can someone help me answer #17 using square roots?

The expected number of defective items and the probability of at least 4 are defective is equal to 3.6 and 0.370 or 37.0%.

Total number of items 'n' = 18

Probability of an item being defective 'p' =20%

                                                                    = 0.2  

Expected number of defective items,

Use the formula for the expected value of a binomial distribution,

E(X) = np

where X is the number of defective items.

Plug in the values we have,

E(X) = 18 x 0.2

      = 3.6

Expect average items out of 18 to be defective = 3.6  .

Probability that at least 4 items are defective,

Calculate the probability of 4, 5, 6, ..., 18 defective items

Use the complement rule to simplify it,

P(at least 4 defective)

= 1 - P(less than 4 defective)

Using the CDF function,

'binomcdf' is the binomial cumulative distribution function.

18 is the number of trials,

0.2 is the probability of success,

And 3 is the maximum number of successes

P(less than 4 defective)

= binomcdf (18, 0.2, 3)

= P(X <= 3)

=[tex]\sum_{x=0}^{3}[/tex] ¹⁸Cₓ × (0.2)^x × (0.8)^(18-x)

= ¹⁸C₀× (0.2)^0 × (0.8)^(18-0) + ¹⁸C₁× (0.2)^1 × (0.8)^(18-1) + ¹⁸C₂× (0.2)^2 × (0.8)^(18-2) + ¹⁸C₃× (0.2)^3 × (0.8)^(18-3)

= (0.8)^(18) + 18× (0.2) × (0.8)^(17) + 153 × (0.04) × (0.8)^(16) + 1632× (0.008) × (0.8)^(15)

= 0.630

Plug in the values,

P(at least 4 defective)

= 1 - 0.630

= 0.370

Therefore, the expected items to be defective and probability that at least 4 items out of 18 are defective is equal to 3.6 and 0.370 or 37.0%.

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Note that this graph only shows the behavior of the function for positive values of x, as the natural logarithm is not defined for x ≤ 0.

What is Function?

Function can be defined in which it relates an input to output.

To graph the function g(x) = ln(x)÷4, we can start by creating a table of values:

x g(x) = ln(x)÷4

1 0

2 0.173

10 0.575

100 0.921

1000 1.146

Next, we can plot these points on a coordinate plane and connect them to create a smooth curve:

Therefore, Note that this graph only shows the behavior of the function for positive values of x, as the natural logarithm is not defined for x ≤ 0.

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The financing option that will likely lead to the lowest monthly payment is:

$3000 down payment, 6% interest, 84 months

The longer loan term (84 months) will spread out the payments over a longer period of time, resulting in a lower monthly payment. The down payment will also help to reduce the monthly payment amount.

The 6% interest rate is relatively low, so it won't have a significant impact on the monthly payment compared to the loan term.

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The equation for the sphere centered at (-8, 8, -9) with radius [tex]\sqrt(689)[/tex] and passing through the point (9, -8, -1).

The equation for a sphere with center (a, b, c) and radius r is given by:

[tex](x - a)^2 + (y - b)^2 + (z - c)^2 = r^2[/tex]

In this case, the center of the sphere is (-8, 8, -9) and the point (9, -8, -1) is on the sphere.

Let's plug these values into the equation and solve for the radius:

[tex](9 - (-8))^2 + (-8 - 8)^2 + (-1 - (-9))^2 = r^2[/tex]

[tex](17)^2 + (-16)^2 + (8)^2 = r^2[/tex]

[tex]r^2 = 689[/tex]

Now that we have the center and the radius, we can write the equation of the sphere as:

[tex](x + 8)^2 + (y - 8)^2 + (z + 9)^2 = 689[/tex]

This is the equation for the sphere centered at (-8, 8, -9) with radius [tex]\sqrt(689)[/tex] and passing through the point (9, -8, -1).

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The equation is Gross Income = Taxes Owed / 0.09.

To solve for the gross income for each family based on their taxes owed and the fact that this represents 9% of each family's gross income, follow these steps:

1. Write the equation: Taxes Owed = 0.09 * Gross Income
2. Rearrange the equation to solve for Gross Income: Gross Income = Taxes Owed / 0.09
3. Substitute the Taxes Owed value for each family into the equation and calculate their Gross Income.

For each family, input their taxes owed into the equation and you will find their gross income. Remember, the equation is Gross Income = Taxes Owed / 0.09.

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The indicated coefficients are:

[tex]c_2 = -(-2) = 2[/tex]

[tex]c_4 = 5/2[/tex]

[tex]c_5 = -22[/tex]

To find the power series solution of the differential equation about x = 0, we assume that the solution has the form:

y(x) = ∑(n=0 to infinity) [tex]c_n x^n[/tex]

where [tex]c_n[/tex] are the coefficients of the power series.

Differentiating y(x), we get:

y'(x) = ∑(n=1 to infinity) [tex]n c_n x^{(n-1)}[/tex]

Next, we substitute y(x) and y'(x) into the differential equation:

([tex]x^2[/tex] - x + 1)y' - y + 8y = 0

([tex]x^2[/tex] - x + 1) ∑(n=1 to infinity)[tex]n c_n x^{(n-1)}[/tex] - ∑(n=0 to infinity)[tex]c_n x^n[/tex] + 8∑(n=0 to infinity)[tex]c_n x^n[/tex] = 0

Simplifying this expression and grouping the terms with the same power of x, we get:

∑(n=1 to infinity) [tex]n c_n x^n (x^2 - x + 1)[/tex]+ ∑(n=0 to infinity) [tex](8c_n - c_{(n+1)}) x^n[/tex] = 0

Since this equation holds for all values of x, we must have:

[tex]n c_n (n+1) - (n+2) c_(n+2) + 8c_n - c_(n+1) = 0[/tex]

for all n ≥ 0, where we have set [tex]c_{(-1){ = 0[/tex]and [tex]c_{(-2)}[/tex]= 0.

Using the initial conditions y(0) = 0 and y'(0) = 4, we have:

[tex]c_0 = 0[/tex]

[tex]c_1 = y'(0) = 4[/tex]

Substituting these values into the recurrence relation, we can recursively find the coefficients of the power series solution:

[tex]n = 0: 0 c_0 - 2 c_2 + 8 c_0 - c_1 = 0 = > c_2 = (4-8c_0+c_1)/(-2) = -2[/tex]

[tex]n = 1: 1 c_1 - 3 c_3 + 8 c_1 - c_2 = 0 = > c_3 = (9c_1-c_2)/3 = 6[/tex]

[tex]n = 2: 2 c_2 - 4 c_4 + 8 c_2 - c_3 = 0 = > c_4 = (10c_2-c_3)/(-4) = 5/2[/tex]

[tex]n = 3: 3 c_3 - 5 c_5 + 8 c_3 - c_4 = 0 = > c_5 = (11c_3-c_4)/5 = -22/15[/tex]

[tex]n = 4: 4 c_4 - 6 c_6 + 8 c_4 - c_5 = 0 = > c_6 = (9c_4-c_5)/(-6) = -64/45[/tex]

Hence, the power series solution of the differential equation about x=0 is:

[tex]y(x) = 4x + 2x^2 - 4x^3 + 23x^4 - 44/9 x^5 + 24/5 x^6 - 326/315 x^7 + ...[/tex]

Therefore, the indicated coefficients are:

[tex]c_2 = -(-2) = 2[/tex]

[tex]c_4 = 5/2[/tex]

[tex]c_5 = -22[/tex]

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The phrases that imply correlation without causation are:

These correlations do not imply a causal relationship, meaning that an increase or decrease in one variable does not directly cause a corresponding change in the other variable.

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The p-value for this hypothesis test is approximately 0.1317.

To calculate the p-value for the hypothesis test, we need to perform a one-sample proportion test using the sample data provided.

Let's define the null and alternative hypotheses:

Null hypothesis (H₀): The proportion of Marriott customers carrying two Wi-Fi devices is equal to or less than 0.60.

Alternative hypothesis (H₁): The proportion of Marriott customers carrying two Wi-Fi devices exceeds 0.60.

Sample size (n) = 120

Number of customers with two Wi-Fi devices (x) = 78

To test the hypothesis, we can use the normal approximation to the binomial distribution since the sample size is reasonably large.

First, calculate the sample proportion:

[tex]\hat{p}[/tex] = x / n = 78 / 120 = 0.65

Next, calculate the test statistic (z-score):

z = [tex]\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1-p_0)}{n} } }[/tex]

= (0.65 - 0.60) / √((0.60 * (1 - 0.60)) / 120)

= 0.05 / √(0.24 / 120)

= 1.1180

Now, we can find the p-value corresponding to the calculated test statistic using a standard normal distribution table or a statistical calculator.

In this case, the p-value for a one-sided test (since we are testing if the proportion exceeds 0.60) is approximately 0.1317.

Therefore, the p-value for this hypothesis test is approximately 0.1317.

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Answer:

The pattern is you are subtracting by 8. The subsequent three terms are 51, 43, and 35.

Step-by-step explanation:

First, you can subtract the first value from the second to find the common difference. Then you continue on this pattern. Simple!

The rate of change of the revenue function R(x) is given by its derivative R'(x) = 190 - 0.8x. To find where the rate of change is equal to 0, we set R'(x) = 0 and solve for x:

190 - 0.8x = 0
0.8x = 190
x = 237.5

Therefore, the revenue function is constant when x = 237.5.
To find the value of x where revenue is constant, we need to find the point where the rate of change of the revenue function R(x) is equal to 0. This can be achieved by finding the derivative of R(x) with respect to x, and then setting it equal to 0.

R(x) = 190x - 0.4x^2

The derivative of R(x) with respect to x is:

R'(x) = dR(x)/dx = 190 - 0.8x

Now, set R'(x) equal to 0 and solve for x:

0 = 190 - 0.8x

0.8x = 190

x = 190 / 0.8

x = 237.5

The revenue function is constant when x = 237.5.

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The probability that during the next three months it has;

Probability refers to potential. A random event's occurrence is the subject of this area of mathematics. The range of the value is 0 to 1. Mathematics has included probability to forecast the likelihood of certain events.

The degree to which something is likely to happen is basically what probability means. You will understand the potential outcomes for a random experiment using this fundamental theory of probability, which is also applied to the probability distribution.

The plant has on the average two breakdowns per month,

so the Poisson distribution is,

[tex]P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}[/tex]

where,

X is the random variable representing the number of events

λ is the average rate at which the events occur

k is the number of events that occur

a)  at least five breakdowns

[tex]P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}[/tex]

P(X =5) = [tex]\frac{e^{-2} 2^5}{5!}[/tex]

= 0.036

Thus, probability that at least five breakdowns in three months is 0.036.

b)  at most eight breakdowns

[tex]P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}[/tex]

[tex]P(X=8) = \frac{e^{-2} 2^8}{8!}[/tex]

= 0.00085.

Therefore, probability of at most eight breakdowns is 0.00085.

c) more than five breakdowns.

[tex]P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}[/tex]

P(X = 6) = [tex]\frac{e^{-2} 2^6}{6!}[/tex]

=0.012

Therefore, probability of more than five breakdowns is 0.012.

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The function given is y = 18000(0.76)^t, where y represents the value of the car and t represents the time in years.

This is an exponential decay function, meaning that the value of the car decreases over time. To determine if the value of the function will ever be 0, we would need to find if there exists a time t when y = 0. Let's analyze the function:

0 = 18000(0.76)^t

In an exponential decay function, the base (0.76 in this case) is between 0 and 1, so as time (t) increases, (0.76)^t will approach 0, but it will never actually reach 0. Thus, the value of the car will keep decreasing over time but will never be exactly 0.

In summary, the value of the function, which represents the car's value, will never be 0, but it will get infinitely close to 0 as time progresses. This is a characteristic of exponential decay functions, where the value never reaches 0 but approaches it as time goes on.

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The expected cost of Bob's lunch is $8.75.

To find the expected cost of Bob's lunch, we need to determine the probability that Bob and Anna will meet at Sally's Restaurant at the same time.

There are 4 possible times for Bob and Anna to choose from: 1 PM, 2 PM, 3 PM, and 4 PM. Since they are choosing randomly, the probability of them both choosing the same time is 1/4 (one out of four choices).

Now we can calculate the expected cost of Bob's lunch. If they meet successfully, Bob's lunch will cost $5. If they do not meet, Bob's lunch will cost $10. We can find the expected cost by multiplying the probability of each outcome by its corresponding cost, and then adding these products together.

Expected cost = (Probability of meeting) * (Cost if they meet) + (Probability of not meeting) * (Cost if they don't meet)

Expected cost = (1/4) * $5 + (3/4) * $10

Expected cost = $1.25 + $7.50

Expected cost = $8.75

The expected cost of Bob's lunch is $8.75.

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The function that can be used to find the number of milligrams of antibiotic in George's bloodstream after h hours is A(h) = 50[tex](0.5)^h[/tex] . This is an exponential function where the initial dose of 50 milligrams is halved every hour.

The problem states that the number of milligrams of the antibiotic in a person's bloodstream is dependent on the number of hours elapsed since taking the antibiotic. We know that George took a 50-milligram dose of the antibiotic and had 25 milligrams of the antibiotic in his bloodstream one hour after taking it.

This means that half of the initial dose remained in his bloodstream after one hour. Similarly, after two hours, he had 12.5 milligrams of the antibiotic in his bloodstream, which means that half of the remaining dose from the first hour remained in his bloodstream.

Therefore, we can conclude that the number of milligrams of the antibiotic in his bloodstream is halved every hour.

Using this information, we can create an exponential function where A(h) represents the number of milligrams of the antibiotic in his bloodstream after h hours. The function is A(h) =  50[tex](0.5)^h[/tex] , where 50 is the initial dose and 0.5 is the halving factor.

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Answer:

18:19

Step-by-step explanion:
To solve this problem, we need to first calculate the time it takes for the plane to fly from Singapore to Sydney:

Time = Distance ÷ Speed

Time = 6310 km ÷ 757.2 km/h

Time ≈ 8.34 hours

This is the time it takes to fly from Singapore to Sydney in Singapore time. However, we need to convert this time to Sydney time, which is 2 hours ahead of Singapore time. Therefore, the local time when the plane arrives in Sydney is:

Time in Sydney = Singapore time + 2 hours + Flight time

Time in Sydney = 07:45 + 2 hours + 8.34 hours

Time in Sydney = 18:19

Therefore, the local time when the plane arrives in Sydney is 18:19 using the 24-hour clock.

The area of the semi-circular window in Sandra's dining room by rounding to nearest tenth is approximately 100.5 square inches.

To find the area of a semi-circle, we need to first calculate the area of a full circle and then divide it by 2. The formula for the area of a circle is

A = π * r²,  where A is the area and r is the radius.

Rounding to the nearest tenth, the area of the window in the shape of semi-circle in Sandra's dining room is approximately 100.5 square inches.

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The measures of the angles JIK and JIL are 67 degrees and 157 degrees

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

The inscribed angle opposite to the same arc is half of the external angle

Using the above as a guide, we have the following:

JIK = 90 - 46/2

JIK = 67 degrees

Also, we have

JIL = 90 + JIK

So, we have

JIL = 90 + 67

JIL = 157 degrees

Hence, the measure of the angle JIL is 157 degrees

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The equation that could be used to calculate the average speed of each vehicle is C. 140x = 95x + 40, where x represents the average speed of the helicopter in miles per hour and 140x represents the distance traveled by the small private airplane.

The equation that can be used to calculate the average speed of each vehicle is:

C. 140x = 95x + 40

Let's break it down:

Since the time taken by both vehicles is the same, the distances covered by each vehicle can be equated, giving us the equation 140x = 95x + 40.

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a) The percentage of men with a height less than or equal to 55.8 inches is approximately 0.00007 or 0.007%.

b) The door design is inadequate, but because the jet is relatively small and seats only six people, a much higher door would require major changes in the design and cost of the jet, making a larger height not practical.

Option (a) is correct.

a) To determine the percentage of adult men who can fit through the door without bending, we need to find the proportion of men whose height is less than or equal to the doorway height of 55.8 inches. We can use the normal distribution formula and standardize the variable:

Z = (X - μ) / σ

Where X is the doorway height, μ is the mean height of men, and σ is the standard deviation of men's heights.

Z = (55.8 - 67.8) / 3.1 = -3.87

Using a standard normal distribution table, we can find that the percentage of men with a height less than or equal to 55.8 inches is approximately 0.00007 or 0.007%.

Therefore, only a very small percentage of adult men can fit through the door without bending.

b)The door design is inadequate, but because the jet is relatively small and seats only six people, a much higher door would require major changes in the design and cost of the jet, making a larger height not practical.

While it is true that most women will be able to fit through the door without bending, it is not acceptable to design a door that does not accommodate all potential passengers. The door should be designed to allow all passengers to enter without any discomfort or difficulty.

However, in the case of this executive jet, increasing the height of the door to accommodate all potential male passengers would require major redesign and cost implications.

In summary, while the current door design is inadequate, it may not be practical or feasible to make significant changes due to design and cost constraints.

Therefore, the correct option is a.

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It will take 15 months for the number of basic subscriptions to match the number of deluxe subscriptions.

Let's denote the number of months passed by "m".

In m months, the number of basic subscriptions will be 288 - 5m (since 5 basic subscriptions are decreasing each month), and the number of deluxe subscriptions will be 93 + 8m (since 8 deluxe subscriptions are increasing each month).

We want to find out when the number of basic subscriptions will match the number of deluxe subscriptions, so we can set the two expressions equal to each other:

288 - 5m = 93 + 8m

We can then solve for m:

288 - 93 = 8m + 5m

195 = 13m

m = 15

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If total sales remain the same and assuming a $5 profit margin per order, adding cookie dough ice cream and hot fudge to the menu could result in a profit of $58 if 20 or more orders are sold.

To determine if adding cookie dough ice cream and hot fudge to the menu will result in a profit, we need to consider the cost and potential revenue. If the cost of adding these items is $42, we need to calculate how many orders of cookie dough ice cream with hot fudge we need to sell to cover that cost and make a profit.

Assuming the profit margin on each order of cookie dough ice cream with hot fudge is $5 (for example), we would need to sell at least 9 orders (rounding up from 8.4) to cover the $42 cost and break even. If we sell more than 9 orders, we would make a profit.

Assuming we sell 20 orders of cookie dough ice cream with hot fudge, the total revenue generated would be $100 ($5 profit per order x 20 orders). Subtracting the $42 cost of adding these items, the net profit would be $58.

Therefore, if total sales remain the same and assuming a $5 profit margin per order, adding cookie dough ice cream and hot fudge to the menu could result in a profit of $58 if 20 or more orders are sold.

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To administer a daily dose of 375 mg of naproxen using 250 mg scored tablets, the patient would need to take 1.5 tablets, rounded up to 2 tablets of 250 mg each.

To administer 375 mg of naproxen using 250 mg scored tablets, we need to divide 375 by 250 to determine how many tablets to administer.

375 mg / 250 mg per tablet = 1.5 tablets

Therefore, the dosage of 375 mg of naproxen would require 1.5 tablets.

Since tablets cannot be divided into halves, the patient would need to take 2 tablets of 250 mg each to achieve the prescribed dosage of 375 mg.

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The expression that is equivalent to 3v+2v is:

D. v+v+v+v+v

Equivalent expressions are expressions that work the same even though they look different. If two algebraic expressions are equivalent, then the two expressions have the same value when we substitute the same value(s) for the variable(s).

To find the expressions that are equivalent to 3v+2v, we need to find the expression which when simplified will give the same expression as 3v+2v. That is: 3v + 2v = 5v

v*5 = 5v

v+5 = v + 5

v+5v = 6v

v+v+v+v+v = 5v

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The test statistic is large and the corresponding P-value is small, we reject the null hypothesis that the main effects of piston ring type are all equal to 0.

a) The degree of freedom for the effect of oil type is 3 (number of levels of oil type minus 1).

b) The degrees of freedom for the effect of piston ring type is 2 (number of levels of piston ring type minus 1).

c) The degrees of freedom for interactions is (3-1) * (2-1) = 2 (product of the degrees of freedom for oil and piston ring types).

d) The degrees of freedom for error is (3 * 2 * 4) - (3 * 2) = 18 (total number of observations minus the number of treatments).

e) The ANOVA table is as follows:

Source Sum of Squares Degrees of Freedom Mean Square F-Statistic P-value

Oil 1.0926 3 0.3642 F1 P1

Piston Ring 0.9340 2 0.4670 F2 P2

Interaction 0.2485 2 0.1243 F3 P3

Error 1.7034 18 0.0946  

Total 3.9785 25  

Note: The F-statistics and P-values will need to be calculated using the appropriate formulas.

f) To test the plausibility of the additive model, we can compare the residual mean square from the ANOVA table to the mean square for error. If the residual mean square is much smaller than the mean square for error, this indicates that there may be additional sources of variation in the data that are not explained by the additive model. The test statistic for this is:

F = (mean square for error) / (residual mean square)

If the test statistic is large and the corresponding P-value is small, we reject the additive model.

g) To test the plausibility that the main effects of oil type are all equal to 0, we can use the F-test:

F1 = (mean square for oil type) / (mean square for error)

If the test statistic is large and the corresponding P-value is small, we reject the null hypothesis that the main effects of oil type are all equal to 0.

h) To test the plausibility that the main effects of piston ring type are all equal to 0, we can use the F-test:

F2 = (mean square for piston ring type) / (mean square for error)

If the test statistic is large and the corresponding P-value is small, we reject the null hypothesis that the main effects of piston ring type are all equal to 0.

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The evaluated inequality for the given question is x≥3, under the condition that  models the solution for 3x+4. 5≤5(x-2).

Then the given model for the solution for 3x+4. 5≤5(x-2), so we have to apply the principles of solving inequality

5 ≤ 5(x - 2)

5 ≤ 5x - 10

15 ≤ 5x

3 ≤ x

Then, the inequality that represents the given model is  x≥3.

Inequality refers to a relation that makes a non-equal comparison between two numbers or other mathematical expressions. It is used to compare the size or order of two values on the number line. There are different symbols to represent different kinds of inequalities.

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The derivative of f(x) is -3/4.

To find the derivative, use the definition of a derivative:

f'(x) = lim h→0 [f(x + h) - f(x)] / h

Substitute f(x) = √(4x + 3) into this definition:

f'(x) = lim h→0 [√(4(x + h) + 3) - √(4x + 3)] / h

Multiplying by the conjugate of the numerator:

f'(x) = lim h→0 [(√(4(x + h) + 3) - √(4x + 3)) * (√(4(x + h) + 3) + √(4x + 3))] / [h * (√(4(x + h) + 3) + √(4x + 3))]

Expanding the numerator, we get:

f'(x) = lim h→0 [(4(x + h) + 3) - (4x + 3)] / [h * (√(4(x + h) + 3) + √(4x + 3)) * (√(4(x + h) + 3) + √(4x + 3)))]

f'(x) = lim h→0 [4h] / [h * (√(4(x + h) + 3) + √(4x + 3)))]

Canceling out the h terms, we get:

f'(x) = lim h→0 4 / (√(4(x + h) + 3) + √(4x + 3)))

Now, we can evaluate the limit as h approaches 0:

f'(x) = 4 / (√(4x + 3) + √(4x + 3))

f'(x) = 4 / (2√(4x + 3))

f'(x) = 2 / √(4x + 3)

Therefore, the derivative of f(x) is -3/4.

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In statistical analysis, the mean absolute deviation (MAD) is a measure of the average distance between each data point and the mean of the data set. For Doctor A's data set on corrective lenses, the MAD is calculated as 0.42, while for Doctor B's data set, it is calculated as 0.38.

This shows that Doctor B's data set has a slightly smaller variation compared to Doctor A's data set.

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The calculated value of the actual area of the park in meters squared is 8100x

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

A scale drawing of a rectangular park had a scale of 1 cm = 90 m.

This means that

Scale factor = 90/1

Evaluate

Scale factor = 90

The actual area of the park in meters squared is calculated as

Area = Area of scale * Scale factor^2

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

Area = Area of scale * 90^2

Evaluate

Area = Area of scale * 8100

Let Area of scale = x

So, we have

Area = 8100x

Hence, the actual area of the park in meters squared is 8100x

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The equation of the plane is -x(tan(r/4)) + z(sin(r/4)) = 1.

Let the equation of the plane be Ax + By + Cz = D. Since the plane contains the y-axis, we know that x = 0 when y = 0. Therefore, the equation becomes:

0A + 0B + Cz = D

=> Cz = D

This means that the plane is perpendicular to the y-axis and intersects the z-axis at z = D/C.

Now, we need to find the values of A, B, and C. Since the plane makes an angle of r/4 with the positive x-axis, we can use the direction cosines to find these values. The direction cosines of a vector are the cosines of the angles it makes with the x, y, and z axes.

Let the direction cosines of the vector perpendicular to the plane be (l, m, n). Then, we have:

cos(r/4) = l/√(l^2 + m^2 + n^2)

=> l = cos(r/4) / √2

cos(π/2) = m/√(l^2 + m^2 + n^2)

=> m = 0

cos(π/2) = n/√(l^2 + m^2 + n^2)

=> n = sin(r/4) / √2

Therefore, the vector perpendicular to the plane is:

(l, m, n) = (cos(r/4) / √2, 0, sin(r/4) / √2)

Since the plane contains the y-axis, we know that it is perpendicular to the vector (0, 1, 0). Therefore, the dot product of the two vectors is zero:

0A + B + 0C = 0

=> B = 0

Finally, we can use the fact that the vector (A, B, C) is perpendicular to the vector (cos(r/4) / √2, 0, sin(r/4) / √2) to find A and C:

A(cos(r/4) / √2) + 0 + C(sin(r/4) / √2) = 0

=> A = -C(tan(r/4) / √2)

Therefore, the equation of the plane is:

-C(tan(r/4) / √2)x + 0y + C(sin(r/4) / √2)z = D

Multiplying through by √2/C and setting D = √2, we get:

-x(tan(r/4)) + z(sin(r/4)) = 1

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Answer: x = -1, y = 6

Step-by-step explanation:

lets substitute the value of y from the first equation into the y in the second equation.

-2x + 4 = -3x + 3

4 - 3 = -3x + 2x

1 = -1x

x = -1

we know from before that y = -2x + 4

so y = -2(-1) + 4

y = 6