L'Shanda can choose between 3 sweaters and 4 skirts. If she selects 1 sweater and 1 skirt, how many possible outcomes are in the sample space?

Answer:

[tex]5.5-2.701\frac{1.1}{\sqrt{42}}=5.04[/tex]    

[tex]5.5+2.701\frac{1.1}{\sqrt{42}}=5.96[/tex]    

So on this case the 99% confidence interval would be given by (5.04;5.96)

And the best option would be:

a. 5.04 and 5.96

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

[tex]\bar X=5.5[/tex] represent the sample mean for the sample  

[tex]\mu[/tex] population mean (variable of interest)

s=1.1 represent the sample standard deviation

n=42 represent the sample size  

Solution to the problem

The confidence interval for the mean is given by the following formula:

[tex]\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex]   (1)

In order to calculate the critical value [tex]t_{\alpha/2}[/tex] we need to find first the degrees of freedom, given by:

[tex]df=n-1=42-1=41[/tex]

Since the Confidence is 0.99 or 99%, the value of [tex]\alpha=0.01[/tex] and [tex]\alpha/2 =0.005[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.005,41)".And we see that [tex]t_{\alpha/2}=2.701[/tex]

Now we have everything in order to replace into formula (1):

[tex]5.5-2.701\frac{1.1}{\sqrt{42}}=5.04[/tex]    

[tex]5.5+2.701\frac{1.1}{\sqrt{42}}=5.96[/tex]    

So on this case the 99% confidence interval would be given by (5.04;5.96)

And the best option would be:

a. 5.04 and 5.96

A)Yes they should market the wax because it thaws before 55 seconds

C) Null hypothesis: mean <55 vs Alternative =55

D) The value is 51.33 moments

When The selected samples for the champion are: 57.9, 62.9, 50.6, 50.5,48.2,47.2,50.2 and 43.1

The mean is ;

Then, Sum =57.9 + 62.9 + 50.6 + 50.5+48.2+47.2+50.2+ 43.1 =410.6

After that, Mean is = 410.6/8 =51.33= mean

Considering Significant is =0.05, therefore, applying this level will give you 51.28-51.38

if particularly, the meantime to complete their standard test course should be less than 55 seconds for a former Olympic champion, then the nullified hypothesis is correct

After that; Null hypothesis: mean <55

Hence, Alternative hypothesis: mean =55

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Answer:

[tex]P(29<X<89)=P(\frac{29-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{89-\mu}{\sigma})=P(\frac{29-59}{10}<Z<\frac{89-59}{10})=P(-3<z<3)[/tex]

And we can find this probability with this difference:

[tex]P(-3<z<3)=P(z<3)-P(z<-3)[/tex]

And in order to find these probabilities using the normal standard distribution or excel and we got.  

[tex]P(-3<z<3)=P(z<3)-P(z<-3)=0.9987-0.00135=0.99735[/tex]

So we expect about 99.735% of values between $29 and $89

Step-by-step explanation:

Let X the random variable that represent the amount of Jens monthly phone of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(59,10)[/tex]  

Where [tex]\mu=59[/tex] and [tex]\sigma=10[/tex]

We are interested on this probability first in order to find a %

[tex]P(29<X<89)[/tex]

The z score is given by:

[tex]z=\frac{x-\mu}{\sigma}[/tex]

If we apply this formula to our probability we got this:

[tex]P(29<X<89)=P(\frac{29-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{89-\mu}{\sigma})=P(\frac{29-59}{10}<Z<\frac{89-59}{10})=P(-3<z<3)[/tex]

And we can find this probability with this difference:

[tex]P(-3<z<3)=P(z<3)-P(z<-3)[/tex]

And in order to find these probabilities using the normal standard distribution or excel and we got.  

[tex]P(-3<z<3)=P(z<3)-P(z<-3)=0.9987-0.00135=0.99735[/tex]

So we expect about 99.735% of values between $29 and $89

Approximately 99.7% of Jen's phone bills are between $29 and $89, as this range lies within three standard deviations from the mean on a normally distributed curve with mean $59 and standard deviation $10.

To determine the percentage of Jen's phone bills that are between $29 and $89, we must calculate the z-scores for each value and then use the standard normal distribution to find the corresponding percentages.

The z-score is given by the formula:

Z = (X - μ)/σ

Where:

Z is the z-score,

X is the value in question,

μ is the mean,

σ is the standard deviation.

For X = $29:

Z = (29 - 59)/10

Z = -3

For X = $89:

Z = (89 - 59)/10

Z = 3

Using the standard normal distribution table or a calculator, we find that the probability of a z-score being between -3 and 3 is approximately 99.7%. Therefore, about 99.7% of Jen's phone bills fall between $29 and $89.

Answer:

  1/2

Step-by-step explanation:

Put the term number in the formula to get your answer.

  Term 2 = 1/2

Substituting n with 2 into the explicit sequence formula 1/n gives the second term of the sequence as 0.5 which expressed as a fraction is 1/2.

The explicit formula given in the question is 1/n, where 'n' represents the term number in the sequence.

When we want to find out the second term of the sequence, we substitute n with 2 into the formula.

So the calculation is 1/2 = 0.5.

Expressing 0.5 as a fraction, you get 1/2.

Therefore, the second term of the sequence, expressed as a fraction, is 1/2.

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Answer:

7.1%

Step-by-step explanation:

Current yield is the ratio of coupon payment of a bond to its current market price.  It is calculated by using coupon payment and the current market value of the bond.

As per given data

Coupon rate = 7%

Face value = $1,000

Market Value = $985

Coupon Payment = $1,000 x 7% = $70

Formula for Current yield is as follow

Current Yield = Annual Coupon Payment / Current Market Price

Current Yield = $70 / $985

Current Yield = 7.11%

To predict the number of golden retrievers G that will be registered in year t, use the linear equation G = -6.4t + 12868.9.

To predict the number of golden retrievers G that will be registered in year t, we can use a linear equation. We have two data points: (2001, 62.5) and (2002, 56.1). We can use the slope-intercept form of a linear equation, y = mx + b, where m is the slope and b is the y-intercept.

Step 1: Determine the slope (m) using the formula m = (y2 - y1) / (x2 - x1) = (56.1 - 62.5) / (2002 - 2001) = -6.4

Step 2: Choose one of the data points and substitute the values into the equation to solve for b. Using (2001, 62.5), we can substitute x = 2001 and y = 62.5.

62.5 = -6.4 * 2001 + b
62.5 = -12806.4 + b
b = 12868.9

Step 3: The linear equation to predict the number of golden retrievers G is G = -6.4t + 12868.9.

Answer:

Therefore, on the graph;

The height = 7.5 units

And base = 5.0 units

Attached is an image for further information;

Step-by-step explanation:

Given that;

1 unit on the grid represent 2m of the garden;

Ratio = 1/2 unit/m

For the height;

height h = 15m

On the graph;

h = 15m × 1/2 unit/m

h = 7.5 units

For the base;

Base b = 10m

On the graph;

b = 10m × 1/2 unit/m

b = 5 units

Therefore, on the graph;

The height = 7.5 units

And base = 5.0 units

Answer:

a) Null hypothesis: [tex]p \leq 0.44[/tex]

b) Alternative hypothesis: [tex]p > 0.44[/tex]

c) For this case our parameter of interest is a proportion "reduced rate of first heart attacks for a new drug". for this reason the answer would be 1 proportion and we can conduct the hypothesis with a 1 z proportion test  

Step-by-step explanation:

For this case the pharmaceutical company thinks they have a drug that will be more effective than aspirin and on this case that means a better rate in order to reduce the heart attacks (that represent the alternative hypothesis since that;s what they want to proof), and the complement would be the null hypothesis.

Part a

Null hypothesis: [tex]p \leq 0.44[/tex]

Part b

Alternative hypothesis: [tex]p > 0.44[/tex]

Part c

For this case our parameter of interest is a proportion "reduced rate of first heart attacks for a new drug". for this reason the answer would be 1 proportion and we can conduct the hypothesis with a 1 z proportion test  

0.7878 is terminating.

Terminating decimals are the numbers that have a fixed or a finite number of digits after the decimal point. Decimal numbers are used to represent the partial amount of whole, just like fractions. In this lesson, we will focus on the type of decimal numbers, that is, terminating decimal numbers.

As, per the definition

0.12 repeats

0.44 repeats

0.56 repeats

But, 0.7878 does not repeat and get terminated.

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Answer:

B. Since P-value is greater than the significance level, we fail to reject the null hypothesis

Explanation:

Given Significance Level is 0.05 and the P-Value is 0.078

Since P-value greater than the significance level the best explanation is given by

Option B i.e.,

Since P-value is greater than the significance level, we fail to reject the null hypothesis

Answer:

(2,1), (1,2) is your base step

if (a,b)   is in the set (a+1,b+1) will be in  the set

if  (a,b) is in the set (a+2,b) will be in the set

if (a,b) is in the set (a,b+2) will be in the set.

Step-by-step explanation:

Think about how to solve this problem in general. How can you assure that the sum a+b is odd ?

Think about this, what happens when you sum two even numbers ? The result is even or odd ?

2+6 = 8 (even  )

10+12 = 22 (even)

And what happens when you sum two odd numbers ? The result will be even or odd ?  Look

3+7 = 10 (even)

5+11 = 16 (even)

Therefore to assure that a+b is odd, one of them has to be odd and one of them has to be even, that is why

(2,1), (1,2) is your base step

if (a,b)   is in the set (a+1,b+1) will be in  the set

if  (a,b) is in the set (a+2,b) will be in the set

if (a,b) is in the set (a,b+2) will be in the set.

The set of ordered pairs that are positive integers and have an odd sum can be defined recursively by starting with the set {(1,2)}, and then adding 2 to either a or b for every pair in the set to generate additional pairs that satisfy the conditions.

The set of ordered pairs of positive integers S given by S = {(a, b) | a ∈ Z+ , b ∈ Z+ , and a + b is odd} can be defined recursively as follows: Start with S = {(1,2)}, which is the smallest possible set that satisfies the conditions. For every (a,b) in S, (a+2, b) and (a, b+2) are also in S, since adding 2 to either a or b results in a sum that is still odd (since an even number plus an odd number equals an odd number).

As a result, the set S of all ordered pairs that result from these operations will satisfy the conditions given, namely being positive integers and having an odd sum. This demonstrates a recursive method for defining the set of ordered pairs.

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The chip amount that represents the 67th percentile is 48.588.and this can be determined by using the formula of z-score.

Given :

The amount of corn chips dispensed into a bag by the dispensing machine has been identified as possessing a normal distribution with a mean of μ = 48.5 ounces and a standard deviation of σ = 0.2 ounces.

To determine the chip amount that represents the 67th percentile, the below formula can be used:

[tex]\rm z = \dfrac{x-\mu}{\sigma}[/tex]

Now, substitute the values of known terms in the above formula:

[tex]\rm 0.44 = \dfrac{x - 48.5}{0.2}[/tex]

Cross multiply in the above equation.

[tex]\rm 0.44\times 0.2 = x - 48.5[/tex]

Now further, simplify the above equation.

0.088 = x - 48.5

x = 48.5 + 0.088

x = 48.588

So, the chip amount that represents the 67th percentile is 48.588.

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The weight that represents the 67th percentile of the corn chip bags, with a given mean of 48.5 ounces and a standard deviation of 0.2 ounce, is 48.59 ounces, calculated using the z-score provided.

To determine the amount of corn chips that represents the 67th percentile, p67, for a bag's weight distribution with a mean of μ = 48.5 ounces and a standard deviation of σ = 0.2 ounce. Given that the z-score for the 67th percentile is z = 0.44, we can use the percentile to z-score formula to find the corresponding weight.

To convert a z-score to a specific value within a normal distribution, we use the formula:

X = μ + zσ

For the 67th percentile:

X = 48.5 + (0.44 × 0.2)

X = 48.5 + 0.088

X ≈ 48.59 ounces (rounded to two decimal places)

This means that the weight that represents the 67th percentile of corn chip bag weights, to the nearest hundredth, is 48.59 ounces.

Answer:

[tex]t_{1} \approx 9.943\,s[/tex] and [tex]t_{2} \approx 0.057\,s[/tex]

Step-by-step explanation:

The following polynomial is needed to be solved:

[tex]-16\cdot t^{2} + 160\cdot t - 9 = 0[/tex]

The roots are found by means of the General Equation for Second-Order Polynomials:

[tex]t_{1} \approx 9.943\,s[/tex] and [tex]t_{2} \approx 0.057\,s[/tex]

Physically speaking, both solutions are reasonable.

Answer:

t = 0.0625

Step-by-step explanation:

Given that,

Height, h = -16t + 160t

To obtain time,t at height,h = 9feet

We substitute h = 9 into the given equation to have:

9 = - 16t + 160t

: 9 = 144t

t = 9/ 144 = 0.0625

Answer:

In solid geometry, a face is a flat (planar) surface that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by faces is a polyhedron. (OR) A face is a 2D shape that makes up one surface of a 3D shape, an edge is where two faces meet and a vertex is the point or corner of a geometric shape.

Step-by-step explanation:

Step-by-step explanation:

Total no of rows = 34

No of seats in first row = 12

No of seats in second row = 14

Third row = 16

If we continue this pattern of even numbers the last row will have 78 seats

Answer:

Step-by-step explanation:

We would set up the hypothesis test.

For the null hypothesis,

µ = 98

For the alternative hypothesis,

µ ≠ 98

This is a 2 tailed test.

Since no population standard deviation is given, the distribution is a student's t.

Since n = 270

Degrees of freedom, df = n - 1 = 270 - 1 = 269

t = (x - µ)/(s/√n)

Where

x = sample mean = 97

µ = population mean = 98

s = samples standard deviation = 7

t = (97 - 98)/(7/√270) = - 2.35

We would determine the p value using the t test calculator. It becomes

p = 0.02

Since alpha, 0.1 > than the p value, 0.02, then we would reject the null hypothesis. Therefore, At a 10% level of significance, there sufficient evidence to support the claim that the technique performs differently than the traditional method.

Answer:

The correct answer is that Jon needs approximately 11.19 gallons of gas to take his children to and from school each week.

Step-by-step explanation:

To solve this problem, we first need to figure out how many miles Jon drives per week.  To do this, we need to multiply the number of miles Jon drives per day (47 miles) by the number of days Jon drives per week (5).

47 * 5 = 235

This means that Jon drives 235 miles per week.  Now, to figure out how many gallons of gas Jon uses, we need to divide the number of miles Jon drives (235) by the number of miles per gallon Jon's vehicle gets (21).

235/21 = 11.19

Therefore, the answer is that Jon needs about 11 gallons of gas to take his children to and from school each week.

Hope this helps!

Answer:

75 cm (if b=10 and h=15)

Step-by-step explanation:

The formula to calculate the area of a triangle is bh1/2. So you must do (10)(15)1/2, 10 multiplied by 15 equals 150. And 150 multiplied by 1/2 equals 75. Hope this helped!

Answer:

85 feet by 170 feet

Step-by-step explanation:

Let the dimension of the dog park be x and y

Since only three sides will be fenced,

Perimeter, x+2y=340

Our goal is to determine the dimension of the park which maximizes the area.

Substituting x=340-2y into A(x,y)

[tex]A(y)=y(340-2y)\\A(y)=-2y^2+340y[/tex]

To maximize the area, we find the vertex using the equation of line of symmetry. Note that you can also find the critical points instead.

Equation of symmetry, [tex]y=-\dfrac{b}{2a}[/tex]

a=-2, b=340

[tex]y=-\dfrac{340}{2(-2)}=85[/tex]

Recall that: x=340-2y

x=340-2(85)=340-170=170 feet

Since x=170 feet, y=85 feet

The dimension of the park which maximizes the area are: 85 feet by 170 feet.

Furthermore, the part opposite the existing fence is 170 feet.

Final answer:

To maximize the area of the dog park with 340 feet of fencing using one existing fence side, an optimization problem is solved where the park's width and length are calculated. The area is maximized by setting the park length to be the longest along the existing fence and finding the width accordingly.

Explanation:

The question asks for the dimensions of the dog park Patricia can build with 340 feet of fencing and utilizing one side of the existing city park's fence to maximize the area. This is a problem of optimization that involves finding the maximum area of a rectangle given the perimeter. Since one side is already fenced, we only need to fence three sides. The perimeter P of three sides is 2w + l = 340 (where w is the width and l is the length we need to find and fencing for). To maximize the area, A = w * l, we use calculus or recognize this as a problem of a fixed perimeter rectangle, where the area is maximized when the rectangle is a square, i.e., the width equals the length.

However, since one side is already existing, Patricia can only maximize the area by setting 2w + l = 340, meaning the park would be longest along the existing fence. By rearranging, l = 340 - 2w, and substituting in the area formula, A = w(340 - 2w), we get a quadratic equation which represents a parabola that opens downwards, meaning its vertex represents the maximum point. Completing the square or using calculus to find the derivative and set it to zero will give us the optimal width, and thus, the optimal length to maximize area.

Answer:

Probability of event

0.34

Step-by-step explanation:

Answer:

The probability density function of X is:

[tex]f_{X}(x)=\frac{1}{15-5}=\frac{1}{10};\ 5<X<15[/tex]

Step-by-step explanation:

A continuous Uniform distribution is the probability distribution of a random outcome of an experiment that lies with certain specific bounds.

Consider that random variable X follows a continuous Uniform distribution and the value of X lies between a and b.

The probability density function of the random variable X is:

[tex]f_{X}(x)=\frac{1}{b-a};\ a<X<b,\ a<b[/tex]

Now, in this case it is provided that the amount of salad taken is uniformly distributed between 5 ounces and 15 ounces.

The random variable X is defined as:

Χ = Salad plate filling weight.

The probability density function of the salad plate filling weight is:

[tex]f_{X}(x)=\frac{1}{15-5}=\frac{1}{10};\ 5<X<15[/tex]

In mathematics at a college level, this answer explains how to find the probability density function of a uniformly distributed random variable representing the amount of salad taken by customers at TAB.

Given: Customers at TAB take salad that is uniformly distributed between 5 ounces and 15 ounces. Let X = Salad plate filling weight.

i. Find the probability density function of X: Since the salad amount is uniformly distributed, the probability density function is a horizontal line, given by f(x) = 1/(b-a), where a = 5 and b = 15, so f(x) = 1/10 for 5 ≤ x ≤ 15.

Answer:

We conclude that the proportion of dropouts has changed from the historical value of 0.081.

Step-by-step explanation:

We are given that in 2009, the high school dropout rate was 8.1%. A polling company recently took a survey of 1000 people between the ages of 16 and 24 and found 6.5% of them are high school dropouts.

The polling company would like to determine whether the proportion of dropouts has changed from the historical value of 0.081.

Let p = proportion of school dropouts rate

SO, Null Hypothesis, [tex]H_0[/tex] : p = 0.081   {means that the proportion of dropouts has not changed from the historical value of 0.081}

Alternate Hypothesis, [tex]H_A[/tex] : p [tex]\neq[/tex] 0.081   {means that the proportion of dropouts has changed from the historical value of 0.081}

The test statistics that will be used here is One-sample z proportion statistics;

             T.S.  = [tex]\frac{\hat p-p}{{\sqrt{\frac{\hat p(1-\hat p)}{n} } } } }[/tex]  ~ N(0,1)

where, [tex]\hat p[/tex]  = sample proportion of high school dropout rate = 6.5%

            n = sample of people = 1000

So, test statistics  =   [tex]\frac{0.065-0.081}{{\sqrt{\frac{0.065(1-0.065)}{1000} } } } }[/tex] 

                               =  -2.05

Also, P-value is given by the following formula;

         P-value = P(Z < -2.05) = 1 - P(Z [tex]\leq[/tex] 2.05)

                                              = 1 - 0.97982 = 0.0202 or 2.02%

Now at 5% significance level, the z table gives critical values between -1.96 and 1.96 for two-tailed test. Since our test statistics does not lies within the range of critical values of z so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region.

Therefore, we conclude that the proportion of dropouts has changed from the historical value of 0.081.

At a 5% significance level, the survey's dropout rate of 6.5% does not significantly differ from the historical rate of 8.1%, based on the calculated test statistic and p-value.

To determine whether the proportion of dropouts has changed from the historical rate of 0.081, we set up hypotheses:

Null Hypothesis (H0): The dropout rate in the survey (p) equals the historical rate (0.081).

Alternative Hypothesis (Ha): The dropout rate in the survey (p) is not equal to the historical rate (0.081).

Using a z-test for proportions, we calculate the test statistic:

Z = (0.065 - 0.081) / sqrt((0.081 * (1 - 0.081)) / 1000) ≈ -1.81

Next, we find the p-value associated with Z, which is approximately 0.0708.

Since 0.0708 > 0.05 (the significance level), we fail to reject the null hypothesis.

Conclusion: At the 5% significance level, there's insufficient evidence to suggest that the dropout rate in the survey differs from the historical rate of 0.081.

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Answer:

[tex]z=\frac{0.40 -0.45}{\sqrt{\frac{0.45(1-0.45)}{370}}}=-1.933[/tex]  

[tex]p_v =2*P(z<-1.933)=0.0532[/tex]  

Step-by-step explanation:

Information given

n=370 represent the sample selected

[tex]\hat p=0.4[/tex] estimated proportion of  readers owned a laptop

[tex]p_o=0.45[/tex] is the value that we want to test

z would represent the statistic

[tex]p_v[/tex] represent the p value

Creating the hypothesis

We need to conduct a hypothesis in order to test if the true proportion of readers owned a laptop is different from 0.45, the system of hypothesis are:  

Null hypothesis:[tex]p=0.45[/tex]  

Alternative hypothesis:[tex]p \neq 0.45[/tex]  

The statistic is:

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

Replacing we got:

[tex]z=\frac{0.40 -0.45}{\sqrt{\frac{0.45(1-0.45)}{370}}}=-1.933[/tex]  

Calculating the p value  

We have a bilateral test so then the p value would be:

[tex]p_v =2*P(z<-1.933)=0.0532[/tex]  

Answer:

17

Step-by-step explanation:

(X-4)2+(y+12)2=17^2

radius is square root of  17^2

Answer:

Blank 1: 20 cups of flour

Blank 2: 18.75 cups of flour or 18 3/4 cups of flour.

Step-by-step explanation:

Blank 1:  It tells you for every 8 tablespoons of honey they need 10 cups of flour. 16 table spoons of honey is double the 8 tablespoons of honey so you would also double the amount of flour.

2 x 10= 20.

Blank 2: You can find out how much cups of flour is needed for one table spoon of honey by dividing the cups of flour by 8 which gives you 1 and 1/4 or 1.25. Then you can multiply that by 15 for the 15 tablespoons of honey.

1.25 x 15 = 18.75

1 1/4 x 15 = 75/4 or 18 3/4.

Using the given ratio of 8 tablespoons of honey to 10 cups of flour, 16 tablespoons of honey requires 20 cups of flour and 15 tablespoons of honey requires 18.75 cups of flour.

The bakery's recipe uses a constant ratio of 8 tablespoons of honey to 10 cups of flour. To find how much flour is needed for different amounts of honey, we use this ratio as a guide.

For 16 tablespoons of honey (which is twice the original amount), we also double the amount of flour, which gives us 20 cups of flour (10 cups*2).

For 15 tablespoons of honey, the situation is not as straightforward and we need to establish a proportion. This requires us to set up the equation as follows: (8 tablespoons of honey : 10 cups of flour = 15 tablespoons of honey : X cups of flour). Solving for X, we get X = 18.75 cups of flour.

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Answer:

Required total charge is [tex]256\pi[/tex] coulombs per square meter.

Step-by-step explanation:

Given electric charge is dristributed over the disk,

[tex]x^2=y^2\leq 16[/tex] so that the charge density at (x,y) is,

[tex]\rho (x,y)=2x+2y+2x^2+2y^2[/tex]

To find total charge on the disk let Q be the total charge and [tex]x=r\cos\theta,y=r\sin\theta[/tex] so that,

[tex]Q={\int\int}_Q\rho(x,y) dA[/tex]                where A is the surface of disk.

[tex]=\int_{0}^{2\pi}\int_{0}^{4}(2x+2y+2x^2+2y^2)dA[/tex]

[tex]=\int_{0}^{2\pi}\int_{0}^{4}(2r\cos\theta+2r\sin\theta+2r^2 \cos^{2}\theta+2r^2\sin^2\theta)rdrd\theta[/tex]

[tex]=2\int_{0}^{2\pi}\int_{0}^{4}r^2(\cos\theta+\sin\theta)drd\theta+2\int_{0}^{2\pi}\int_{0}^{4}r^3drd\theta[/tex]

[tex]=\frac{2}{3}\int_{0}^{2\pi}(\sin\theta+\cos\theta)\Big[r^3\Big]_{0}^{4}d\theta+2\int_{0}^{2\pi}\Big[\frac{r^4}{4}\Big]d\theta[/tex]

[tex]=\frac{128}{3}\int_{0}^{2\pi}(\sin\theta+\cos\theta)d\theta+128\int_{0}^{2\pi}d\theta[/tex]

[tex]=\frac{128}{3}\Big[\sin\theta-\cos\theta\Big]_{0}^{2\pi}+128\times 2\pi[/tex]

[tex]=\frac{128}{3}\Big[\sin 2\pi-\cos 2\pi-\sin 0+\cos 0\Big]+256\pi[/tex]

[tex]=256\pi[/tex]

Hence total charge is [tex]256\pi[/tex] coulombs per square meter.

To find the total charge on a disk with a given charge density, we need to integrate the charge density over the entire area of the disk. In this case, the charge density is not constant, so we convert the cartesian coordinates to polar coordinates which simplifies the integral. The final solution is obtained by performing a double integral over r and θ.

The student is asking for the total electric charge on a disk with a given charge density. The disk being referred to is defined by the equation x² + y² ≤ 16, i.e., a disk of radius 4 units, and the charge density at any point (x, y) on the disk is given by rho(x, y) = 2x + 2y + 2x² + 2y².

In such problems, we find the total charge by integrating the charge density over the entire area of the disk. But in this case, we first need to convert the cartesian coordinates (x, y) to polar coordinates. which simplifies the integral. In polar coordinates, the area element is r dr dθ and the charge density funciton becomes rho(r, θ) = 2r cosθ + 2r sinθ + 2r². We perform a double integral of this function over r from 0 to 4 and θ from 0 to 2π.

This is a classic example of a problem in electrostatics, particularly involving the calculation of electric charge given a non-uniform charge density.

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