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Construct a scatterplot and identify the mathematical model that best fits the data. Assume that the model is to be used only for the scope of the given data and consider only linear, quadratic, logarithmic, exponential, and power models. Use a calculator or computer to obtain the regression equation of the model that best fits the data. You may need to fit several models and compare the values of R2.

The table below shows the weight of crops of oranges and the resulting gallons of juice. Using the weight as the independent variable, find the regression equation of the best model.




y = –13.07 + 0.079x


y = 264.43 e1.0001x


y = 0.079x0.998


y = –7511.9 + 923.99 ln x

Both x² and mx + b are differentiable functions of x (they are both polynomials), so if f(x) is also differentiable, we need to pay special attention at x = 2 where the two pieces of f meet.

Continuity means that the limit

[tex]\displaystyle \lim_{x\to2} f(x)[/tex]

must exist.

From the left side, we have x < 2 and f(x) = x², so

[tex]\displaystyle \lim_{x\to2^-} f(x) = \lim_{x\to2} x^2 = 4[/tex]

From the right, we have x > 2 and f(x) = mx + b, so

[tex]\displaystyle \lim_{x\to2^+} f(x) = \lim_{x\to2} (mx+b) = 4m+b[/tex]

It follows that 4m + b = 4.

Differentiability means that the limit

[tex]\displaystyle \lim_{x\to2} \frac{f(x) - f(2)}{x - 2}[/tex]

must exist.

From the left side, we again have x < 2 and f(x) = x². Then

[tex]\displaystyle \lim_{x\to2^-}\frac{f(x)-f(2)}{x-2} = \lim_{x\to2} \frac{x^2-4}{x-2} = \lim_{x\to2} (x+2) = 4[/tex]

From the right side side, we have x > 2 so f(x) = mx + b. Then

[tex]\displaystyle \lim_{x\to2^+}\frac{f(x)-f(2)}{x-2} = \lim_{x\to2} \frac{(mx+b)-(2m+b)}{x-2} = \lim_{x\to2} \frac{mx-2m}{x-2} = \lim_{x\to2}m = m[/tex]

The one-sided limits must be equal, so m = 4, and from the other constraint it follows that 16 + b = 4, or b = -12.

2sin^2xcos^2x

Using trig functions we know:

sin^2(x) = 1- cos(2x)/2

cos^2(x) = 1 + cos(2x) /2

Now we have:

2sin^2xcos^2x  =  2 * 1- cos(2x)/2 * 1 + cos(2x) /2

Simplify to: (1- cos(2x) * 1+cos(2x))/2

Difference of squares is (a-b) (a+b) = a^2 -b^2

(1- cos(2x) * 1+cos(2x))/2 = 1^2 -cos^2(x)/2 *1^2 +cos^2(x) /2

Multiply to get 1-cos(4x) /4

The answer is D.

Answer:

B) 15/91

Step-by-step explanation:

Probability of a girl as first choice is ...

number of girls / total number of students = 6/14 = 3/7

Probability of a girl as second choice (given a girl as first choice) is ...

number of girls / remaining number of students = 5/13

Then the probability of the two events is the product of their individual probabilities:

(3/7)·(5/13) = 15/91 . . . . . matches choice B

Answer:

C. (5, 3)

Step-by-step explanation:

this point is within the answer area... look below at the graph of both linear inequalities

For this case we have the following system of inequations:

[tex]y\leq2x-5\\y> -3x[/tex]

We must replace each of the points and verify that the inequalities are met:

Point A: (1,1)

[tex]1 \leq2 (1) -5\\1 \leq2-5\\1 \leq-3[/tex]

It is not fulfilled!

Point B: (-2,2)

[tex]2 \leq2 (-2) -5\\2 \leq-4-5\\2 \leq-9[/tex]

It is not fulfilled!

Point C: (5,3)

[tex]3 \leq2 (5) -5\\3 \leq10-5\\3 \leq5[/tex]

Is fulfilled!

[tex]3> -3 (5)\\3> -15[/tex]

Is fulfilled!

Point D: (-4,4)

[tex]4 \leq2 (-4) -5\\4 \leq -8-5\\4 \leq -13[/tex]

It is not fulfilled!

Thus, the point that is the solution of inequalities is:

(5,3)

ANswer:

(5,3)

Answer:

C 3

Step-by-step explanation:

5x^2 + 6x + 7 = 2(x + 2)

Distribute the 2

5x^2 + 6x + 7 = 2x + 4

Subtract 2x from each side

5x^2+6x-2x +7 = 2x-2x+4

5x^2 + 4x + 7 = 4

Subtract 4 from each side

5x^2 +4x+7-4 = 4-4

5x^2 +4x+3 = 0

This is in the form ax^2 + bx +c  where c is 3

Answer:

Step-by-step explanation:

[tex]ax^2+bx+c=0\\\\5x^2+6x+7=2(x+2)\qquad\text{use the distributive property}\\\\5x^2+6x+7=2x+4\qquad\text{subtract}\ 2x\ \text{and 4 from both sides}\\\\5x^2+4x+\boxed{3}=0[/tex]

Answer:

48 ft³

Step-by-step explanation:

The volume (V) of the pyramid is

V = [tex]\frac{1}{3}[/tex] area of base × perpendicular height (h)

Calculate h using the right triangle formed by a segment from the vertex to the midpoint of the base and across to the slant face ( the hypotenuse )

Using Pythagoras' identity on the right triangle then

h² + 3² = 5²

h² + 9 = 25 ( subtract 9 from both sides )

h² = 16 ( take the square root of both sides )

h = [tex]\sqrt{16}[/tex] = 4

area of square base = 6² = 36, hence

V = [tex]\frac{1}{3}[/tex] × 36 × 4 = 12 × 4 = 48 ft³

Answer:

She's being paid six hours per hour

Step-by-step explanation:

She's being paid six hours per hour

72/6 = 6

Answer:

Step-by-step explanation:

The goal to solving any equation is to have x = {something}.  That means we need to get the x out from underneath that radical.  It's a square root, so we can "undo" it by squaring.  Square both sides because this is an equation.  Squaring both sides gives you

[tex]x^2=-3x+40[/tex]

Get everything on one side of the equals sign and set the quadratic equal to 0:

[tex]x^2+3x-40=0[/tex]

Throw this into the quadratic formula to get that the solutions are x = 5 and -8.  We need to see if only one works, both work, or neither work in the original equation.

Does [tex]5=\sqrt{-3(5)+40}[/tex]?

[tex]5=\sqrt{-15+40}[/tex] and

[tex]5=\sqrt{25}[/tex]

and 5 = 5.  So 5 works.  Let's try -8 now:

[tex]-8=\sqrt{-3(-8)+40}[/tex] and

[tex]-8=\sqrt{24+40}[/tex] so

[tex]-8=\sqrt{64}[/tex]

-8 = 8?  No it doesn't.  So only 5 works. Your choice is the third one down.

[tex]\bf x^{-1}-1\div x-1\implies \implies \cfrac{1}{x}-1\div x-1\implies \cfrac{\frac{1}{x}-1}{~~x-1~~}\implies \cfrac{~~\frac{1-x}{x}~~}{\frac{x-1}{1}} \\\\\\ \cfrac{1-x}{x}\cdot \cfrac{1}{x-1}\implies \cfrac{-(\begin{matrix} x-1 \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix})}{x}\cdot \cfrac{1}{\begin{matrix} x-1 \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}}\implies -\cfrac{1}{x}\implies -x^{-1}[/tex]

Answer:

Where's the list?

Answer:

5 units

Step-by-step explanation:

Let's take point X and X' (it will be the same regardless of the point).

The distance is 3 units to the right then 4 units up.

A direct line will be diagonal and can be found using the pythagorean theorem (refer to visual).

(D: a > 0) 3² + 4² = a² --> 9 + 16 = a² --> a² = 25 --> a = 5

So the distance is 5 units.

When a figure is translated, or moved without rotation or change in size, the distances between corresponding points on the original and new figure remain the same. Therefore, the distance between any two corresponding points on Triangle XYZ and Triangle X'Y'Z' is the same as it was prior to the translation.

In this case, the triangle XYZ is being translated, which means that it is being shifted in the plane. The student has moved the triangle 4 units up and 3 units left. This operation does not alter the size, shape or orientation of the triangle, just its position. Therefore, the distance between any two points on the triangle remains the same before and after the translation. To clarify, if the distance between point X and Y initially is 'd', after the triangle has been translated to yield triangle X'Y'Z', the distance between X' and Y' remains 'd'. This is an important characteristics of translation, a core concept in geometry.

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Answer:

1.  0.75

2. -0.8

mark me brainlyest

First when looking at dialations, ignore the negative signs. They do not affect the expansion or contraction.

An expansion is any number greater than 1 and a contraction is any number less than 1.

Expansion: b) -2 and d) 2

Contraction: b) - 0.8 and d) 0.8

Step-by-step explanation:

0 = x² - 6x - 27

a = 1, b = -6, c = -27  Factor using AC method.

ac = (1)(-27) = -27

Factors of -27 that add up to -6 are -9 and 3.

Therefore:

0 = (x - 9) (x + 3)

x = -3, x = 9

The x-intercepts are (-3, 0) and (9, 0).

Answer:

Do Addition

Step-by-step explanation:

x=76+78

x= 154

Answer:

x = 77°

Step-by-step explanation:

The measure of x is half the sum of the measures of the arcs intercepted by the angle and it's vertical angle, that is

x = 0.5 × (78 + 76)° = 0.5 ×154° = 77°

Question 9:

cos x = 10/36

Take the inverse on both sides.

arccos(cos x) = arccos(10/36)

x = 73.8723797868

x = 73.87

Question 10:

tan 89 = x

57.2899616308 = x

57.29

Hopefully I read the question right, if not I'm sorry

Koen ran for 4 and a half hours, which is also 4.5 hours

Since he averaged a speed of 6 mph, I multiply the number of hours by speed each hour, which is 4.5*6

4.5*6=27

Koen ran 27 miles.

Answer:

38 miles

Step-by-step explanation:

study island

Check the picture below.

so the top and bottom segments are simply 7 units, we can read that off the grid.  Let's find the length of the other two segments, "c".

[tex]\bf \textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2\implies c=\sqrt{a^2+b^2} \qquad \begin{cases} c=hypotenuse\\ a=\stackrel{adjacent}{2}\\ b=\stackrel{opposite}{3}\\ \end{cases} \\\\\\ c=\sqrt{2^2+3^2}\implies c=\sqrt{13} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{perimeter of the parallelogram}}{7+7+\sqrt{13}+\sqrt{13}}\qquad \approx \qquad 21.21[/tex]

Answer:

C: 30 units

Step-by-step explanation:

on edge 2021! hope this helps!!~ d=(´▽｀)=b

Answer:

The distance across the river from Omkar to the big rock is 131343149 meters

Step-by-step explanation:

* Lets study the information in the problem

- Let Omkar position is point A

- Let Melissa position is point B

- Let big rock position is C on the other side of the river

* Now we have triangle ABC

- The distance between Omkar and Melissa is 100100100 meters

  along the river

- The angle between Omkar and the big rock is angle BAC

∴ m∠BAC = 33°

- The angle between Melissa and the big rock is angle ABC

∴ m∠ABC = 98°

- The big rock is at angle C

* Now we can find the distance between Omkar and the big rock

 by finding the length of side AC in the triangle

- By using the sine rule

∵ sin A/BC = sin B/AC = sin C/AB

∵ AB = 100100100 meters

∵ m∠ABC = 98°

- Lets find m∠C

∵ In any triangle the sum of the measures of the interior angles is 180°

∴ m∠A + m∠B + m∠C = 180°

∵ m∠A = 33° , m∠B = 98°

∴ 33° + 98° + m∠C = 180° ⇒ add

∴ 131° + m∠C = 180° ⇒ subtract 131 from both sides

∴ m∠C = 49°

- Now lets use the sine rule

∵ sin ABC/AC = sin C/AB

∴ sin 98/AC = sin 49/100100100 ⇒ by using cross multiplication

∴ AC = (sin 98 × 100100100) ÷ sin 49 = 131343148.8

∴ AC ≅ 131343149 meters

* The distance across the river from Omkar to the big rock is

 131343149 meters

Answer:

The distance across the river from Omkar to the big rock is 72 meters.

Step-by-step explanation:

Using the given information draw as triangle as shown below.

According to angle sum property, the sum of interior angles of a triangle is 180°.

In triangle ABC,

[tex]\angle A+\angle B+\angle C=180^{\circ}[/tex]

[tex]98^{\circ}+33^{\circ}+\angle C=180^{\circ}[/tex]

[tex]131^{\circ}+\angle C=180^{\circ}[/tex]

[tex]\angle C=180^{\circ}-131^{\circ}=49^{\circ}[/tex]

The measure of angle C is 49°.

Sine formula:

[tex]\frac{a}{\sin a}=\frac{b}{\sin b}=\frac{c}{\sin c}[/tex]

Using sine formula in triangle ABC, we get

[tex]\frac{AC}{\sin B}=\frac{AB}{\sin C}[/tex]

[tex]\frac{AC}{\sin 33^{\circ}}=\frac{100}{\sin 49^{\circ}}[/tex]

[tex]AC=\frac{100}{\sin 49^{\circ}}\times \sin 33^{\circ}[/tex]

[tex]AC=\frac{100}{0.7547}\cdot0.544639[/tex]

[tex]AC=72.166[/tex]

[tex]AC\approx 72[/tex]

Therefore the distance across the river from Omkar to the big rock is 72 meters.

Answer:

117

Step-by-step explanation:

182/14=13

13x9=117

Answer:

Step-by-step explanation:

Their point of intersection, assuming there is one, will be somewhere on the line y = x.  This line, y = x, is the line of symmetry between a function and its inverse.  So if the two do in fact intersect, it will be at some point on that line

Answer:

The exact value of [tex]\sin(60\degree)[/tex] is [tex]\frac{\sqrt{3} }{2}[/tex].

Step-by-step explanation:

Recall that   [tex]\sin(60\degree)[/tex]  is a special angle that can be obtained using an equilateral triangle.

The right triangle obtained using one of the lines symmetry was used to find the exact value of  [tex]\sin(60\degree)[/tex]  using SOH-CAH-TOA

The exact value of [tex]\sin(60\degree)[/tex] is [tex]\frac{\sqrt{3} }{2}[/tex].

we know that f(1)=2 so f^-1(2)=1

so B is correct!

Answer:  The correct option is

(B) [tex]f^{-1}(x)=\dfrac{1}{x-1}.[/tex]

Step-by-step explanation:  We are given to select the correct expression that is the inverse of the following function :

[tex]f(x)=\dfrac{x+1}{x}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]

Let y denotes f(x). Then,

[tex]y=f(x)~~~~~~\Rightarrow x=f^{-1}(y).[/tex]

Substituting this value in equation (i), we get

[tex]f(x)=\dfrac{x+1}{x}\\\\\\\Rightarrow y=\dfrac{f^{-1}(y)+1}{f^{-1}(y)}\\\\\\\Rightarrow yf^{-1}(y)=f^{-1}(y)+1\\\\\Rightarrow yf^{-1}(y)-f^{-1}(y)=1\\\\\Rightarrow (y-1)f^{-1}(y)=1\\\\\Rightarrow f^{-1}(y)=\dfrac{1}{y-1}\\\\\Rightarrow f^{-1}(x)=\dfrac{1}{x-1}.[/tex]

Thus, the required inverse of the given function is

[tex]f^{-1}(x)=\dfrac{1}{x-1}.[/tex]

Option (B) is CORRECT.

Answer:

c. (2, 1) and (−1, −2)

Step-by-step explanation:

Assuming you intended y=x²-3 and x-y=1, you can graph the parabola and the straight line (rewrite x-y=1 as y = x-1).

You can then see the intersections at x=-1, y=-2 and x=2, y=1.

Answer:

1)49

2)231

Step-by-step explanation:

1)

4+9=13

4*2=8

8+1=9

2)

2+3+1=6

2+1=3

1*2=2

Answer:

#1. 49 #2. 231

Step-by-step explanation:

#1. 4 x 2 + 1 =

8 + 1 =

9

add 40 + 9 = 49

#2. x + 2x + (x + 2x) = 6

6x = 6  

x = 1  

2x = 2

x + 2x = 3  

(2 x 100) + (3 x 10) + 1 or 231.

Can I get Brainliest☺☺☺☺

Step-by-step explanation:

[tex] { \sin(x) }^{2} + { \cos(x) }^{2} = 1 \\ { \cos(theta) }^{2} = \frac{9}{16} \\ \: { \sin(theta) }^{2} = \frac{6}{16} = \frac{3}{8} \\ \: \sin(theta) = \frac{ \sqrt{3} }{ \sqrt{8} } \\ \sin(theta ) = \frac{ \sqrt{24} }{8}

= \frac{ 2\sqrt{6} }{8} = \frac{ \sqrt{6} }{4}

[/tex]

Answer:

[tex]\frac{\sqrt{7} }{4}[/tex]

Step-by-step explanation:

Using the Pythagorean identity

sin²x + cos²x = 1, then

sinx = [tex]\sqrt{1-cos^2x}[/tex]

sinΘ = [tex]\sqrt{1-(3/4)^2}[/tex]

        = [tex]\sqrt{1-\frac{9}{16} }[/tex] = [tex]\sqrt{\frac{7}{16} }[/tex] = [tex]\frac{\sqrt{7} }{4}[/tex]

Answer:

6.5 Gallons

Step-by-step explanation:

288 miles / 12 gallons = 24 mpg

156 miles / 24 mpg = 6.5 gallons

Answer:0.76

0.58

Conclusion you are More likely to win by playing regular defense

Step-by-step explanation:

Probability of winning when playing regular defense is greater than probability of winning when playing prevent defense.

Probability is simply the possibility of getting an event. Or in other words, we are predicting the chance of getting an event.

The value of probability will be always in the range from 0 to 1.

Given that,

Taking the strategy of "regular" defense,

P(winning) = 42

P(losing) = 8

Total regular defense games = 50

Taking the strategy of "prevent" defense,

P(winning) = 35

P(losing) = 15

Total regular prevent games = 50

Total games = 100

Probability of winning when playing regular defense = 42 / 50 = 0.84

Probability of winning when playing prevent defense = 35 / 50 = 0.7

Hence, Probability of winning when playing regular defense is greater than probability of winning when playing prevent defense.

Learn more about Probability here :

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Answer:

A. 196 m²

Step-by-step explanation:

The two similar solids have volumes 1728 m³ and 343 m³.

The ratio of their side lengths is :

[tex]\sqrt[3]{1728}:\sqrt[3]{343}[/tex]

This simplifies to:

[tex]12:7[/tex]

If the surface area of the larger solid is  576 m², then the surface area of the smaller solid is given by:

[tex]\frac{7^2}{12^2}\times 576=196[/tex]

The correct answer is A

Answer:

slope = 5

Step-by-step explanation:

To calculate the slope m use the slope formula

m = (y₂ - y₁ ) / (x₂ - x₁ )

with (x₁, y₁ ) = (1, 3) and (x₂, y₂ ) = (2, 8) ← ordered pairs from the table

m = [tex]\frac{8-3}{2-1}[/tex] = 5

The slope of a line can be calculated using the formula (y2-y1) / (x2-x1) using any two points on the line. The result you get from this calculation is the slope of the line.

To find the slope of a line from a set of coordinates, you need two points from the line. Let's consider these two points as (x1, y1) and (x2, y2). The formula for the slope is (y2-y1) / (x2-x1). This formula shows the change in y-values divided by the change in x-values, often referred to as 'rise over run'.

If, for instance, from your table the two points are (3,4) and (5,8), substitute these values into the formula: Slope = (8-4) / (5-3) = 4/2 = 2. So the slope of the line is 2.

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Answer:

x = 29.91

Step-by-step explanation:

Cos (53) = 18/x

x = 18 / Cos(53)

x = 18/ 0.6018

x = 29.91

For this case we have that by definition of trigonometric relations of a rectangular triangle, that the cosine of an angle is given by the leg adjacent to the angle on the hypotenuse of the triangle. Then, according to the figure we have:

[tex]cos (53) = \frac {18} {x}[/tex]

Clearing x:

[tex]x = \frac {18} {cos (53)}\\x = \frac {18} {0.60181502}\\x = 29,9095226969[/tex]

Rounding:

[tex]x = 29.91[/tex]

Answer:

29.91