y=−16x ^2 +240x+115 ?

In a simple linear regression model , if we change the input variable by 1 unit , output variable will change by its slope.

What is linear regression?

According on the value of another variable, a variable's value can be predicted using a linear regression analysis. The dependent variable is the one you're trying to forecast. The term "independent variable" refers to the variable you are using to forecast the value of the other variable.

Main Body:

For linear regression

Y=a + bx + error

If we neglect error

then Y=a + bx.

If x increases by 1

then Y = a +b(x+1) which implies

Y=a + bx + b.

So Y increases by its slope.

Hence their is an increase in the slope in y.

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245.31 (dollars) is the cost (in dollars) of materials for the least expensive such container.

Calculus maxima and minima are found  using the concept of derivatives. Knowing that the derivative concept gives information about the slope/slope of a function, we find the point where the slope is zero. These points are called inflection points/stationary points. These are the points associated with the maximum or minimum (local) values ​​of the function.

Knowledge of maxima and minima is essential for our everyday problems. In addition, this article also explains how to find the absolute maximum and minimum values.

Solvable maximum and minimum arithmetic problems  are discussed in this article.

Suppose the width is x (m), length of the base is 2x (m), the base area is 2x^2 (m^2).

Since the volume is 10 (m^3), the height has to be 10/2x^2 (m) = 5/x^2.

The cost of making such container is

cost of base: 2x^2*15 = 30x^2

cost of sides: (2*2x*5/x^2 + 2*x*5/x^2)*9 = 270/x

The overall cost is hence the sum of the base and the sides: f(x) = 30x^2 + 270/x

The get the minimum,

df(x)/dx = 30*(2x - 9/x^2) = 0

so x = (9/2)^(1/3) = 1.651 (m)

f(x) = 245.31 (dollars)

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The axis of symmetry has the equation x=2, and the vertex is (2,4). The chart is displayed below.

For example,

Find the vertex of y = x^2 - 4x + 8 and sketch its graph.

Solution

Completing the square, we have

y  = x^2 - 4x + 8

= (x-2)^2+4 .

Hence the vertex is (2,4) and the axis of symmetry has equation x=2.

The graph is shown below.

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A parabola that have a focus of (7,3) and a directrix of y = 5 has these attributes

The parabola is a vertical parabola and facing down

the vertex of the parabular is v(7, 4)

p value of the parabola = -1

A parabola refers to  shape that traces equation of the form

ax² + bx + c

The directrix of a parabola refers to a vertical or horizontal line that never touches the parabola and at distance p to the vertex of the parabola., and 2p to the focus of the parabola.

when the directrix is on the y coordinate, the parabola formed is a vertical parabola.

If the value of the directrix is more than the corresponding focus or vertex there will be negative p and the parabola will face downward.

conversely, If the value of the directrix is less than the corresponding focus or vertex there will be positive p and the parabola will face up.

from definition of the directrix we calculate for p

2p = y coordinate of focus - equation of directrix

2p = 3 - 5

2p = -2

p = -1

vertex, v(h, k) compared with f(h, k + p)

h + p = 3

h = 4

therefore vertex is (7, 4)

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Answer:

216, 343

Step-by-step explanation:

1, 8, 27, 64, 125 ← are perfect cubes , that is

1³ = 1 × 1 × 1 = 1

2³ = 2 × 2 × 2 = 8

3³ = 3 × 3 × 3 = 27

4³ = 4 × 4 × 4 = 64

5³ = 5 × 5 × 5 = 125

continuing the pattern for 6 and 7

6³ = 6 × 6 × 6 = 216

7³ = 7 × 7 × 7 = 343

thus the next 2 terms in the sequence are 216, 343

After evolution of log_6(36)  value obtain is 2..

What is Logarithm?

The logarithm is exponentiation's opposite function in mathematics. This indicates that the exponent to which b must be raised in order to obtain a number x is the logarithm of x to the base b. For instance, because 1000 = 103, the logarithm of 1000 in base 10 is 3, or log10 = 3.

Given that;

log6 36

Formula we used to solve this logarithm equation is :

logb(x) = y, if b^y = x

So,

     y = log6 36

     = log6 62

     = 2 log6 6 = 2 x 1

     log6 36 = 2

Hence ,the value log6(36) is  2.

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The pair of polygons in the diagram are not similar.

What are polygons?

A closed polygonal chain made up of a finite number of straight-line segments is what defines a polygon in geometry. A polygon can be defined as a region that is circumscribed by a bounding circuit, a plane, or both. The edges or sides of a polygon circuit are its segments.

Solution Explained:

Similar polygons have proportional matching sides and all corresponding angles that are congruent.

Since the polygons' side lengths ensure that they are parallelograms, the congruence of just one pair of related angles is sufficient to guarantee the congruence of all the others.

The ratio of side lengths in the left polygon is 13/8, whereas in the right polygon it is 5.2/2.4, which equals 13/6. The matching sides are NOT proportionate since their ratios are different from one another.

Hence, the pair of polygons are not similar.

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The number of  blueberry muffins she is left with is 9.

The quantitative relation between two amounts, showing the number of times one value contains or is contained within the other.

Given that, Sarah baked blueberry and chocolate muffins in the ratio of 5:2. she gave 6 blueberry muffins to her brother, and now the ratio of blueberry to chocolate muffins has changed to 3:2.

Let the number of muffins before be 5x and after be 3x, then

5x-6 = 3x

2x = 6

x = 3

Therefore, the number of muffins after = 3*3 = 9

Hence, The number of  blueberry muffins she is left with is 9.

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The volume of a cylinder in terms of r is √V/πh.

Volume is defined as the space occupied within the boundaries of an object in three-dimensional space.

Given that, the volume of a cylinder = πr²h

Solving in terms of r,

r = √V/πh

Hence, The volume of a cylinder in terms of r is √V/πh.

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The value of y-coordinate of the solution of the system of equation

2x + 3y = 6 and 2x + y = -2 is y = 4.

Given, a system of equation

2x + 3y = 6 and 2x + y = -2

On subtracting both the equations, we get

3y - y = 6 - (-2)

2y = 6 + 2

2y = 8

On dividing both the sides by 2, we get

2y/2 = 8/2

y = 4

So, the value of y-coordinate of the solution is y = 4.

Hence, the value of y-coordinate of the solution of the system of equation 2x + 3y = 6 and 2x + y = -2 is y = 4.

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Answer:

By counting, we see that the point travels down 3, over 6. This is a slope of -3/6. This then simplifies down to -1/2.

The slope is -1/2

Answer:

LCM of 12 and 42 is 84. it is correct

A recipe says it produces the best summer drink. The recipe calls for 12 cup of lemonade for every 112 cups of iced tea. If 16 cups of the summer drink are made

then 8 cups of lemonade are needed.

What is unitary method?

It is a method where we find the value of a single unit from the value of multiple units and the value of multiple units from the value of a single unit.

Steps to Use Unitary Method

First, let us make a note of the information we have. There are 5 ice-creams.

5 ice-creams cost is $125.

Step 1: Let's find the cost of 1 ice cream.

In order to do that, divide the total cost of ice-creams by the total number of ice-creams.

The cost of 1 ice-cream = Total cost of ice-creams/Total number of ice-

creams = 125/5 = 25.

Therefore, the cost of 1 ice cream is $25.

Step 2: To find the cost of 3 ice-creams, multiply the cost of 1 ice cream by the number of ice-creams.

The cost of 3 ice-creams is cost of 1 ice-cream × number of ice-creams = 25 × 3 = $75.

Finally, we have the cost of 3 ice-creams i.e. $75.

lemonade: tea : total

1                 1    1+1=2

We need a total of 16 and divide it

= 16/2 = 8

Then the composition will be

lemonade:    tea:        total

1*8 = 8      1*8 = 8      2*8 = 16

Hence, 8 cups of lemonade are needed.

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Answer:

Step-by-step explanation:

(4 + 15.6a) − (7 + 3.7a) =

4 + 15.6a - 7 + 3.7a =

19.3a - 3

Answer:

[tex] \sf \: 11.9a - 3 [/tex]

Step-by-step explanation:

Now we have to,

→ simplify the given expression.

The expression is,

→ (4 + 15.6a) - (7 + 3.7a)

Let's simplify the expression,

→ (4 + 15.6a) - (7 + 3.7a)

→ 4 + 15.6a - 7 - 3.7a

→ (15.6a - 3.7a) + (4 - 7)

→ (11.9a) + (-3)

→ 11.9a - 3

Hence, the answer is 11.9a - 3.

Answer:

X= 1/2 y= -2/5. X+2y= -1/5.

The equation of the line passing through the point with coordinates (1/2, -2/5) and having slope -1/2 is:

y = -1/2x + b.

Substituting coordinates of the point in the equation, we get:

-2/5 = -1/2(1/2) + b => b = 1/5.

Therefore, the equation of the line is:

y = -1/2x + 1/5.

Step-by-step explanation:

Answer: y = -5x+2

Step-by-step explanation:

Slope = [tex]\frac{rise}{run}[/tex] or [tex]\frac{y_{2}-y_{1}}{x_{2}-x{1}}[/tex]

therefore, given those two points,

Slope = [tex]\frac{(-3)-7}{1-(-1)}[/tex]

Slope = [tex]\frac{-10}{2}[/tex]

Slope = -5

Since the m value is the slope, m = -5

In order to find the b value, just substitute the values in the equation with any of the two points given. I'll use (-1,7) but you can use the other one just fine.

let x =1, y = -3

-3=-5(1)+b

-3 = -5+b

-3+5 = b

b = 2

Plug b in the equation:

y=-5x+2

James is 24 years old.

Let John's age now is J.

Let james's age now is A.

From the question, we have

John is 3/4 the age of  james.

J=3/4*A

J=3A/4

Three years ago, John was 5/7 the age of james.

J-3=5/7*(A-3)

J-3=(5A-15)/7

J=(5A-15)/7+3

3A/4=(5A-15)/7+3

21A=4(5A-15)+84

21A=20A-60+84

A=24

James is 24 years old.

Multiplication:

Finding the product of two or more numbers in mathematics is done by multiplying the numbers. It is one of the fundamental operations in mathematics that we perform on a daily basis. Multiplication tables are the main use that is obvious. In mathematics, the repeated addition of one number in relation to another is represented by the multiplication of two numbers. These figures can be fractions, integers, whole numbers, natural numbers, etc. When m is multiplied by n, either m is added to itself 'n' times or the other way around.

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Answer:

∴ x = 32

Step-by-step explanation:

For J to be parallel to K,

(2x)° + (4x-12)° = 180°

   2x + 4x - 12 = 180

                  6x = 192

                 ∴ x = 32

The value are x=1,y=3 and z=-1 when the equation are x+5y-3z=19, 2x-3y+4z=-11 and 4x+y-2z=9.

Given that,

The equation x+5y-3z=19

2x-3y+4z=-11

4x+y-2z=9

We have to find the x,y ad z values. Its is nothing but the zeros of the polynomial.

Take the equations as,

x+5y-3z=19----->equation(1)

2x-3y+4z=-11----->equation(2)

4x+y-2z=9----->equation(3)

Subtracting equation (2) with 2×equation(1)

(x+5y-3z=19)×2⇒ 2x+10y-6z=38

                            2x-3y+4z=-11

                        ----------------------------

                            13y-10z=49----->equation(4)

Now,

Subtracting equation (3) with 2×equation(2)

(2x-3y+4z=-11)×2⇒ 4x-6y+8z=-22

                               4x+y-2z=9

                          ---------------------------
                                 -7y+10z=-31-------->equation(5)

Now,

Adding equation (4) and equation(5)

13y-10z=49

-7y+10z=-31

-----------------

6y=18

y=18/6

y=3

Now,

Substitute y=3 in equation(5)

-7(3)+10z=-31

-21+10z=-31

10z=-31+21

10z=-10

z=-10/10

z=-1

Now,

Substitute y=3 and z=-1 in equation(1)

x+5y-3z=19

x+5(3)-3(-1)=19

x+15+3=19

x+18=19

x=19-18

x=1

Therefore, The value are x=1,y=3 and z=-1 when the equation are x+5y-3z=19, 2x-3y+4z=-11 and 4x+y-2z=9.

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Answer:

The scale factor is 3

Step-by-step explanation:

Answer:

1.  x = 2

2.  x = 360

3.  x = -41 and x = 49

4.  All real numbers.

5.  x = 1

Step-by-step explanation:

[tex]\begin{aligned}&\textsf{Given}: & \dfrac{x+16}{3}&=3x\\&\textsf{Multiply both sides by 3}: & x+16&=9x\\&\textsf{Subtract $x$ from both sides}: & 16&=8x\\&\textsf{Divide both sides by 8}: & x&=2\end{aligned}[/tex]

Therefore, the solution is x = 2.

[tex]\begin{aligned}&\textsf{Given}: & x &= 0.8x + 72\\&\textsf{Subtract $0.8x$ from both sides}: & 0.2x&=72\\&\textsf{Divide both sides by 0.2}: & x&=360\end{aligned}[/tex]

Therefore, the solution is x = 360.

[tex]\begin{aligned}&\textsf{Given}: & \dfrac{1}{5}|x-4|-3&=6\\&\textsf{Add 3 to both sides}: & \dfrac{1}{5}|x-4|&=9\\&\textsf{Multiply both sides by 5}: &|x-4|&=45 \\\end{aligned}[/tex]

Set the contents of the absolute value equal to both the positive and negative value of the number on the other side of the equation, and solve both equations.

[tex]\begin{aligned}\underline{\sf Equation\; 1}&&\underline{\sf Equation\; 2}\\x-4&=45 & x-4&=-45\\x-4+4&=45+4 & \quad \quad \quad x-4+4&=-45+4\\x&=49 & x&=-41\\\end{aligned}[/tex]

Therefore, the solutions are x = -41 and x = 49.

[tex]\begin{aligned}&\textsf{Given}: & -2(x + 3) &= -2x-6\\&\textsf{Distribute}: &-2x-6&=-2x-6\\&\textsf{Add 6 to both sides}: & -2x&=-2x\\&\textsf{Divide both sides by -2}: & x&=x\end{aligned}[/tex]

Therefore, the solution is all real numbers.

[tex]\begin{aligned}&\textsf{Given}: & \dfrac{3}{2}+\dfrac{1}{2}x & = 2x\\&\textsf{Subtract $\dfrac{1}{2}x$ from both sides}: & \dfrac{3}{2}&=\dfrac{3}{2}x\\&\textsf{Multiply both sides by 2}: & 3&=3x\\&\textsf{Divide both sides by 3}: & 1&=x\end{aligned}[/tex]

Therefore, the solution is x = 1.

Answer: Answer: 1m = 60s, what is the equation?

Step-by-step explanation: 1 minute = 60 seconds so 60s = 1m

The random variable used in this situation is discrete random variable.

What is a random variable?

Suppose there is an event and the outcome of an event has to be represented by a variable. That variable is called random variable.

50 students were randomly sampled and asked questions about their exercise habits. one of the questions they were asked concerned the frequency of exercise, defined to be the number of times they exercised in a week

Since there is a notation about frequency in this question, it cannot be continuous. So it is a discrete random variable.

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According to the given coordinates, there are 4 set of points will appear on the Kelsey’s graph and the graph is attached below.

Coordinates.

Coordinates refers the locations of points on a grid known as the cartesian plane or coordinate plane.

Given,

Kelsey is going to graph the ordered pairs that are represented by this table on a coordinate plane.

Here we need to find the number of points should appear in Kelsey’s graph.

In order to find the number of points, first we have to plot the table vales on the graph.

Once we plotted the values on the graph then the graph is look like the following graph.

While we looking through the graph we have identified that the points are (-6,-4), (-4, -6), (3, -6) and (5, 0) are presented on the graph.

Therefore, there are 4 points are plotted on Kelsey’s graph.

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The value of [tex]x_4[/tex] = 5.8284.

Given:

An iterative formula:

[tex]x_{n+1} = 19/x_n + 3[/tex]

and [tex]x_1[/tex] = 9

for x_2 n = 1 substitute

[tex]x_{1+1} = 19/x_1 + 3[/tex]

[tex]x_{2} = 19/9 + 3[/tex]

= 19 + 3*9 / 9

= 19+27 / 9

= 46/9

[tex]x_3 = 19/x_2 + 3\\[/tex]

= 19/46/9 + 3

= 19*9 / 46 + 3

= 171+138 / 46

= 309/46

[tex]x_4 = 19/x_3 + 3[/tex]

= 19/309/46 + 3

= 19*46+3*309 / 309

= 1801/309

= 5.8284

Therefore The value of [tex]x_4[/tex] = 5.8284.

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Solutions for the given expression in the given interval will be {0, π/2, 3π/2}.

What are trigonometric expressions?

Trigonometric identities are, by definition, reciprocal. These formulas show the connections between the sine/cosine functions and tangents.

Refer to the attachment for the Question:

[tex]\sin^2{(\theta)}=2\sin^2{(\frac{\theta}{2})}[/tex]

[tex][2\sin{(\frac{\theta}{2})}\cos{(\frac{\theta}{2})}]^2=2\sin^2{(\frac{\theta}{2})}[/tex]

[tex][4\sin^2{(\frac{\theta}{2})}\cos^2{(\frac{\theta}{2})}]=2\sin^2{(\frac{\theta}{2})}[/tex]

[tex][4\sin^2{(\frac{\theta}{2})}\cos^2{(\frac{\theta}{2})}]-2\sin^2{(\frac{\theta}{2})}=0[/tex]

[tex]2\sin^2{(\frac{\theta}{2})}[2\cos^2{(\frac{\theta}{2})}-1]=0[/tex]

[tex]\sin{(\frac{\theta}{2})}=0[/tex]

[tex]{(\frac{\theta}{2})}=n\pi[/tex]

[tex]\theta=2n\pi[/tex] [Here, n = integer]

[tex]\theta=0,\pi,2\pi,....[/tex]

Therefore,[tex]\theta=0[/tex]  [Since,[tex]0\le \theta < 2\pi[/tex] ]

[tex][2\cos^2{(\frac{\theta}{2})}-1]=0[/tex]

[tex]\cos{\theta}=0[/tex]

[tex]x=\frac{\pi}{2}+\pi n[/tex]  [Here, n = integer]

Therefore,[tex]\theta=\frac{\pi}{2}, \frac{3\pi}{2}[/tex]  [Since, [tex]0\le \theta < 2\pi[/tex]]

Hence, solutions for the given expression will be  {0, π/2, 3π/2}.

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Answer:

The factors of 2000 are 1, 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 125, 200, 250, 400, 500, 1000, 2000

&

The multiples of 2000 are 2000, 4000, 6000, 8000, 10000, 12000, 14000, 16000, 18000, 20000, ...

Option a is correct.

$2 more is Addison’s holiday hourly rate than her regular hourly rate

What are Time and Work?

According to fundamental mathematical concepts, time may be described as the amount of time that a specific activity occurs, whereas work can be described as the collection of actions carried out in order to attain a specific activity or outcome. It should go without saying that time is needed to accomplish a task. It implies that time and work have a certain relationship.

The task completed is calculated as the sum of the number of people participating in the work and the number of days it took to finish. The fundamental time and work equation is W = N X D.

W is the amount of work done.

N is the number of people.

D is the number of days or hours needed.

Let r be the regular rate and h the holiday rate.

Then you can write these two equations:

From the first statement:

188.50 = 5h + 16r

From the second statement:

254.00 = 8h + 20r.

There you have a system of two independent equations with two variables, which you can solve by several methods.

If you multiply the first by 8 and the second by 5, you get:

40h + 128r = 1508

40h + 100r = 1270

Subtract the second equation from the first one:

28r = 238

Divide by 28

r = 238/28 = 8.5

You can use now any of the two original statements to find h

254.00 = 8h + 20r

8h = 254 -20(8.5) = 84

h = 84/8 = 10.5

h - r = 10.50 - 8.50 = 2.00

The holiday hourly rate is $2.00 more than the regular hourly rate.

Hence,

$2 more is Addison’s holiday hourly rate than her regular hourly rate

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