When choosing a sample size for a population proportion a practitioner is considering either setting p = 0.5, or, using a preliminary estimate of p-hat=0.7. Which is true? a. Using the preliminary estimate of p-hat -0.7 yields a larger sample size than when p0.5 is used. b. Using the value 0.3 (the complement of p-hat in this case) yields a larger sample than when p0.5 is used. c. The sample size is maximized (thus making it a more conservative approach) with the value of p vs. the value of p-hat. d. Both the preliminary estimate of phat -0.7 and p 0.5 yield the same sample size.

The smallest possible value of f ( t ) = 9 based on the values of p , q , r , s.

Given :

Let f ( x ) be a polynomial with integer coefficients. Suppose there are four distinct integers p , q , r , s such that f ( p ) = f ( q ) = f ( r ) =f ( s ) = 5. If t is an integer and f ( t ) > 5,

Let g(x) = f(x) − 5.

g(x) = (x−p)(x−q)(x−r)(x−s)h(x)

The condition f(t) > 5 translates to g(t) > 0.

Since p,q,r,s,t are distinct integers, the smallest possible positive value of (t−p)(t−q)(t−r)(t−s) is 4 :

the four numbers in the parentheses are all distinct integers ≠ 0, so the smallest value we can get from the product (−2)⋅(−1)⋅1⋅2. }

The smallest possible positive value of h(t) is 1, since we must have g(t)≠0.

Thus the smallest possible value of g(t) is 4, and therefore the smallest possible value of f(t) is 9, and it is achieved for t=2 if we have

f(x)=x(x−1)(x−3)(x−4)+5

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Full question ;

Let f(x) be a polynomial with integer coefficients. Suppose there are four distinct integers p,q,r,s such that f(p)=f(q)=f(r)=f(s)=5. If t is an integer and f(t)>5, what is the smallest possible value of f(t)?

By using matrix, it can be calculated that:

[tex]\frac{1}{4}C = \begin{bmatrix}3 & 4 & -5 \\ 1 & -6 & 7\end{bmatrix}[/tex]

The term "matrix" refers to any configuration of numbers in the form of rows and columns. a collection of numbers lined up in rows and columns to form a rectangular array is called a matrix. The elements, or entries, of the matrix are the integers.

In addition to numerous mathematical disciplines, matrices find extensive use in the fields of engineering, physics, economics, and statistics. Solving linear equations is made easier by it. Matrices are incredibly priceless items that are used in a variety of contexts. In addition to mathematical applications, matrices are employed in a wide range of scientific disciplines. Nearly every element of our life uses engineering mathematics.

Here,

[tex]C = \begin{bmatrix} 12 & 16 & -20 \\ 4 & -24 & 28 \end{bmatrix}[/tex]

[tex]\frac{1}{4}C = \begin{bmatrix} 12\times \frac{1}{4} & 16 \times \frac{1}{4} & -20 \times \frac{1}{4} \\ 4 \times \frac{1}{4} & -24 \times \frac{1}{4}& 28 \times \frac{1}{4}\end{bmatrix}\\\\\frac{1}{4}C = \begin{bmatrix}3 & 4 & -5 \\ 1 & -6 & 7\end{bmatrix}[/tex]

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Answer:

x=38

Step-by-step explanation:

linear par

180-134=46

180-84=96

sum of a triangle is 180

96+46+x=180

142+x=180

x=180-142

x=38

percentage rate increase by 206%

What is exponential growth?

An exponential function's curve is created by a pattern of data called exponential growth, which exhibits higher increases with time.

Consider a population of mice that increases exponentially every year by a factor of two, starting with 2 in the first year and increasing to 4 in the second, 8 in the third, 16 in the fourth, and so on. The population is rising by a factor of 2 per year in this example. If mice gave birth to four pups instead, you would then have 4, 16, 64, and 256.

Linear growth, which is additive, and geometric growth can be contrasted with exponential growth, which is multiplicative (which is raised to a power).

This equation represents exponential growth because the base is greater than 1, the function represents growth and  Whenever the base is less than 1, the function represents decay.

The base 1.06 is greater than 1, hence The equation represents exponential growth.

The formula for exponential growth:

[tex]y=a(1+r)^{x}[/tex]

where,

f(x) = exponential growth function

a = initial amount

r = growth rate

x = number of time intervals

In this case, [tex]r=1.06+1 = 2.06[/tex]

which represents percentage rate increase by 206%

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In order for m n to be even, m must be even or n must be an even number.

Let's understood what is integers.

In mathematics, a collection of both positive and negative integers is referred to as an integer. Like whole numbers, integers do not contain a fractional portion. The definition of an integer is that a number that can be either positive, negative, or zero but is not a fraction. On integers, we can carry out all arithmetic operations, including addition, subtraction, multiplication, and division. Examples of integers include 1, 2, 5, 8, -9, and -12. "Z" stands for an integer.

Proof by contraposition:

Suppose that the statement m“ is even or is even n” is not true. Consequently, m and n should both be odd. Let's check to determine if the sum of two odd numbers is even or odd: Let m and n be equal to 2a+1 and 2b+1 respectively, then their product is:

(2a+1) (2b+1) = 4ab+2a+2b+1 =2(2ab+a+b) +1

This shows that the expression 2(2ab+a+b) +1 is of the form 2n+1, thus the product is odd. If the product of odd numbers is odd, then m n is not true to be even. Therefore, in order for m n to be even, m must be even, or n must be an even number.

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Answer:

$22.50

Step-by-step explanation:

Answer:

Step-by-step explanation:

[tex]-x+y\leq -1\\x-y\geq 1\\x+2y\geq 4[/tex]

dark blue is the required region.

The stopping distance table and the analysis of the stopping distance are presented as follows;

1. The stopping distance calculator, an online tool, provides the stopping distance at a specified speed as presented in the table on the following sections.

2. Increasing the vehicle speed by 10 mph, increases the required stopping distance exponentially, which reduces the safety of driving

The stopping distance is the distance traveled by a vehicle, which is the sum of the distance traveled during the reaction time and the distance traveled during braking (the braking distance)

The table in the question using a perception–reaction time of 2.5 seconds, and the online stopping distance calculator is completed as follows;

Speed (mph)       [tex]{}[/tex] Speed (m/s)        Stopping Distance (m)

60 mph [tex]{}[/tex]                 26.82 m/s           119.55 m

50 mph  [tex]{}[/tex]                22.35 m/s           92.34 m

40 mph  [tex]{}[/tex]                17.88 m/s            68.05 m

30 mph   [tex]{}[/tex]               13.41 m/s             46.665 m

20 mph [tex]{}[/tex]                 8.94 m/s             28.197 m

10 mph [tex]{}[/tex]                  4.47 m/s             12.642 m

2. The details from the above table indicates that as the speed increases, the stopping distance increases exponentially, such that increasing the speed by 10 mph increases the required stopping distance when the vehicle is moving at an already high speed, thereby reducing safety by increasing the speed by 10 mph.

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Answer:

[tex]\dfrac{1}{36n^2+6n}[/tex]

Step-by-step explanation:

Given factorial expression:

[tex]\dfrac{(6n-1)!}{(6n+1)!}[/tex]

[tex]\boxed{\begin{minipage}{6cm}\underline{Factorial Rule}\\\\$n!=\:n\cdot \left(n-1\right) \cdot \left(n-2\right) \cdot ... \cdot 3 \cdot 2\cdot 1$\\ \end{minipage}}[/tex]

Apply the factorial rule to the numerator and denominator of the given rational factorial expression:

[tex](6n-1)!=\left(6n-1\right)\cdot \left(6n-2\right)\cdot \left(6n-3\right)\cdot... \cdot 3 \cdot 2\cdot 1[/tex]

[tex]\left(6n+1\right)!=\left(6n+1\right)\cdot \:6n \cdot (6n-1) \cdot...\cdot 3 \cdot 2\cdot 1[/tex]

Therefore:

[tex]\begin{aligned}\implies \dfrac{(6n-1)!}{(6n+1)!}&=\dfrac{\left(6n-1\right)\cdot \left(6n-2\right)\cdot \left(6n-3\right)\cdot... \cdot 3 \cdot 2\cdot 1}{\left(6n+1\right)\cdot \:6n \cdot (6n-1) \cdot...\cdot 3 \cdot 2\cdot 1}\\\\&=\dfrac{1}{(6n+1) \cdot 6n}\\\\&=\dfrac{1}{6n(6n+1)}\\\\&=\dfrac{1}{36n^2+6n}\end{aligned}[/tex]

Answer:

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

We know that:

Therefore:

Therefore:

Answer:

The Rate of return would then be 192 / 273 ≈ 70.32%

The difference of the given polynomials 8x³+5x+6 and 2x²+3x is 8x³-2x²+2x+6. So, the correct answer is D.

To subtract polynomials from another, we should change the signs (from '+' to '-' or from '-' to '+') of all the terms of the expression which is to be subtracted and then the two expressions are added.

Given that, 8x³+5x+6-(2x²+3x)

Group the like terms are perform the addition or subtraction

= 8x³+5x+6-2x²-3x

= 8x³+(5x-3x)-2x²+6 (Here, like terms are 5x and 3x)

= 8x³+2x-2x²+6

= 8x³-2x²+2x+6

So, the standard form of obtained polynomial is 8x³-2x²+2x+6.

The polynomials difference is 8x³-2x²+2x+6. Therefore, option D is the correct answer.

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"Your question is incomplete, probably the complete question/missing part is:"

Which polynomial represents the difference below?

8x³ + 5x+6-(2x² + 3x)

A. 10x+ 8x² +6

B. -2x² + 8x³ + 8x+6

C. 6x¹⁰+2x+6

D. -2x²+ 8x³+2x+6

The value of the equation h(x) = -5x-4 is - 19 when h = (3).

An equation is a mathematical statement that contains the symbol "equal to" between two expressions with identical values.

As in 3x + 5 = 15, for example.

There are many different types of equations, including linear, quadratic, cubic, and others.

The three primary forms of linear equations are point-slope, standard, and slope-intercept.

So, the equation we have is:

h(x) = -5x-4

Now, solve the equation when h = (3)

h(x) = -5x-4

h(3) = -5(3) -4

h(3) = -15 - 4

h(3) = - 19

Therefore, the value of the equation h(x) = -5x-4 is - 19 when h = (3).

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Answer:

a = 4

Step-by-step explanation:

Step 1: Find two numbers that multiply to give 16 and add to give -8.

t²-8t+16 = 0

t² -4t -4t + 16 = 0

The two numbers are 4 and -4.

Step 2: Rewrite the equation in the form (t + 4)(t - 4).

t² - 8t + 16 = (t + 4)(t - 4)

Step 3: Factor the equation to get (t + 4)².

(t + 4)² = (t + 4)(t + 4)

Therefore, a = 4.

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Answer:

A.[tex] \frac{7}{6} \sqrt{x} [/tex]

Step-by-step explanation:

Solution Given:

[tex] \sqrt{98{x}^{3} } \div \sqrt{72 {x}^{2} } [/tex]

Bye using indices formula

[tex] \sqrt{x} \div \sqrt{y} = \sqrt{ \frac{x}{y} } [/tex]

we get

[tex] \sqrt{ \frac{98{x}^{3} }{72 {x}^{2} } } [/tex]

[tex] \sqrt{ \frac{49 {x}^{3} }{ 36 {x}^{2} } }[/tex]

[tex] \sqrt{ \frac{{7}^{2} {x}^{3 - 2} }{{6}^{2} } } [/tex]

[tex] \frac{7}{6} \sqrt{x} [/tex]

The measures of the lengths of the sides and angles of the triangles found using the law of cosines and the law of sines are presented as follows;

Question 1

∠A = 47.35°

∠B = 78.65°

c = 3.3

Question  2

∠B = 78.24°

∠C = 29.76°

a = 67.03

Question 3

∠A = 45.77°

∠B = 86.417°

∠C = 47.813°

Question 4

∠A = 36.15°

∠B = 60.48°

∠C = 93.37°

Question 5

a = 258.98

b = 327.41

∠C = 53°

Question 6

a = 34.11

b = 73.987

∠C = 53°

The law of cosines is a relationship between two sides (b and c) and the included angle, (∠A) and the third side (a) of the triangle.

Mathematically; a² = b² + c² - 2·b·c·cos(A)

Question 1

The dimensions of the triangle ΔABC are;

a = 3.0, b = 4.0, ∠C = 54°

The law of cosines indicates that we get;

c² = b² + a² - 2·b·a·cos(∠C)

Therefore;

c² = 3.0² + 4.0² - 2 × 3.0 × 4.0 × cos(54°) ≈ 10.893

The law of sines indicates that we get;

sin(54°)/3.3 = sin(∠A)/3.0

Question 2

b = 69, c = 35, ∠A = 72°

a² = 69² + 35² - 2 × 69 × 35 × cos(72°) ≈ 4493.45

The law of sines indicates that we get;

sin(72°)/67.03 = sin(∠B)/69

∠B = arcsine(69 × sin(72°)/67.03) ≈ 78·24°

∠C = 180° - 72° - 78.24° ≈ 29.76°

Question 3

a = 28, b = 39, c = 29

a² = b² + c² - 2·b·c·cos(A)

cos(A) = (a² - (b² + c²)) ÷ (2·b·c)

Therefore; cos(A) = (28² - (39² + 29²)) ÷ (-2 × 39 × 29) ≈ 0.6976

sin(45.77)/28 = sin(B)/39

sin(B) = 39 × sin(45.77)/28 ≈ 0.998

Question 4

a = 13, b = 17, c = 22

cos(A) = (13² - (17² + 22²)) ÷ (-2 × 17 × 22) ≈ 0.807

sin(36.15)°/13 = sin(∠B)/17

sin(∠B) = 17 × sin(36.15)°/13

Question 5

The parameters of the triangle are; ∠A = 50°, ∠B = 77°, c = 270

Please find attached the sketch of the triangle in the correct option created with MS Word

a/sin(50°) = 270/sin(53°)

a = sin(50°) × 270/sin(53°) ≈ 258.98

b = sin(77°) × 270/sin(53°) ≈ 329.41

Question 6

The parameters of the triangle are;

∠A = 27°, ∠B = 100°, c = 60

Please find attached the drawing of the correct triangle

∠C = 180° - 27° - 100° = 53°

∠C = 53°

60/sin(53°) = a/sin(27°)

a = sin(27°) × 60/sin(53°) ≈ 34.11

b = sin(100°) × 60/sin(53°) ≈ 73.987

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The lines that carry the rectangle onto itself are x = 0 and y = 1

The graph that completes the question is added as an attachment

From the question, we have the following parameters that can be used in our computation:

The rectangular graph

The coordinates of one end of the graph are

(-3, 3) and (-3, -1)

Next, we calculate the midpoint of these ends

So, we have

Midpoint = 1/2(x₁ + x₂, y₁ + y₂)

Substitute the known values in the above equation, so, we have the following representation

Midpoint = 1/2(-3 + 3, -1 + 3)

Evaluate the like terms

Midpoint = 1/2(0, 2)

So, we have

Midpoint = (0, 1)

So, we have

x = 0 and y = 1

Hence, the reflection lines are x = 0 and y = 1

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Verified that during the first 3 seconds of fall the instantaneous velocity equals the average velocity, the time is 1.5 seconds

The height of an object t seconds after it is dropped from a height of 300 meters is

s(t) = -4.9t^2 + 300

The average velocity of object during first 3 seconds

= s(3) - s(0) / 3 - 0

s(3) = -4.9 × (3)^2 + 300

= -44.1 + 300

= 255.9 meters

s(0) = -4.9(0)^2 + 300

s(0) = 300

The average velocity = (255.9 - 300) / 3 - 0

= -44.1 / 3

= -14.7 meter per second

To find  instantaneous velocity differentiate the function

= -9.8t

The instantaneous velocity = The average velocity

-9.8t = -14.7

t = -14.7/-9.8

t = 1.5 seconds

Therefore, time is 1.5 seconds

I have answered the question in general, as the given question is incomplete

The complete question is :

The height of an object t seconds after it is dropped from a height of 300 meters is s(t) = -4.9t^2 + 300.

(b) Use the Mean Value Theorem to verify that at some time during the first 3 seconds of fall, the instantaneous velocity equals the average velocity. Find that time.

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The equation of given graph is y = 2x + 6.

What is equation of line?

The formula for a straight line is y = mx + c where c is the height at which the line intersects the y-axis, also known as the y-intercept, and m is the gradient.

Given:

The graph of the line is given.

From graph we have to find the equation of line.

Let the graph passes through the points (0, 6) and (2, 10).

From these two points to find the slope.

Slope = [tex]\frac{y_2 - y_1}{x_2 - x_1}[/tex]

Here, [tex](x_1, y_1) = (0, 6), (x_2, y_2) = (2, 10)[/tex]

⇒ Slope = m = [tex]\frac{10-6}{2-0}= \frac{4}{2} = 2[/tex]

So, the slope is 2.

Now to find the equation of line.

Consider, the point - slope form of the line,

[tex]y-y_1=m(x-x_1)[/tex]

Plug [tex]m = 2, (x_1, y_1) = (0, 6)[/tex]

⇒

[tex]y-6=2(x-0)\\y-6=2x\\y=2x+6[/tex]

Hence, the equation of given graph is y = 2x + 6.

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The volume of each brick is 0.75m³

Volume is defined as the space occupied within the boundaries of an object in three-dimensional space. It is also known as the capacity of the object. It is measured in cubic units.

A brick has a cuboidal shape. This means the volume of a cuboid is given as:

V = l×b×h

This means volume = area × height

The volume of the whole wall is calculated as:

15× 6× 20 = 18000m³

Since there are 24000 bricks needed for the wall. The volume of each brick is therefore calculated as:

V =18000/24000

= 0.75m³

Therefore the volume of each brick is 0.75m³

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Resulting box has the greatest volume for the values  (25 ± 5√7)/6 .

This is a problem that can be solved using derivatives , maxima & minima and common logic.

Hence , going by logic :

Creating a flap of 'a' inches in width, the base of the box will be

 (10 - 2a) by (15 - 2a)

and the depth of the box will be the width of the fold-up flap: a.

Then the volume of the box is

 v = [tex]a(10 -2a)(15 -2a) = 150a -50a^2 +4a^3[/tex]

Using the derivative of the volume will be zero at the maximum volume.

 0 = [tex]dv/da = 150 -100a +12a^2[/tex]

This has roots at

 a = (100 ±√(100² - 4(12)(150)))/(2·12)

 a = (100 ± √2800)/24 = (25 ± 5√7)/6

Only the smaller of these solutions gives a maximum volume.

You should cut (5/6)(5-√7) ≈ 1.962 inches to obtain the greatest volume.

Similarly , replacing the values of 10 by A and 15 by B , a generalized solution can be formed .

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Answer:

non-violence and kindness with animals

Step-by-step explanation:

Answer:

75

Step-by-step explanation:

I hour and 15 minutes in minutes is 75 minutes

1 hour = 60 minutes

you add 15 and you get

60+15=75

The answer would be 75 Minutes.

1 Hour = 60 Minutes

[tex]60 + 15 = 75[/tex] Minutes.

The value of width will be;

⇒ 18.5 cm

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

Given that;

The volume of a rectangular prism = 6,618.375 cm³

The height is 13.25 cm and the length is 27 cm.

Now,

We know that,

The volume of rectangular prism = Length x Width x Height

Substitute all the values, we get;

⇒ 6,618.375 = 27 × x × 13.25

⇒ 6,618.375 / 357.75 = x

⇒ x = 18.5 cm

Thus, The value of width = 18.5 cm

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a) The number alive after 7 years is given as follows: 48.

b) The percentage of herd that dies each year is of 10%.

The exponential function in the context of this problem is defined as follows:

N(t)=100(0.9)^t.

The parameters of the function are defined as follows:

The amount of herd alive after 7 years is found with the numeric value at t = 7, replacing the lone instance of t in the function by 7, hence:

N(7) = 100 x (0.9)^7 = 48.

(rounding to the nearest whole number).

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(a) C(p) is not defined since we can't divide by 0.

(b) The domain of C(p) as given by the equation is (-∞, 100) ∪ (100, ∞)

(c) The domain of C(p) in the context of the application is [0, 100)

(d) The percentage of particulate pollution removed from an industrial plant can't be negative or higher or equal than 100

Here we have given that Suppose that the cost C (in dollars) of removing p percent of the particulate pollution from the smokestacks of an industrial plant is given by

C(p) = 8100p / (100 − p)

And we need to find the following.

(a) In general ma the we know that dividing by zero is not possible one.

Therefore, C(p) is undefined when we try to divide it by 0.

(b) The domain of C(p) is the function is not defined at p =100 the domain would be written as,

=> D = (-∞, 100) ∪ (100, ∞)

(c) The domain of C(p) in the context of the application  is calculated as,

=> D = [0, 100)

Therefore, the percentage of particulate pollution removed from an industrial plant can't be negative or higher or equal than 100.

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A)

A manufacturer of banana chips would like to know whether its bag-filling machine works correctly at the 414-gram setting.

So, Null hypothesis: [tex]H_{0}[/tex] : μ < 414

It is believed that the machine is underfilling the bags.

So, Alternate hypothesis: [tex]H_{1}[/tex] : μ < 414

Given,

n= 8

Population standard deviation (б) = 18

x= 407

We will use the t-test since n > 8 and we are given the population standard deviation.

t=x-μ / (б/[tex]\sqrt{n-1}[/tex])

t= [tex]\frac{407-414}{\frac{18}{\sqrt{7} } }[/tex]

t= -1.028

Use the t table to find p value

p-value = 12.706

Level of significance α = 0.025

p-value>α

It is a two-tailed test.

So, we fail to reject the null hypothesis.

So, its bag-filling machine works correctly at the 414-gram setting.

B)

Let μ be the population mean amount of ozone in the upper atmosphere.

As per the given, we have

[tex]H_{0}[/tex]    : μ = 4.8

[tex]H_{1}[/tex] : μ ≠ 4.8

Sample size: n= 26

Sample mean = 4.6

Standard deviation = 1.2

Since population standard deviation is now given, so we use a t-test.

t= [tex]\frac{4.6-4.8}{\frac{1.2}{\sqrt{25} } }[/tex]

t= -0.2/0.24

t= -0.833

It is a two-tailed test.

We are accepting the null hypothesis.

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Answer:

Step-by-step explanation:

3.14 x 3=9.42

a. [tex]\frac{dy}{dx} \ at \ x=0 \ is \ c.[/tex]

b.  [tex]\frac{dy}{dx} \at x=-L \ is \ 3aL^2+2bL+c.[/tex]

a. The derivative of a cubic function

[tex]y=ax^3+bx^2+cx+d[/tex]  is  [tex]y'=3ax^2+2bx+c[/tex].

Plugging in x=0, we get y'=c. Thus, [tex]\frac{dy}{dx}[/tex] at x=0 is c.

b. Plugging in x=-L, we get [tex]y'=3a(-L)^2+2b(-L)+c[/tex].

Thus, [tex]\frac{dy}{dx}[/tex] at [tex]x=-L \ is\ 3a(-L)^2+2b(-L)+c.[/tex]

A derivative is a financial instrument that derives its value from an underlying asset. It is a contract between two or more parties that specifies conditions (such as the date, price, and quantity of the underlying asset) under which payments, or payoffs, are to be made between the parties. Derivatives can be used for a variety of purposes, such as hedging risk or speculating on the future price of an asset.

A function is a mathematical relation between two sets of numbers that assigns each element in one set to exactly one element in the other set. For example, the function f(x) = 2x+3 assigns each real number x to the real number 2x+3

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Answer:

x = 15°

Step-by-step explanation:

⭐the exterior angle of a triangle is equal to the sum of the angles that is opposite of said exterior angle

Thus, to solve for x, we have to make an equation in regards of the exterior angle.

132° = x° + 117°

∴ 15 = x° . . . .  subtract 117 from LHS and RHS

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