A book normally costs $21.50. Today it was on sale for $15.05. What percentage discount was offered during the sale?

Answer:

[tex]t_{1/2} \approx 9.6\,days[/tex]

Step-by-step explanation:

The time constant of the radioactive constant is:

[tex]\tau = \frac{1}{0.072}\,days[/tex]

[tex]\tau = 13.889\,days[/tex]

The half-life of the substance is:

[tex]t_{1/2} = \tau \cdot \ln 2[/tex]

[tex]t_{1/2} = (13.889\,days)\cdot \ln 2[/tex]

[tex]t_{1/2} \approx 9.6\,days[/tex]

Answer:7^6

Step-by-step explanation:

Answer:

7^6

Step-by-step explanation:

Answer:

  10.2

Step-by-step explanation:

Apparently, you want to find the solution for ...

  132.2 = 23.2√(5 +2.7t)

  132.2/23.2 = √(5 +2.7t) . . . . divide by 23.2

  (132.2/23.2)² = 5 +2.7t . . . . . square both sides

  (132.2/23.2)² -5 = 2.7t . . . . . subtract 5

  ((132.2/23.2)² -5)/2.7 = t . . . . divide by the coefficient of t

  10.2 ≈ t

It will take about 10.2 years for the demand for computers to reach 132.2 million.

Answer:

the correct graph is pictured below

Step-by-step explanation:

the graph is below

Answer:

  about 100 cm²

Step-by-step explanation:

The side length of the rhombus is 1/4 of the perimeter so is 10 cm. The length of half of the other diagonal will be the length of the leg of a right triangle with hypotenuse 10 and leg 7 (half the given diagonal).

  d= √(10² -7²) = √51

Then the area of the rhombus is the product of this and the given diagonal:

  A = (14 cm)(√51 cm) ≈ 99.98 cm²

The area of the rhombus is about 100 cm².

Answer: x=  −6 /5  is the answer

Answer:

x= -1 1/5 or x= -1.2

Step-by-step explanation:

-4x+-3=x+3

-5x-3=3

-5x=6

-x=6/5

x=-6/5

x=-1 1/5

or x=-1.2

Hope this helps! Pls rate brainliest

Answer:

720

Step-by-step explanation:

96 * 7.5

Answer:

  5.892×10^-2

Step-by-step explanation:

Scientific notation has one digit to the left of the decimal point. To write the number in scientific notation, it can work to start by writing the number with that as one of the factors:

  0.05891 = 5.891 × 0.01

  = 5.891 × 1/10^2

  = 5.891 × 10^-2

__

You can also enter this number into your calculator and change the display mode to SCI.

_____

It helps to understand the decimal place-value number system in terms of the power of 10 that multiplies each number place.

Answer: 300

We know that 5×3=15. Since we are trying to get 1500 we Multiply 300×5 and get 1500.

How do we know?

As you see when you still multiply either of those problems you have 15 in it. So we should know 300×5=1500.

Note: We can also multiply 500×3 and get 1500. You still get the same answer but just switched numbers.

Answer:

[tex]300[/tex]

Step-by-step explanation:

[tex] \frac{1500}{5} = 300[/tex]

It's very easy to find if you use a calculator.

To know whether that the answer is correct or wrong you can do like this.

[tex]300 \times 5 = 1500[/tex]

hope this helps

thanks.

Answer:

24 guppies

Step-by-step explanation:

Assuming a linear relationship,  the volume required per guppy is given by:

[tex]g=\frac{576\ in^3}{3\ guppies}\\ g= 192\ in^3/guppy[/tex]

The volume of the fish tank is given by the product of its length, by its width and its height:

[tex]V = 24*12*16\\V=4,608\ in^3[/tex]

The number of guppies that this tank can safely hold is:

[tex]n=\frac{V}{g}=\frac{4,608}{192} \\n=24\ guppies[/tex]

The tank can safely hold 24 guppies.

Answer:

The z-score (value of z) for an income of $1,100 is 1.

Step-by-step explanation:

We are given that the mean of a normally distributed group of weekly incomes of a large group of executives is $1,000 and the standard deviation is $100.

Let X = group of weekly incomes of a large group of executives

So, X ~ N([tex]\mu=1,000 ,\sigma^{2} = 100^{2}[/tex])

The z-score probability distribution for a normal distribution is given by;

               Z = [tex]\frac{X-\mu}{\sigma}[/tex]  ~ N(0,1)

where, [tex]\mu[/tex] = mean income = $1,000

            [tex]\sigma[/tex] = standard deviation = $100

The Z-score measures how many standard deviations the measure is away from the mean. After finding the Z-score, we look at the z-score table and find the p-value (area) associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X.

Now, we are given an income of $1,100 for which we have to find the z-score (value of z);

So, z-score is given by = [tex]\frac{X-\mu}{\sigma}[/tex] = [tex]\frac{1,100-1,000}{100}[/tex] = 1

Hence, the z-score (value of z) for an income of $1,100 is 1.

The z-score for an income of $1,100 is 1.0.

The z-score is a measure of how many standard deviations a particular value, in this case, income, is from the mean of a normally distributed dataset. The formula to calculate the z-score is:

z = (x - μ) / σ

Where:

Given the mean (μ) is $1,000 and the standard deviation (σ) is $100, we can substitute these values into the formula to find the z-score of an income of $1,100.

z = ($1,100 - $1,000) / $100

z = $100 / $100

z = 1

Therefore, the z-score for an income of $1,100 is 1.0. This means the income of $1,100 is one standard deviation above the mean.

Answer:

   look for the x-intercepts

Step-by-step explanation:

A "zero" is a value of x where the function value is zero. On a graph, that point is where the graph meets the x-axis. Every x-intercept is a zero of the function.

__

If there are no x-intercepts, then there are no real zeros. The roots (zeros) will be complex.

Answer:

1/2

Step-by-step explanation:

Answer:

11/12

Step-by-step explanation:

If you convert 3/4 and 1/6,

you will get 9/12+2/12.

If you add both of them, you will get your answer.

Answer:

Step-by-step explanation:

So, we do not have enough evidence to conclude that the number of adolescent relationships is equal between those with an older sibling and those without.

check the attached file for explanation and solution

Answer:

90% confidence interval for the true proportion of air travelers who prefer the window seat is (0.575, 0.625)

Step-by-step explanation:

We have the following data:

Sample size = n = 1000

Proportion of travelers who prefer window seat = p = 60%

Standard Error = SE = 0.015

We need to construct a 90% confidence interval for the proportion of travelers who prefer window seat. Therefore, we will use One-sample z test about population proportion for constructing the confidence interval. The formula to calculate the confidence interval is:

[tex](p-z_{\frac{\alpha}{2}}\sqrt{\frac{p(1-p)}{n}}, p+z_{\frac{\alpha}{2}}\sqrt{\frac{p(1-p)}{n}})[/tex]

Since, standard error is calculated as:

[tex]SE=\sqrt{\frac{p(1-p)}{n} }[/tex]

Re-writing the formula of confidence interval:

[tex](p-z_{\frac{\alpha}{2}} \times SE, p+z_{\frac{\alpha}{2}} \times SE)[/tex]

Here, [tex]z_{\frac{\alpha}{2}}[/tex] is the critical value for 90% confidence interval. From the z-table this value comes out to be 1.645.

Substituting all the values in the formula gives us:

[tex](0.6 - 1.645 \times 0.015, 0.6 + 1.645 \times 0.015)\\\\ = (0.575, 0.625)[/tex]

Therefore, the 90% confidence interval for the true proportion of air travelers who prefer the window seat is (0.575, 0.625)

Answer:

[tex]y=x-2[/tex]

Step-by-step explanation:

So we are given the formula for the slope of a hyperbola in this form:

[tex]\frac{x^2}{a^2}-\frac{y^2}{b^2}=1[/tex].

That formula for the slope is [tex]m=\frac{b^2x}{a^2y}[/tex]

If we compare the following two equations, we will be able to find [tex]a^2[/tex] and [tex]b^2[/tex]:

[tex]\frac{x^2}{a^2}-\frac{y^2}{b^2}=1[/tex]

[tex]\frac{x^2}{8}-\frac{y^2}{4}=1[/tex]

We see that [tex]a^2=8[/tex] while [tex]b^2=4[/tex].

So the slope at [tex](x,y)=(4,2)[/tex] is:

[tex]m=\frac{b^2x}{a^2y}=\frac{4(4)}{8(2)}=\frac{16}{16}=1[/tex].

Recall: Slope-intercept form of a linear equation is [tex]y=mx+b[/tex].

We just found [tex]m=1[/tex]. Let's plug that in.

[tex]y=1x+b[/tex]

[tex]y=x+b[/tex]

To find [tex]b[/tex], the [tex]y[/tex]-intercept, we will need to use a point on our tangent line. We know that it is going through [tex](4,2)[/tex].

Let's enter this point in to find [tex]b[/tex].

[tex]2=4+b[/tex]

Subtract 4 on both sides:

[tex]2-4=b[/tex]

Simplify:

[tex]-2=b[/tex]

The equation for the tangent line at [tex](4,2)[/tex] on the given equation is:

[tex]y=x-2[/tex]

Answer: y = x - 2

Step-by-step explanation:

First you take the derivative of each term. d/dx(x²/8) - d/dx(y²/4) = d/dx(1)

x/4 - (y/2)dy/dx = 0

Then you solve for dy/dx: dy/dx = x/2y

Plug in the values: dy/dx = 1

To find the y-intercept, plug in values for y = mx+ b. 2 = 4 + b, b = -2

The equation is y = x - 2

Answer:

Increasing:

[tex](\frac{\pi}{3},\frac{5\pi}{3})[/tex]

Decreasing

[tex](0,\frac{\pi}{3})[/tex] U [tex](\frac{5\pi}{3},2\pi)[/tex]  

Step-by-step explanation:

Increasing and Decreasing Intervals

To find if a function is increasing in a point x=a, we evaluate the first derivative in x=a and if:

For continuous functions, we can safely assume between a given critical point and the next one, the function keeps its behavior, i.e. it's increasing or decreasing in the interval formed by both points.

So, we find the critital points of  

[tex]f(x)=x-2sinx[/tex]

Taking the derivative

[tex]f'(x)=1-2cosx[/tex]

Equating to 0

[tex]1-2cosx=0[/tex]

Solving

[tex]\displaystyle cosx=\frac{1}{2}[/tex]

There are two solutions in the interval  [tex](0,2\pi)[/tex]

[tex]\displaystyle x=\frac{\pi}{3},\ x=\frac{5\pi}{3}[/tex]

Now we compute the second derivative

[tex]f''(x)=2sinx[/tex]

Evaluating for both critical points

[tex]\displaystyle f''(\frac{\pi}{3})=2sin\frac{\pi}{3}=\sqrt{3}[/tex]

Since it's positive, the point is a minimum

[tex]\displaystyle f''(\frac{5\pi}{3})=2sin\frac{5\pi}{3}=-\sqrt{3}[/tex]

Since it's negative, the point is a maximum

In the interval  

[tex](0,\frac{\pi}{3})[/tex]

the function is decreasing

In the interval  

[tex](\frac{\pi}{3},\frac{5\pi}{3})[/tex]

the function is increasing

In the interval  

[tex](\frac{5\pi}{3},2\pi)[/tex]  

the function is decreasing

First, you need to find the derivative of the function f(x) = x - 2sin(x).

Using the rules of calculus, the derivative of x is 1 and the derivative of -2sin(x) is -2cos(x). Therefore, the derivative of the function, denoted as f'(x), is given by:

f'(x) = 1 - 2cos(x).

To determine where the function is increasing or decreasing, we need to find the critical points, the x-values where the derivative of the function is equal to zero or undefined.

Setting the derivative equal to zero and solving:

1 - 2cos(x) = 0.

2cos(x) = 1.

cos(x) = 1/2.

The solutions to this equation on the interval 0 to 2π are x = π/3 and x = 5π/3.

With these critical points, we have divided the entire interval into three subintervals:

(0, π/3), (π/3, 5π/3), and (5π/3, 2π).

We now determine the sign of the derivative on each of these intervals. We pick a "test point" from each interval and substitute it into the derivative.

From (0, π/3), we pick x = π/6, and find that f'(π/6) is positive.
From (π/3, 5π/3), we pick x = π, and find that f'(π) is negative.
From (5π/3, 2π), we pick x = 3π/2, and find that f'(3π/2) is positive.

By the First Derivative Test, a positive derivative indicates that the function is increasing on that interval and a negative derivative indicates that the function is decreasing on that interval.

Therefore, the function increases on the intervals (0, π/3) and (5π/3, 2π),
and decreases on the interval (π/3, 5π/3).

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Answer:

(a) The least-square regression line is: [tex]y=4.662+2.709x[/tex].

(b) The number of households using online banking at the beginning of 2007 is 31.8.

Step-by-step explanation:

The general form of a least square regression line is:

[tex]y=\alpha +\beta x[/tex]

Here,

y = dependent variable

x = independent variable

α = intercept

β = slope

(a)

The formula to compute intercept and slope is:

[tex]\begin{aligned} \alpha &= \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} \\\beta &= \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} \end{aligned}[/tex]

The values of ∑X, ∑Y, ∑XY and ∑X² are computed in the table below.

Compute the value of intercept and slope as follows:

[tex]\begin{aligned} \alpha &= \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 68.6 \cdot 55 - 15 \cdot 218.9}{ 6 \cdot 55 - 15^2} \approx 4.662 \\ \\\beta &= \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 6 \cdot 218.9 - 15 \cdot 68.6 }{ 6 \cdot 55 - \left( 15 \right)^2} \approx 2.709\end{aligned}[/tex]

The least-square regression line is:

[tex]y=4.662+2.709x[/tex]

(b)

For the year 2007 the value of x is 10.

Compute the value of y for x = 10 as follows:

[tex]y=4.662+2.709x[/tex]

  [tex]=4.662+(2.709\times10)\\=4.662+27.09\\=31.752\\\approx 31.8[/tex]

Thus, the number of households using online banking at the beginning of 2007 is 31.8.

Answer:

At 95% confidence level, the difference between the proportion of freshmen using steroids in Illinois and the proportion of seniors using steroids in Illinois is -7.01135×10⁻³ < [tex]\hat{p}_1-\hat{p}_2[/tex] < 1.237

Step-by-step explanation:

Here we are required to construct the 95% confidence interval of the difference between two proportions

The formula for the confidence interval of the difference between two proportions is as follows;

[tex]\hat{p}_1-\hat{p}_2\pm z^{*}\sqrt{\frac{\hat{p}_1\left (1-\hat{p}_1 \right )}{n_{1}}+\frac{\hat{p}_2\left (1-\hat{p}_2 \right )}{n_{2}}}[/tex]

Where:

[tex]\hat{p}_1 = \frac{34}{1679}[/tex]

[tex]\hat{p}_2 = \frac{24}{1366}[/tex]

n₁ = 1679

n₂ = 1366

[tex]z_{\alpha /2}[/tex] at 95% confidence level = 1.96

Plugging in the values, we have;

[tex]\frac{34}{1679}- \frac{24}{1366} \pm 1.96 \times \sqrt{\frac{ \frac{34}{1679}\left (1- \frac{34}{1679}\right )}{1679}+\frac{\frac{24}{1366} \left (1-\frac{24}{1366} \right )}{1366}}[/tex]

Which gives;

-7.01135×10⁻³ < [tex]\hat{p}_1-\hat{p}_2[/tex] < 1.237.

At 95% confidence level, the difference between the proportion of freshmen using steroids in Illinois and the proportion of seniors using steroids in Illinois = -7.01135×10⁻³ < [tex]\hat{p}_1-\hat{p}_2[/tex] < 1.237.

Answer:

18.86 feet

Step-by-step explanation:

The circumference of the inner circle is 88 ft.

Circumference of a Circle[tex]=2\pi r[/tex]

Therefore:

[tex]2\pi r =88\\r=88 \div (2*\frac{22}{7})=14 ft[/tex]

Radius of the inner circle=14 feet

If the distance between the inner circle and the outer circle is 3 ft, the radius of the outer circle=14+3 =17 feet.

Therefore. circumference of the outer Circle[tex]=2\pi r[/tex]

[tex]=2*\frac{22}{7}*17=106.86 ft[/tex]

Difference in Circumference=106.86-88 =18.86 feet

The circumference of the outer circle is greater than that of the inner circle by 18.86 feet.

18.9

Step-by-step explanation:

Answer:

Heather rode the farthest

She rode 0.1km farther than Hailey.

Step-by-step explanation:

This is a conversion of units question.

Heather rode her horse 2 kilometers.

Hailey rode 1900 meters on her horse.

Each km has 1000 meters.

So 1900 meters = 1900/1000 = 1.9 km

This means that Hailey rode for 1.9 km.

Heather rode the farthest(2km is greather than 1.9km)

2 - 1.9 = 0.1km

She rode 0.1km farther than Hailey.

Answer:

A:0.055

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

[tex]\mu = 2.5, \sigma = 0.25[/tex]

Which of the following is closest to the probability that the selected battery will have a life span of at most 2.1 hours?

This is the pvalue of Z when X = 2.1. So

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{2.1 - 2.5}{0.25}[/tex]

[tex]Z = -1.6[/tex]

[tex]Z = -1.6[/tex] has a pvalue of 0.0548

So the correct answer is:

A:0.055

Answer:

4x+5y = 320

x+y = 70

Step-by-step explanation:

We need one equation for the total number of shirts, and one for the total cost.

Total number = 70

So that means medium shirts + large shirts = 70

So our first equation is x+y = 70

Total cost = $320

So that means $4 times medium shirts + $5 times large shirts = 320

So our second equation is 4x+5y = 320

B alone can empty the pool in 5.744 hours, if A alone can empty the pool in 4.75 hours and  it takes 2.6 hours when both drains are turned on.

Step-by-step explanation:

The given is,

                   A alone can empty the pool in 4.75 hours.

                   It takes 2.6 hours when both drains are turned on.

Step:1

            One hour work drains A and B =

                  One hour work of drain A + One hour work of Drain B.........(1)

            One hour work of Drain A = [tex]\frac{1}{4.75}[/tex]

                     One hour of ( A + B ) = [tex]\frac{1}{2.6}[/tex]

            Equation (1) becomes,

                     One hour work of B = One hour work of ( A + B )

                                                                  - One hour work of A

            Substitute the values,

                     One hour work of B =  [tex]\frac{1}{2.6} - \frac{1}{4.75}[/tex]

                                                       = [tex]\frac{4.75-2.6}{(4.75)(2.6)}[/tex]

                                                       = [tex]\frac{2.15}{12.35}[/tex]

                                                       = [tex]\frac{1}{5.744}[/tex]

                    One hour work of B  = [tex]\frac{1}{5.744}[/tex]

             B alone can empty the pool in 5.744 hours

Result:

           B alone can empty the pool in 5.744 hours, if A alone can empty the pool in 4.75 hours and  it takes 2.6 hours when both drains are turned on.

Answer: Top right, a rectangle has all the properties of a square

Step-by-step explanation: A rectangle does not have all the properties of a square.

The equation we've developed to represent the rabbit population over time, where 't' is the number of months since January, is P(t) = 19sin((2π/12)t - π/2) + (160t/12) + 650. This equation covers the oscillations in the population and the steady yearly increase.

The subject matter falls under the discipline of Mathematics, particularly in the topics involving functions. We can create a sinusoidal (sine or cosine) function to represent the oscillation of the population of rabbits.

Given that the population fluctuates 19 above and below the average, and the average increases by 160 each year, this suggests a sinusoidal period of 12 months (a year) with a vertical shift (midline) that increases linearly.

Considering t as the number of months since January, the equation for the population P in terms of the months since January t would be:

P(t) = 19sin((2π/12)t - π/2) + (160t/12) + 650

The 19 is the amplitude, (2π/12)t - π/2 represents the sinusoidal oscillation adjusted to start at the minimum in January, (160t/12) is the yearly change in population that increases per month, and 650 is the average population at the start.

https://brainly.com/question/32049339

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To find the population equation in terms of months since January considering oscillations and growth, use the formula p(t) = 650 + 160t + 19sin(2πt/12).

Population Equation: The equation for the population p in terms of the months since January t can be written as p(t) = 650 + 160t + 19sin(2πt/12). This equation takes into account the initial population of 650 rabbits, an increase of 160 rabbits per year, and the oscillation of 19 above and below the average population.

Answer:

The probability of students scored less than 60 = .0768

Step-by-step explanation:

Given -

Mean score [tex](\nu )[/tex] = 70

standard deviation [tex](\sigma )[/tex] = 7

Let X be the score of students

the probability that students scored less than 60 =

[tex]P(X< 60)[/tex]  = [tex]P(\frac{X - \nu }{\sigma}< \frac{60 - 70}{7})[/tex]

                  =  [tex]P(z < \frac{60 - 70}{7})[/tex]   put[ Z= [tex]\frac{X - \nu }{\sigma}[/tex]]

                   =  [tex]P(z < -1.428)[/tex]  using z table

                    =  .0768

The student must sell at least 70 glasses of lemonade at $0.25 each, considering the initial costs of $16.00 and the additional $0.02 cost per glass for sugar, to begin to make a profit.

To calculate the minimum number of glasses of lemonade that must be sold to start making a profit, we need to consider the total costs and the revenue per glass. The student spent $12.00 on lemons and $4.00 on glasses, totaling $16.00 in costs. Additionally, each glass of lemonade has an added cost of $0.02 for sugar. The revenue from selling one glass of lemonade is $0.25.

To break even, the total revenue must equal the total costs. As costs are fixed at $16.00 and the variable cost is $0.02 per glass, we can set up the equation: (Number of glasses × $0.25) - (Number of glasses × $0.02) = $16.00. This simplifies to (Number of glasses × $0.23) = $16.00, and solving for Number of glasses gives us Number of glasses = $16.00 / $0.23, which is approximately 69.57 glasses. Since you can't sell a fraction of a glass, rounding up means the student needs to sell at least 70 glasses to begin to make a profit.

Step-by-step explanation:

a) 24 / 250 = 0.096

b) Standard error for a proportion is:

σ = √(pq/n)

σ = √(0.096 × 0.904 / 250)

σ = 0.0186

c) At 98% confidence, the critical value is 2.326.  The margin of error is therefore:

2.326 × 0.0186 = 0.0433

d) At 95% confidence, the critical value is 1.960.  The margin of error is therefore:

1.960 × 0.0186 = 0.0365

So the confidence interval is:

(0.0960 − 0.0365, 0.0960 + 0.0365)

(0.0595, 0.1325)

e) 0.10 is within the 95% confidence interval, so the null hypothesis would not be rejected.

Final answer:

a) The point estimate of the proportion of components in the shipment that fail to meet the company's specifications is 9.6%. b) The standard error of the estimated proportion is 1.9%. c) The margin of error at the 98% confidence level is 4.4%. d) The 95% confidence interval estimate for the true proportion is approximately 5.9% to 13.3%. e) The null hypothesis that the proportion is 0.10 is rejected if the z-test statistic falls outside the range (-1.96, 1.96).

Explanation:

a) Point estimate:
The point estimate of the proportion of components in the shipment that fail to meet the company's specifications is the number of failed components divided by the total number of components tested. In this case, the point estimate is 24/250 = 0.096, or 9.6%.

b) Standard error:
The standard error of the estimated proportion is calculated using the formula SE = sqrt((phat * (1 - phat)) / n), where phat is the point estimate and n is the sample size. In this case, the standard error is sqrt((0.096 * (1 - 0.096)) / 250) = 0.019, or 1.9%.

c) Margin of error:
The margin of error is determined by multiplying the standard error by the appropriate critical value from the standard normal distribution. For a 98% confidence level, the critical value is approximately 2.33. Therefore, the margin of error is 2.33 * 0.019 = 0.044, or 4.4%.

d) Confidence interval:
The 95% confidence interval estimate for the true proportion of components that fail to meet the specifications is given by the formula phat +/- z * SE, where phat is the point estimate, z is the appropriate critical value from the standard normal distribution (for 95% confidence, z is approximately 1.96), and SE is the standard error. Therefore, the confidence interval is 0.096 +/- 1.96 * 0.019, or approximately 0.059 to 0.133.

e) Hypothesis test:
To test the null hypothesis H_0: p = 0.10 against the alternative hypothesis H_a: p != 0.10, we can use a two-tailed z-test. The test statistic is calculated as (phat - p_0) / sqrt((p_0 * (1 - p_0)) / n), where p_0 is the null hypothesis value (0.10), phat is the point estimate, and n is the sample size. The critical value for a significance level of 0.05 is approximately 1.96 from the standard normal distribution. If the test statistic is outside the range (-1.96, 1.96), we reject the null hypothesis. In this case, if the test statistic falls outside the range (-1.96, 1.96), we would reject the null hypothesis and conclude that the true proportion of components that fail to meet the specifications is not 0.10.

Answer:

Required results are [tex]\nabla .\vec{F}[/tex]=8\cos(8x+5z)-8ye^x[/tex]  and  [tex]\nabla\times \vec{F}=-8e^xz\uvec{i}+(8ye^xz+5\sin(8x+5z))\uvec{j}[/tex]

Step-by-step explanation:

Given vector function is,

[tex]\vec{F}=\sin(8x+5z)\uvec{i}-8ye^xz\uvec{k}[/tex]

To find [tex]\nabla .\vec{F}[/tex] and [tex]\nabla \times \vec{F}[/tex] .

[tex]\nabla .\vec{F}[/tex]

[tex]=(\frac{\partial}{\partial x}\uvec{i}+\frac{\partial}{\partial y} \uvec{j}+\frac{\partial}{\partial z} \uvec{k})(\sin(8x+5z)\uvec{i}-8ye^xz\uvec{k})[/tex]

[tex]=\frac{\partial}{\partial x}(\sin(8x+5z))-\frac{\partial}{\partial z}(8ye^xz)[/tex]

[tex]=8\cos(8x+5z)-8ye^x[/tex]

And,

[tex]\nabla \times \vec{F}[/tex]

[tex]=(\frac{\partial}{\partial x}\uvec{i}+\frac{\partial}{\partial y} \uvec{j}+\frac{\partial}{\partial z} \uvec{k})\times(\sin(8x+5z)\uvec{i}-8ye^xz\uvec{k})[/tex]

[tex]\end{Vmatrix}[/tex]

[tex]=\uvec{i}\Big[\frac{\partial}{\partial y}(-8ye^xz)\Big]-\uvec{j}\Big[\frac{\partial}{\partial x}(-8ye^xz)-\frac{\partial}{\partial z}(\sin(8x+5z))\Big]+\uvec{k}\Big[-\frac{\partial}{\partial y}(-\sin(8x+5z))\Big][/tex]

[tex]=-8e^xz\uvec{i}+(8ye^xz+5\sin(8x+5z))\uvec{j}[/tex]

Hence the result.