How can I derive the first equation to get the second equation?

Answer:

6:40

Step-by-step explanation:

(a) Calculate the standard error of the sample proportion.

(b) Determine the z-score for the sample proportion.

(c) Find the p-value for the given z-score.

To find the standard error, z-score, and P-value for the given test, follow these steps:

(a) Calculate the standard error using the formula:

[tex]\[ \text{Standard error} = \sqrt{\frac{p \times (1 - p)}{n}} \][/tex]

Given [tex]\( p = 0.50 \) (the population proportion) and \( n = 100 \) (the sample size), substitute these values into the formula:[/tex]

[tex]\[ \text{Standard error} = \sqrt{\frac{0.50 \times (1 - 0.50)}{100}} \]\[ = \sqrt{\frac{0.50 \times 0.50}{100}} \]\[ = \sqrt{\frac{0.25}{100}} \]\[ = \sqrt{0.0025} \]\[ = 0.05 \][/tex]

(b) Calculate the z-score using the formula:

[tex]\[ \text{z-score} = \frac{\text{sample proportion} - \text{population proportion}}{\text{standard error}} \]Given that the sample proportion is \( 0.47 \), substitute this value along with the previously calculated standard error into the formula:\[ \text{z-score} = \frac{0.47 - 0.50}{0.05} \]\[ = \frac{-0.03}{0.05} \]\[ = -0.6 \][/tex]

(c) Finally, find the P-value using a z-table or statistical software corresponding to the calculated z-score. The P-value represents the probability of observing a sample proportion at least as extreme as the one obtained, assuming the null hypothesis is true.

Final answer:

The marketing firm's survey, where a sample of people is asked if they use a product, is an example of survey research aimed at observing and measuring characteristics without attempting modification. This cross-sectional study helps understand consumer behavior. (Option C)

Explanation:

The survey described by the marketing firm where one hundred people are contacted and fifteen confirm that they use the product is an example of survey research, which is a quantitative research method. In this survey, the firm is observing and measuring specific characteristics of the subjects, but not attempting to modify the subjects. This type of survey can help the firm in understanding consumers' preferences, tastes, attitudes, and behaviors regarding the use of the product. They are not applying any treatment to the subjects or observing them over time as would be done in a longitudinal study. Instead, they're conducting a cross-sectional survey at a single point in time. (Option C)

The shape and the circumference of the base created by the rotation is; Cone and Circumference of 63 mm

We are given;

Each side of equilateral triangle = 20 mm

Now, when the triangle is rotated about the line ‘a’ which passes through the midpoint of any of the 3 sides of the equilateral triangle we will get a conical shape.

The distance from the line a which cuts one side of a triangle to one vertex of the triangle is the radius = 10 mm

We know that;

Circumference of a circle = 2πr

Thus;

C = 2 × 3.14 × 10

C = 6.28 × 10

C = 62.8 mm

C = 63 mm

Thus, the approximate circumference of the base is 63 mm.

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Answer:

9

Step-by-step explanation:

The quadratic equation 3x² − 15x + 20 = 0 has a radicand of -15, indicating no real solutions. The equation 3x² + 5x − 8 = 0 can be solved using the quadratic formula, yielding solutions of x = 1 and x = -8/3.

For the quadratic equation 3x² − 15x + 20 = 0, the radicand can be found as part of the quadratic formula process, which is b^2 - 4ac. Here, a=3, b= -15, and c=20. Substituting these values in, we get the radicand as (-15)² - 4(3)(20) = 225 - 240 = -15. Since the radicand is negative, this indicates that the equation has no real solutions; the solutions are complex numbers.

To solve the equation 3x² + 5x − 8 = 0, we will use the quadratic formula, x = −b ± √(b^2 - 4ac) / (2a), since we have a quadratic with a, b, and c all non-zero. Substituting a=3, b=5, and c= -8, we find the radicand to be (5)² - 4(3)(-8) = 25 + 96 = 121. Calculating further, x = (-5 ± √121) / 6, which simplifies to x = (-5 ± 11) / 6. Thus, we have two solutions: x = (11 - 5) / 6 = 1 and x = (-5 - 11) / 6 = -16/6 = -8/3.

To completely factorize [tex]\(x^2 - 3x + c\),[/tex]the correct constant term is 10. With this, the expression becomes [tex]\((x - 2)(x - 5)\),[/tex] achieving complete factorization.

To find the correct constant term that would allow for complete factorization of the quadratic expression [tex]\(x^2 - 3x + c\),[/tex]  let's consider what it means to factorize a quadratic expression:

A quadratic expression can be factored if it can be represented in the form [tex]\((x - a)(x - b)\),[/tex] where a and b are the roots of the expression. Given the original expression [tex]\(x^2 - 3x + c\),[/tex] we can expand [tex]\((x - a)(x - b)\)[/tex] and then compare coefficients to determine the constant term c.

Expanding [tex]\((x - a)(x - b)\)[/tex] :

  - [tex]\((x - a)(x - b) = x^2 - (a + b)x + a \cdot b\).[/tex]

Comparing Coefficients :

  - By comparing with [tex]\(x^2 - 3x + c\),[/tex] we can identify that [tex]\(a + b = 3\)[/tex]  (the coefficient for [tex]\(x\)[/tex]  and [tex]\(a \cdot b = c\)[/tex] (the constant term).

  - Given \(a + b = 3\), let's find possible values for a and b that would yield a correct factorization:

    - Consider [tex]\(a = 1\), \(b = 2\):[/tex] Then [tex]\(a \cdot b = 1 \times 2 = 2\),[/tex] which is different from the given constant \(c\).

    - Consider [tex]\(a = -2\), \(b = -5\):[/tex]  Then [tex]\(a \cdot b = -2 \times -5 = 10\),[/tex] suggesting that [tex]\(c = 10.[/tex]

    - Consider [tex]\(a = 5\), \(b = 2\):[/tex] Then [tex]\(a \cdot b = 5 \times 2 = 10\),[/tex] also suggesting [tex]\(c = 10\).[/tex]

Considering this process, the correct answer that would allow for complete factorization of the given expression [tex]\(x^2 - 3x + c\)[/tex] is 10:

Thus, the factorization of [tex]\(x^2 - 3x + 10\)[/tex] results in [tex]\((x - 2)(x - 5)\),[/tex] suggesting that the constant term in question should be 10.  

The complete question is : Which constant term in the expression [tex]\(x^2 - 3x + c\)[/tex]  would allow it to be completely factored? Consider the possible values of c and determine which would result in a fully factored form. Options: -10, 0, 10.

The correct constant term that completes the factoring of the expression [tex]\( x^2 - 3x + \ ? \)[/tex]  is 10.

To find the constant term that completes the factoring of the quadratic expression [tex]\( x^2 - 3x + \ ? \)[/tex], we can follow these steps:

Understand that when factoring a quadratic expression of the form [tex]\( ax^2 + bx + c \)[/tex], we are looking for two numbers that multiply to ( ac ) and add to ( b ).

In our case, ( a = 1 ), ( b = -3 ), and ( c ) is the constant term we're looking for.

Since ( a ) is 1, we need to find two numbers that multiply to ( c ) and add up to ( -3 ).

These two numbers are the factors of ( c ).

Given that the constant term ( c ) is the term that doesn't include ( x ), it will be the product of these two factors.

To find the constant term, we can factorize the expression ( ac ), where ( a = 1 ) and ( c ) is the constant term.

Once we find the factors of ( c ), we can test different values until we find the correct one that makes the expression factorable.

Let's start by factoring ( ac ):

Since ( a = 1 ), and ( b = -3 ), we have ( ac = c ).

Given that the product of the factors should be ( c ), and the factors should add up to ( -3 ), we can find the factors of ( c ) by trial and error.

Let's try different values of ( c ) and see which one works:

If ( c = 10 ), the factors of ( c ) would be 1 and 10. However, 1 + 10 = 11, not -3.

If ( c = -10 ), the factors of ( c ) would be -1 and 10. However, -1 + 10 = 9, not -3.

If ( c = -10 ), the factors of ( c ) would be 1 and -10. However, 1 + (-10) = -9, not -3.

If ( c = 10 ), the factors of ( c ) would be -1 and -10. However, -1 + (-10) = -11, not -3.

If ( c = 0 ), the factors of ( c ) would be 0 and 0. However, 0 + 0 = 0, not -3.

Based on these trials, we see that none of the values satisfy the condition of adding up to -3.

Therefore, the correct constant term that completes the factoring of the expression [tex]\( x^2 - 3x + \ ? \)[/tex]  is 10.

The student initially had 42 ounces of rice, bought 58 more for a total of 100 ounces. These were divided into 10 portions, so each portion will contain 10 ounces of rice.

This problem is a basic arithmetic question. The student starts with 42 ounces of rice and then adds 58 more ounces, for a total of 100 ounces. She then divides these 100 ounces into 10 equal portions. The key here is to understand the concept of division, which basically means splitting up a total amount (100 ounces) evenly into a certain number of parts (10 portions). To do this, simply use the operation of division: 100 divided by 10 equals 10. Thus, each portion will contain 10 ounces of rice.

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There is a unique solution for a triangle with A=30 degrees, a=20 units, and b=16 units. By using the Law of Sines, sin(B) is computed as 0.4. Since there is no obtuse angle with the same sine, there is only one possible angle B, leading to one possible triangle.

To determine the number of possible solutions for a triangle with A=30 degrees, a=20 units (side opposite angle A), and b=16 units (another side), we need to apply the Law of Sines and explore the possible cases. According to the Law of Sines:

a/sin(A) = b/sin(B)

For the given values, we have:

20/sin(30 degrees) = 16/sin(B)

Calculating sin(B) gives us:

sin(B) = 16 * sin(30 degrees) / 20 = 8/20 = 0.4

Now, if sin(B) is less than 1, which it is in our case, there are two possible scenarios:

B is acute: There will be one solution for B, meaning B could be angle whose sine is 0.4.

B is obtuse: We must also check if there is a possible obtuse angle that also has a sine of 0.4. However, since the sine function is positive and less than or equal to 1 for angles between 0 and 180 degrees and it's symmetric with respect to 90 degrees, there can't be an obtuse angle with the same sine value as an acute angle.

Therefore, we only have one possible angle B, which implies we have a unique solution for the triangle.

Additionally, we should check whether side b is larger than the altitude from A; otherwise, there would be no solution for the triangle. To do this, we can use the extended Law of Sines to calculate the diameter (D) of the triangle's circumcircle:

D = a / sin(A)

And thus, the altitude (h) from A would be:

h = D * sin(B)

If b > h, we have a valid triangle and a unique solution.

There is exactly one possible solution for a triangle with the given side lengths a = 20, b = 16, and angle A = 30°.

To determine the number of possible solutions for a triangle given the side lengths a, b, and the angle A, we can use the Law of Sines. The Law of Sines states:

[tex]\[\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\][/tex]

where:

a, b, c - side lengths of the triangle

A, B, C - angles opposite to the respective sides

Given:

A = 30°

a = 20

b = 16

[tex]\[\frac{a}{\sin(A)} = \frac{b}{\sin(B)}\][/tex]

[tex]\[\frac{20}{\sin(30^\circ)} = \frac{16}{\sin(B)}\][/tex]

sin (B) = [tex]\frac{16 \times \sin(30^\circ)}{20}[/tex]

         = [tex]\frac{16 \times 0.5}{20}[/tex]sin

         = [tex]\frac{8}{20}[/tex]

         = 0.4

To find the angle B, we take the inverse sine:

B = [tex]\sin^{-1}(0.4)[/tex]

B = 23.58°

C = 180° - A - B

C = 180° - 30° - 23.58°

C = 126.42°

Now, let's check if the side lengths satisfy the triangle inequality theorem:

a + b > c

20 + 16 >

36 > c

Since 36 is greater than c, the triangle inequality theorem is satisfied.

So, there is exactly one possible solution for a triangle with the given side lengths a = 20, b = 16, and angle A = 30°.

Answer:

Favorable Outcomes

Step-by-step explanation:

Using synthetic division and the Remainder Theorem, we find that P(a) for P(x) when a=2 is P(2)=4, which is the remainder of the synthetic division.

To find P(a) for the given polynomial P(x)=x⁴+3x³-6x²-10x+8 when a=2, we can use synthetic division. The Remainder Theorem states that the remainder of the division of a polynomial by a linear divisor (x - a) is equal to P(a).

Let's perform the synthetic division:

The synthetic division should look like this:

The final number in the bottom row is the remainder, which is also P(2). So, P(2)=4.

Answer:

The zeros to the quadratic equation are:

[tex]x= -4+\sqrt{\frac{41}{2}}\\\\x= -4-\sqrt{\frac{41}{2}}[/tex]

Step-by-step explanation:

A quadratic function is one of the form [tex]f(x) = ax^2 + bx + c[/tex], where a, b, and c are numbers with a not equal to zero.

The zeros of a quadratic function are the two values of x when [tex]f(x) = 0[/tex] or [tex]ax^2 + bx +c = 0[/tex].

To find the zeros of the quadratic function [tex]f(x)= 2x^2 + 16x -9[/tex] , we set [tex]f(x) = 0[/tex], and solve the equation.

[tex]2x^2+16x\:-9=0[/tex]

[tex]\mathrm{For\:a\:quadratic\:equation\:of\:the\:form\:}ax^2+bx+c=0\mathrm{\:the\:solutions\:are\:}\\\\x_{1,\:2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]

[tex]\mathrm{For\:}\quad a=2,\:b=16,\:c=-9:\quad x_{1,\:2}=\frac{-16\pm \sqrt{16^2-4\cdot \:2\left(-9\right)}}{2\cdot \:2}\\\\x=\frac{-16+\sqrt{16^2-4\cdot \:2\left(-9\right)}}{2\cdot \:2}= -4+\sqrt{\frac{41}{2}}\\\\x=\frac{-16-\sqrt{16^2-4\cdot \:2\left(-9\right)}}{2\cdot \:2}= -4-\sqrt{\frac{41}{2}}[/tex]

The equation of the line with a slope of -3/5 and a y-intercept of 7/10 is y = (-3/5)x + (7/10).

To write an equation of a line with a given slope (m) and y-intercept (0,b), we use the slope-intercept form of a linear equation which is y = mx + b. In this case, the slope is -3/5 and the y-intercept is 7/10.

Substituting these values into the slope-intercept formula, the equation of the line is y = (-3/5)x + (7/10).

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The surface area (S) of a rectangular prism is given in terms of its length (L), width (W), and height (H) as

... S = 2(LW +WH + LH)

Substitute your numbers and do the arithmetic.

... S = 2(7.2·10 + 10·6.3 + 7.2·6.3)

... = 2(72 + 63 + 45.36)

... = 2·180.36

... = 360.72 . . . . square inches

-4x - 5y = 7

3x + 5y = -14

You can add these two equations together straightaway since the y-terms have opposite coefficients.

-4x - 5y = 7

3x + 5y = -14

+___________

-x - 0 = -7

-x = -7

x = 7

Substitute 7 for x into either of the original equations and solve algebraically to find y.

3x + 5y = -14

3(7) + 5y = -14

21 + 5y = -14

21 = -14 - 5y

35 = -5y

-7 = y

Finally, check work by substituting both x- and y-values into both original equations.

-4x - 5y = 7

-4(7) - 5(-7) = 7

-28 + 35 = 7

7 = 7

3x + 5y = -14

3(7) + 5(-7) = -14

21 - 35 = -14

-14 = -14

Answer:

x = 7 and y = -7; (7, -7).

The circumference of an Oreo is 6.28 inches what is the area of the top cookie?

Solution:

Circumference= 6.28 inches

The formula of circle =2πr

So, 2πr=6.28

2*3.14*r=6.28

6.28r=6.28

To find the value of r, Let us divide by 6.28 on both sides

6.28 r /6.28 =6.28/6.28

r=1

Area of circle=πr²

=π1²

=π

=3.14 inches²

Area of the top cookie=3.14 inches²