ft^{2}[/tex], then, a second-order polynomial is formed:

[tex]2\cdot w^{2}-3\cdot w - 44 = 0[/tex]

The roots of this equation are found via General Equation for Second-Order Polynomials:

[tex]w_{1} = \frac{11}{2}\,ft[/tex] and [tex]w_{2} = -4\,ft[/tex]

Only the first roots is a physically reasonable solution. Then, the length of the shed is:

[tex]l = 2\cdot \left(\frac{11}{2}\,ft \right)-3\,ft[/tex]

[tex]l = 8\,ft[/tex]

The length and width of the floor of the shed are 8 feet and 5.5 feet, respectively.

Answer:

A. 1 4/6 cups of blueberries

Step-by-step explanation:

1 -- 2/3                      

Proportion, Batches to Blueberries

1*(2 1/2) -- (2/3)( 2 1/2)              

Because we are now multiplying the 1 batch to 2 1/2 batches. So to keep the proportion balanced/equal we are using the same operation on the right side of the proportion

2 1/2 -- (2/3)( 5/2 )      

2 1/2 -- 5/3

2 1/2 -- 1 2/3

Simplify

On the right side shows the blueberries for 2 1/2 batches. 1 2/3 = 1 4/6    

Hope that helps! Tell me if you need more info

Answer:

HI = 13

Step-by-step explanation:

The triangle that is shown is a 45-45-90 triangle, so we know that GH = GJ = 9 and IJ = 13, we are able to solve for HI.

Technically, IJ = HI, since both triangles are congruent. Both IJ and HI will be 13.

Answer:

Option D.

Step-by-step explanation:

The given quadratic equation is

[tex]y=x^2+7x+10[/tex]

We need to draw the graph of given equation by using graphing calculator as shown below.

From the graph it is clear that the parabola intersect the x-axis at points (-2,0) and (-5,0). So, the x-intercepts are (-2,0) and (-5,0).

Therefore, the correct option is D.

Answer:

The best first step would be to multiply the second equation by -2

Step-by-step explanation:

The best first step would be to multiply the second equation by -2

then you would have the following

[tex]\ \ \ \ \ \ 4x - 5y = 2 \\-2*(2x)+ (-2)*y = (-2)*3[/tex]

and when you multiply it is easy to eliminate because you will get

   [tex]4x - 5y = 2 \\-4x -2y = 6[/tex]

and if you sum the equations you get

-7y = 8

so that is a single variable equation which is easier to solve.

Answer:

F(x) = [tex]\frac{x^3}{3} - \frac{7x^2}{2} + 5x + c[/tex]

Step-by-step explanation:

The antiderivative of a function (also called the integration of a function) is the reverse of the differentiation of that function. Given a function f(x), its integration, F(x), can be calculated as follows;

F(x) = [tex]\int\limits{f(x)} \, dx[/tex]

From the question, f(x) = x² - 7x + 5

Therefore,

F(x) = [tex]\int\limits {(x^2 - 7x + 5)} \, dx[/tex]

F(x) = [tex]\frac{x^3}{3} - \frac{7x^2}{2} + 5x + c[/tex]

Where c is the constant of the integration (antiderivative).

PS: The constant of integration is used for indefinite integrals and allows to express integration of a function in its most general form.

Answer:

58 people

Step-by-step explanation:

174 people ate lunch.

1/3 of the 74 people had dessert after lunch.

Multiplying 1/3 and 174.

1/3 × 74

= 58

58 people had desert after lunch.

Answer:

[tex]\boxed{ 58\ people}[/tex]

Step-by-step explanation:

People who ate lunch = 174 people

People among among them who had desserts = 1/3 of the total

(Remember "of" means to "multiply")

=> 1/3 * 174

=> 1 * 58

=> 58 people

Answer:

It will take Mrs Chandler 18 hours 45 minutes or 18.75 hours to catch up Mr Chandler

Step-by-step explanation:

What we want to know here is that at how many hours will they have traveled same distance.

Let the total time taken by Mrs Chandler to catch up be x hours

Since Mr Chandler left 2.5 hours earlier , then the total time taken by him would be x + 2.5 hours

Now, we know that distance = speed * time

Since it’s same distance covered;

For Mr Chandler, his distance is calculated as 75(x + 2.5)

For Mrs Chandler, her distance is calculated as 85x

We equate both since they are equal;

75(x + 2.5) = 85x

75x + 187.5 = 85x

85x -75x = 187.5

10x = 187.5

x = 18.75 hours or 18 hours 45 minutes

Answer:

[tex] 24x^3\sqrt{5x} - 4x^3\sqrt{10x} [/tex]

Step-by-step explanation:

The product [tex]2\sqrt{8x^3} (3\sqrt{10x^4} - x\sqrt{5x^2})[/tex] can be simplified as follows:

Step 1: Use the distributive property of multiplication

[tex]2\sqrt{8x^3}(3\sqrt{10x^4)} - 2\sqrt{8x^3}(x\sqrt{5x^2})[/tex]

[tex] 2*3\sqrt{8x^3*10x^4} - 2*x\sqrt{8x^3*5x^2} [/tex]

[tex] 6\sqrt{80x^7} - 2x\sqrt{40x^5} [/tex]

Step 2: simplify further

[tex] 6\sqrt{16*5*x^3*x^3*x} - 2x\sqrt{4*10*x^4*x} [/tex]

[tex] 6*4*x^3\sqrt{5*x} - 2x*2*x^2\sqrt{10*x} [/tex]

[tex] 24x^3\sqrt{5x} - 4x^3\sqrt{10x} [/tex]

Answer:

Step-by-step explanation:

No figure supplied, so lots of assumptions needed.

Assume side length of triangle is 4 cm.  

( If = 4 cm  means ??)

Assume ABC is equiangular, all three angles are 60 degrees.

(This is a cross-sectional view, but don't see any)

Side length = 4

altitude of triangle = 4 sin(60) = 2sqrt(3)

radius of circumscribed circle of equilateral triangle

R = (2/3) altitude

= (2/3)*2sqrt(3)

= (4/3)sqrt(3)

Diameter

D = 2R

= (8/3) sqrt(3)

Answer: 8 cm

Step-by-step explanation:

The figure in the image attached below shows that there are two specific angles that are congruent to each other, angles AD and CD.

We are given the length of one of these angles:

AD= 4 cm so we must multiply 4 by 2, since there are TWO angles measuring 4 cm.

4 cm x 2 angles (AD and CD) =8 cm.

Proof of answer is shown below!

Answer:

Step-by-step explanation:

Since the above sequence is a geometric sequence

An nth term of a geometric sequence is given by

[tex]A(n) = a(r)^{n - 1} [/tex]

where a is the first term

r is the common ratio

n is the number of terms

From the question

a = 10

To find the common ratio divide the previous term by the next term

That's

r = 30/10 = 3 or 90/30 = 3 or 270/90 = 3

Since we are finding the 12th term

n = 12

So the 12th term is

[tex]A(12) = 10( {3})^{12 - 1} [/tex]

[tex]A(12) = 10 ({3})^{11} [/tex]

Hope this helps you

We are given the equation y = 3x - 6

The slope-intercept form of a line is y = mx + b where m is the slope and b is the y-intercept.

The b value in this equation is -6, thus the y-intercept is -6.

Let me know if you need any clarifications, thanks!

Answer:

Campaign Finance Reform

Gender Gap among Voters in the District

There is a gender gap among women and men who favor campaign finance reform.

Step-by-step explanation:

In issues such as the above, a gender gap always exist between women and men who think that there is the need to reform the campaign finance.  Women ordinarily favor a reduction in the campaign finance.  On the other hand, men do not mind so much about the candidate expenditure in campaigns.  Reducing the huge campaign finance will ensure that political campaigns and aspiration to political offices are not left to money bags.  Many women would like to get involved, but they are limited by funding.  So, anytime the issue of reforming the whole electoral system, especially with respect to campaigns, women favor the reforms more than men.  The gap is always there.  The main issue is how would this gap be measured?

Step-by-step explanation:

81^x² = 27^x

(3^4)^x² = (3^3)^x

3^(4x²) = 3^(3x)

4x² = 3x

4x² − 3x = 0

x (4x − 3) = 0

x = 0 or ¾

Answer:

59°

Step-by-step explanation:

A triangle adds up to 180°. A right triangle has a 90° angle.

1. Set up the equation

90 + 31 + x = 180

2. Simplify

121 + x = 180

3. Solve for x by subtracting 121 from both sides

x = 59

Answer:

a. p`± z₀.₀₂₅[tex]\sqrt{ \frac{p`q`}{n}[/tex]  

b.0.6 ±  1.96 [tex]\sqrt \frac{0.6* 0.4}{1000}[/tex]  

c. { -1.96 ≤  p`± z₀.₀₂₅[tex]\sqrt{ \frac{p`q`}{n}[/tex]     ≥ 1.96} = 0.95  

Step-by-step explanation:

Here the total number of trials is n= 1000

The number of successes is p` = 600/1000 = 0.6. The q` is 1 - p`= 1- 0.6 = 0.4

The degree of confidence is 95 %  therefore z₀.₀₂₅ = 1.96 ( α/2 = 0.025)

a.  The formula used will be

p`± z₀.₀₂₅[tex]\sqrt{ \frac{p`q`}{n}[/tex]       ( z with the base alpha by 2 (α/2 = 0.025))

b. Putting the values

0.6 ±  1.96 [tex]\sqrt \frac{0.6* 0.4}{1000}[/tex]  

c. Confidence Interval in Interval Notation.

{ -1.96 ≤  p`± z₀.₀₂₅[tex]\sqrt{ \frac{p`q`}{n}[/tex]     ≥ 1.96} = 0.95  

{ -z( base alpha by 2) ≤  p`± z₀.₀₂₅[tex]\sqrt{ \frac{p`q`}{n}[/tex]     ≥ z( base alpha by 2)  } = 1- α

Answer:

Step-by-step explanation:

The statement

The value of y varies inversely as the square of x is written as

[tex]y = \frac{k}{ {x}^{2} } [/tex]

where k is the constant of proportionality

To find the value of x when y = 1 first find the formula for the variation

y = 16 x = 3

k = yx²

k = 16(3)²

k = 16 × 9

k = 144

The formula for the variation is

[tex]y = \frac{144}{ {x}^{2} } [/tex]

when y = 1

We have

[tex]1 = \frac{144}{ {x}^{2} } [/tex]

Cross multiply

x² = 144

Find the square root of both sides

We have the final answer as

Hope this helps you

Step-by-step explanation:

The center of a circle is the point in the circle which is equidistant to all the edges of thr circle. The point a is the center, while point b is an arbitrary point in the circle. Find attachment for the diagram.

Answer:

i think that this question is wrong

Step-by-step explanation:

Answer:

-3x^2 -5x +2

Step-by-step explanation:

-2x(x+3)-(x+1)(x-2)=

Distribute

-2x^2 -6x  -(x+1)(x-2)

Foil

-2x^2 -6x  -(x^2 -2x +x -2)

Combine like terms

-2x^2 -6x  -(x^2 -x  -2)

Distribute the minus sign

-2x^2 -6x  -x^2  +x +2

Combine like terms

-2x^2  -x^2  -6x +x +2

-3x^2 -5x +2

Answer:

[tex]\huge\boxed{-2x(x+3)-(x+1)(x-2)=-3x^2-5x+2}[/tex]

Step-by-step explanation:

[tex]-2x(x+3)-(x+1)(x-2)[/tex]

Use the distributive property: a(b + c) = ab + ac

and FOIL: (a + b)(c + d) = ac + ad + bc + bd

[tex]=(-2x)(x)+(-2x)(3)-\bigg[(x)(x)+(x)(-2)+(1)(x)+(1)(-2)\bigg]\\\\=-2x^2-6x-\bigg(x^2-2x+x-2\bigg)=-2x^2-6x-x^2-(-2x)-x-(-2)\\\\=-2x^2-6x-x^2+2x-x+2[/tex]

Combine like terms:

[tex]=(-2x^2-x^2)+(-6x+2x-x)+2=-3x^2+(-5x)+2\\\\=-3x^2-5x+2[/tex]

Answer:

160 adults and 80 students

Step-by-step explanation:

With the information from the exercise we have the following system of equations:

Let x = number of students; y = number of adults

I want to maximize the following:

z = 3 * x + 6 * y

But with the following constraints

x + y = 240

y / 2 <= x

As the value is higher for adults, it is best to sell as much as possible for adults.

So let's solve the system of equations like this:

y / 2 + y = 240

3/2 * y = 240

y = 240 * 2/3

y = 160

Which means that the maximum profit is obtained when there are 160 adults and 80 students, so it is true that added to 240 and or every two adults, there must be at least one student.

The correct answer is 180 oranges

Explanation:

In mathematics, a ratio expresses two or more numbers that are related. In the case fo the ration 9: 40 this expresses 9 oranges are used to serve fruit salad for 40 people. Now, if you need to determine what is the number of oranges not for 40 people but for 800 people you can use cross multiplication. This process is explained below:

[tex]\frac{9}{40} = \frac{x}{800}[/tex]   - 1. Multiply  9 x 800 and 40 x x (cross multiplication)

[tex]7200 = 40x[/tex] - 2. Solve the equation by diving 7200 into 40

[tex]\frac{7200}{40} = x[/tex]

[tex]x = 180[/tex] - 3. 180 represents the number of oranges to serve 800 people, which   can be expressed as 180: 800

Answer:

[tex]\boxed{x=0}[/tex]

Step-by-step explanation:

[tex]|x-5|=|x+5|[/tex]

Solve absolute value.

There are two possibilities.

First possibility:

[tex]x-5=x+5\\0=10[/tex]

No solution.

Second possibility:

[tex]x-5=-(x + 5)\\x-5=-x-5\\2x=0\\x=0[/tex]