true or false: prior probability is the initial probability based on the present level of information.

The theoretical probability of choosing a letter c is  0.24 and (b) the experimental probability for letter c is 0.4

You should be aware that theoretical Probability is defined as the ratio of the number of favorable outcomes to the number of possible outcomes

the given table is

A                 14

C                 5

C                  7

E                 15

L                 9

Total number  = 14+5+7+15+9 = 50

total of C = 5+7 = 12

the theoretical probability = number of requires outcome/number of possible outcomes

P(T) = 12/50 = 0.24

(b) The Experimental Probability is the ratio of the number of times an event occurs to the total number of trials or times the activity is performed

Number of times C occurred = 2

Total number of times = 5

P(E) = 2/5 = 0.4

The experimental probability is higher than the theoretical probabilty

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Answer:

A line graph

I think

Step-by-step explanation:

Answer:

The mean is (x + y + 20)/5, so x + y has to be a multiple of 5, with 6 < x, y < 10. If we let x = 7 and y = 8, the sum of the five numbers is 35, so the mean is 7, which is the same as the median. So x = 7 and

y = 8.

The mean thickness of each paper sheet is 0.0095 inch with a variance of 0.00003225. Approximately 45.45% of the sheets are less than 0.0095 inch thick, and 40% have a thickness greater than 0.0123 inch.

Mean and variance for thickness of each paper sheet

Since the thickness of each paper sheet is uniformly distributed between 0.004 and 0.015 inch, the mean thickness is the average of the minimum and maximum values

Mean = (0.004 + 0.015) / 2 = 0.0095 inch

The variance of a uniform distribution is given by

Variance = (b - a)^2 / 12

where a and b are the minimum and maximum values of the distribution, respectively. Substituting the values, we get

Variance = (0.015 - 0.004)^2 / 12 = 0.00003225

Therefore, the mean thickness of each paper sheet is 0.0095 inch, and the variance is 0.00003225.

Proportion of paper sheets which are less than 0.0095 inch thick

Since the thickness of each paper sheet is uniformly distributed between 0.004 and 0.015 inch, we can find the proportion of sheets that are less than 0.0095 inch by finding the area under the density function of the uniform distribution between 0.004 and 0.0095.

The density function of a uniform distribution is given by

f(x) = 1 / (b - a)

where a and b are the minimum and maximum values of the distribution, respectively. Substituting the values, we get

f(x) = 1 / (0.015 - 0.004) = 1/0.011

The area under the density function between 0.004 and 0.0095 gives the proportion of paper sheets that are less than 0.0095 inch thick:

Proportion = ∫(0.004 to 0.0095) f(x) dx = [x / (b - a)](0.0095 - 0.004) = 0.4545

Therefore, approximately 45.45% of the paper sheets are less than 0.0095 inch thick.

Thickness exceeded by 40 percent of the paper sheets

To find the thickness exceeded by 40% of the paper sheets, we need to find the value of x for which 40% of the area under the density function lies to the right of x.

Since the density function is uniform, the area to the right of x is given by

Area = (b - x) / (b - a)

We want to find the value of x for which this area is equal to 0.4. Substituting the values, we get

0.4 = (0.015 - x) / (0.015 - 0.004)

Solving for x, we get

x = 0.0123

Therefore, 40% of the paper sheets have a thickness greater than 0.0123 inch.

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The value of f(12, 4, 1) is  -11

(a) To evaluate f(12, 4, 1) for the function f(x, y, z) = y - x - 3z, plug in the given values for x, y, and z:

f(12, 4, 1) = 4 - 12 - 3(1) = 4 - 12 - 3 = -8 - 3 = -11.

So, f(12, 4, 1) = -11.

(b) To find the domain of f(x, y, z) = y - x - 3z, consider the possible values of x, y, and z. Since there are no restrictions on x, y, and z, the domain of f is:

-∞ < x < ∞, -∞ < y < ∞, -∞ < z < ∞

These inequalities represent that x, y, and z can be any real numbers.

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41 trees should be planted per acre to obtain the maximum yield per acre, and the maximum yield should be of 697 pound acre. So, the grower should plant 41 trees per acre to obtain the maximum yield of 697 pounds per acre.

To obtain the maximum yield per acre, we need to find the optimal number of trees to plant.

Let x be the number of additional trees planted per acre, so the total number of trees planted per acre will be 18 + x.

From the information given, we know that if x additional trees are planted, the average yield per tree will be reduced by x pounds.

Therefore, the total yield per acre will be (18 + x)(40 - x) pounds of cherries.

To find the maximum yield, we need to find the value of x that maximizes this expression. We can do this by taking the derivative of the expression with respect to x:

d/dx [(18 + x)(40 - x)] = 18 - 2x

Setting this equal to zero and solving for x, we get:

18 - 2x = 0
x = 9

Therefore, the optimal number of additional trees to plant per acre is 9, and the total number of trees to plant per acre is 27. The maximum yield per acre is:

(18 + 9)(40 - 9) = 27 * 31 = 837 pounds of cherries.

A commercial cherry grower estimates from past records that if 18 trees are planted per acre, each tree will yield an average of 40 pounds of cherries per season.

With each additional tree planted per acre (up to 23 additional trees), the average yield per tree is reduced by 1 pound.

To find the maximum yield per acre, we can create an equation for total yield as a function of the number of additional trees (x) planted per acre:

Total Yield (Y) = (18 + x) * (40 - x)

To find the maximum yield, we need to find the value of x that maximizes Y. The equation is a quadratic equation, which has a maximum value when x = -b/2a.

In this case, a = -1, and b = 58.

So: x = -58/(2 * -1) = 29

However, we can only plant up to 23 additional trees. Therefore, the grower should plant all 23 additional trees to maximize the yield per acre.

Now, we can find the maximum yield:

Y = (18 + 23) * (40 - 23) = 41 * 17 = 697 pounds per acre

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F. P(A/B) = P(A intersect B) / P(B)

Since A and B are independent, P(A intersect B) = P(A) × P(B)

P(A/B) = P(A) × P(B) / P(B) = P(A)

So, P(A/B) = P(A) = 0.05.

A. The probability of finding a customer who is single and male can be calculated as:

P(single and male) = number of single and male customers / total number of customers

P(single and male) = 20 / 200

P(single and male) = 0.10 or 10%

B. The probability that a female customer is single can be calculated using Bayes' theorem as:

P(single | female) = P(female and single) / P(female)

P(single | female) = 30 / (30 + 50)

P(single | female) = 0.375 or 37.5%

C. The probability that a customer is male or married can be calculated as:

P(male or married) = P(male) + P(married) - P(male and married)

P(male or married) = (20 + 100) / 200 + (100 + 50) / 200 - 0

P(male or married) = 0.85 or 85%

D. To check if marital status is independent of gender, we need to compare the joint probability of gender and marital status with the product of their marginal probabilities. If they are equal, then the events are independent. Let's calculate the probabilities:

P(male and single) = 20 / 200

P(male) = (20 + 100) / 200

P(single) = (20 + 30) / 200

P(female and single) = 30 / 200

P(female) = (30 + 50) / 200

P(married) = (100 + 50) / 200

P(male and single) = P(male) × P(single) if gender and marital status are independent

20 / 200 = (120 / 200) × (50 / 200)

0.10 = 0.30 × 0.25

Since the probabilities are not equal, gender and marital status are not independent.

E. Gender and marital status are not mutually exclusive because there are customers who can be both male and married, or female and single.

F. P(A/B) = P(A intersect B) / P(B)

Since A and B are independent, P(A intersect B) = P(A) × P(B)

P(A/B) = P(A) × P(B) / P(B) = P(A)

So, P(A/B) = P(A) = 0.05.

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Assuming that the random variable X is normally distributed with a mean (µ) of 100 and a standard deviation (σ) of 10, you are looking to compute the probability P(X < 108). Probability in this case was found to be 0.788

To find this probability, we can utilize the standard normal distribution (Z-distribution) by converting the given value (X = 108) into a Z-score using the following formula:  Z = (X - µ) / σ
plugging in the given values, we get:
Z = (108 - 100) / 10

Z = 0.8

Now, we can find the probability P(X < 108) by looking up the corresponding value in the Z-table, which gives us the area under the standard normal curve to the left of the Z-score. For Z = 0.8, the table value is approximately 0.7881. This means that the probability P(X < 108) is approximately 0.7881, or 78.81%.

Therefore, among the given options, 0.788 is the closest and most accurate answer for the probability P(X < 108) in a normally distributed random variable with a mean of 100 and a standard deviation of 10.

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To convert an arithmetic expression from infix to postfix form, we need to first read the expression in infix form and store it in an infix queue.

Then, we use an operator stack to keep track of the operators as we convert the expression to postfix form. We start by iterating through the infix queue and adding operands directly to the postfix queue. When we encounter an operator, we check its precedence against the operator on top of the operator stack.

If the precedence is higher, we push the operator onto the stack. Otherwise, we pop operators from the stack and add them to the postfix queue until the precedence of the top operator is lower than the current operator.

After we finish iterating through the infix queue, we pop any remaining operators from the stack and add them to the postfix queue. The resulting postfix queue contains the expression in postfix form.

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Answer:

We can use the fact that tan(-θ) = -tan(θ) to find the value of tan(-θ) from the given values of cos(θ) and sin(θ). First, we can use the Pythagorean identity to find the value of cos²(θ) + sin²(θ):

cos²(θ) + sin²(θ) = (0.8)² + (0.3)² = 0.64 + 0.09 = 0.73

Now, we can use the fact that cos²(θ) + sin²(θ) = 1 to find the value of cos(-θ) and sin(-θ):

cos(-θ) = cos(θ) = 0.8

sin(-θ) = -sin(θ) = -0.3

Finally, we can use the definition of tangent as the ratio of sine to cosine to find the value of tan(-θ):

tan(-θ) = -tan(θ) = -(sin(θ) / cos(θ)) = -0.3 / 0.8 = -0.375

Therefore, tan(-θ) = -0.375.

Step-by-step explanation:

Answer:

52mm

Step-by-step explanation:

(4x3)2+(4x4)+4x3= 52mm

triangle sides+square sides+bottom square side= answer

Hope this helps :)

An immediate inference can only be made based on a single premise that is either affirmative or particular.

An immediate inference is a type of logical deduction where a conclusion can be drawn directly from a single premise. This premise can be in the form of an affirmative statement (A) or a particular statement (I). However, it cannot be based on an 'E-statement (i.e. universal negative) as this type of statement does not provide enough information to draw a valid conclusion. Therefore, an immediate inference can only be made based on a single premise that is either affirmative or particular.

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Check the picture below.

so we know the interior angles in the polygons is 144°, so

[tex]\underset{in~degrees}{\textit{sum of all interior angles}}\\\\ n\theta = 180(n-2) ~~ \begin{cases} n=\stackrel{number~of}{sides}\\ \theta = \stackrel{degrees}{angle}\\[-0.5em] \hrulefill\\ \theta =144 \end{cases}\implies n(144)=180(n-2) \\\\\\ 144n=180n-360\implies 144n+360=180n \\\\\\ 360=36n\implies \cfrac{360}{36}=n\implies \boxed{10=n}[/tex]

The given statement "In scientific experiments, we measure a small number of items (a sample) and then attempt to infer the sample characteristics to the the entire population." is True. In scientific experiments, we often cannot measure an entire population, so we use statistical sampling to measure a smaller subset (a sample) and then try to generalize our findings to the entire population using statistical inference.

In scientific experiments, it is often not feasible or practical to measure the entire population. Instead, we collect a representative sample from the population and use statistical methods to infer the characteristics of the population from the sample.

This process is known as statistical inference. By carefully designing the sampling process and using appropriate statistical methods, we can make accurate and reliable inferences about the population.

Therefore, it is important to ensure that the sample is representative and unbiased and that the statistical methods used are appropriate for the data and research question at hand. The statement is true.

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A proper delivery location should have easy accessibility, adequate space for delivery vehicles, loading facilities, level surface, good lighting, and protection from weather conditions. Additional features may depend on the specific type of products being delivered.

Some essential features for a proper delivery location may include easy accessibility, sufficient space for delivery vehicles to maneuver, a loading dock or ramp, a level surface, adequate lighting, and protection from weather conditions. Other features may depend on the specific type of products being delivered, such as specialized equipment for handling fragile or hazardous materials, secure storage, or refrigeration.

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The complete question is :

What are the essential features for a proper delivery location ?

Because Old Navy has a higher standard deviation, we know that there is more variability in the wait times at Old Navy.

Overall, wait times at Old Navy are more spread out from the average; wait times at the Gap are closer to the average.
Based on the given information, we know that both Gap and Old Navy have an average wait time of 5 minutes. However, Gap has a standard deviation of 2 minutes, while Old Navy has a standard deviation of 5 minutes.

Because Old Navy has a higher standard deviation, we know that there is more variability in the wait times at Old Navy. Overall, wait times at Old Navy are more spread out from the average; wait times at the Gap are closer to the average.

In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of data values. It is a widely used measure of the variability or spread of a data set. The standard deviation is the square root of the variance, which is the average of the squared differences from the mean.

A higher standard deviation indicates that the data points are more spread out from the mean, while a lower standard deviation indicates that the data points are closer to the mean. The standard deviation is typically used in combination with the mean as a measure of central tendency to provide a more complete description of the data.

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Answer:

Step-by-step explanation:

Since PQRS is a square, the distance between P and R along the y-axis must be equal to the distance between Q and S along the y-axis.

We know that the side length of the square PQRS is 1 cm, so the distance between P and R (or between Q and S) along the x-axis is also 1 cm.

Therefore, we have:

b - d = distance between P and R along the y-axis

= distance between Q and S along the y-axis

Looking at the grid, we can see that the y-coordinate of point Q is b+1, and the y-coordinate of point S is d+1.

So, the distance between Q and S along the y-axis is:

(b+1) - (d+1) = b - d

Therefore, b - d = 0.

So, the value of b-d is 0.

Answer:

[tex] {x}^{2} + {y}^{2} = 25[/tex]

0 < theta < π

The solution is :

f(x) = (x - 3)(x + 1) → Corresponds with the first (raised higher ) ∪ shaped graph

f(x) = -2(x - 1)((x + 3) → Corresponds with the ∩ shaped graph

f(x) = 0.5(x - 6)((x + 2) → Corresponds with the second (lower) ∪ shaped graph

We have,

For the function f(x) = (x - 3)(x + 1)

We have;

When x = 0, y = -3

When y = 0 x = 3 or -1

Comparing with the graphs, it best suits the first ∪ shaped graph  that rises here than the other ∪ shaped graph

For the function;

f(x) = -2(x - 1)((x + 3)

When x = 0, y = 6

When y = 0, x = 1 or -3

Which corresponds with the ∩ shaped graph

For the function;

f(x) = 2(x + 6)((x - 2)

When x = 0, y = -24

When y = 0, x = -6 or 2

For the function;

f(x) = 0.5(x - 6)((x + 2)

When x = 0, y = -6

When y = 0, x = 6 or -2

Which best suits the second ∪ shaped graph  that is lower than the other (first) ∪ shaped graph

For the function;

f(x) = 0.5(x + 6)((x - 2)

When x = 0, y = -6

When y = 0, x = -6 or 2

For the function;

f(x) = (x + 3)((x - 1)

When x = 0, y = -3

When y = 0, x = -3 or 1

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The rate of cocaine relapse among chronic cocaine users can be tested using a chi-squared test for independence to compare observed frequencies of relapse and no relapse across different treatment groups. A two-sample z-test can be used to compare the relapse rate between Desipramine and placebo. A binomial test can be used to test if the probability of relapse is the same as the probability of no relapse for individuals treated with lithium.

Regarding the question about the rate of cocaine relapse among chronic cocaine users and the effect of different treatments, the hypotheses can be stated as follows:

Null hypothesis (H0): The rate of cocaine relapse is the same across the three treatment groups (desipramine, lithium, and placebo).

Alternative hypothesis (Ha): The rate of cocaine relapse is different across the three treatment groups.

To test these hypotheses, we can collect data on the number of patients who relapse and the number of patients who do not relapse in each treatment group and use a chi-squared test for independence to determine if there is a significant difference in the rate of relapse among the groups.

Regarding the question about the difference in relapse rate between desipramine and placebo, the hypotheses can be stated as follows:

Null hypothesis (H0): The rate of relapse is the same for patients treated with desipramine and placebo.

Alternative hypothesis (Ha): The rate of relapse is different for patients treated with desipramine and placebo.

To test these hypotheses, we can collect data on the number of patients who relapse and the number of patients who do not relapse in each treatment group and use a two-sample z-test for the difference in proportions to calculate a confidence interval for the difference in relapse rates between the two treatments.

Regarding the question about the probability of relapse for individuals treated with lithium, the hypotheses can be stated as follows:

Null hypothesis (H0): The probability of relapse is the same as the probability of no relapse for individuals treated with lithium.

Alternative hypothesis (Ha): The probability of relapse is different from the probability of no relapse for individuals treated with lithium.

To test these hypotheses, we can collect data on the number of individuals who relapse and the number of individuals who do not relapse among those treated with lithium and use a binomial test to compare the observed number of relapses to the expected number assuming equal probabilities of relapse and no relapse.

The rate of cocaine relapse among chronic cocaine users can be tested using a chi-squared test for independence to compare observed frequencies of relapse and no relapse across different treatment groups. A two-sample z-test can be used to compare the relapse rate between Desipramine and placebo. A binomial test can be used to test if the probability of relapse is the same as the probability of no relapse for individuals treated with lithium.

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The complete question is :

Can the rate of cocaine relapse among chronic cocaine users be influenced by treatment with desipramine, lithium, or placebo? Additionally, is there a significant difference in the rate of relapse between Desipramine and placebo? Finally, is the probability of relapse the same as the probability of no relapse for individuals treated with lithium?

To test whether the cocaine addicts taking desipramine, lithium, and placebo have a different rate of relapse, we can perform a hypothesis test using ? = 0.1. We can use a chi-squared test for independence to compare the observed frequencies of relapse and no relapse across the three treatment groups with the expected frequencies assuming no difference in relapse rates.

To compare the rate of relapse for Desipramine and placebo, we can construct a 95% confidence interval for the difference in relapse rates between the two treatments. We can use a two-sample z-test for the difference in proportions to calculate the confidence interval and determine if it includes zero. If the confidence interval does not include zero, then we can conclude that there is a significant difference in the rate of relapse between the two treatments.

To test whether the probability of relapse is the same as the probability of no relapse for individuals treated with lithium, we can perform a hypothesis test using ? = 0.5. We can use a binomial test to compare the observed number of relapses to the expected number assuming equal probabilities of relapse and no relapse. If the p-value is less than or equal to ? = 0.5, then we can reject the null hypothesis and conclude that the probability of relapse is not the same as the probability of no relapse for individuals treated with lithium.

Please note that the significance level alpha (α) is typically set at 0.05 or 0.01, not 0.1, for hypothesis testing.

Of the 80 vehicles that boarded at Woodport, the number of lorries is given as follows:

33 lorries.

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

Considering that there is a 26% probability of a vehicle on the ferry being a lorry, the outcomes are given as follows:

Hence the value of x is obtained as follows:

(58 + x)/350 = 0.26

58 + x = 0.26 x 350

x = 350(0.26) - 58

x = 33.

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Answer:

x = 8 cm

Step-by-step explanation:

Given:

A right triangle

The length of hypotenuse is equal to 10

The length of the shorter leg is equal to 6

Find: x - ? (the longer leg)

According to the Pythagorean theorem:

[tex] {x}^{2} = {10}^{2} - {6}^{2} [/tex]

[tex] {x}^{2} = 100 - 36 = 64[/tex]

[tex]x > 0[/tex]

[tex]x = \sqrt{64} = 8 \: cm[/tex]

Answer:

d. correct

Step-by-step explanation:

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A random variable is a numerical measure of the outcome from a probability experiment, so its value is determined by chance. Random variables are typically denoted using letters such as X. When experiments are conducted in a way such that the outcome is a numerical​ result, it is said that the outcome is a random variable. A numerical variable is a numerical measure of the outcome of a probability​ experiment, so its value is determined by chance.

A random variable is a numerical measure of the outcome from a probability experiment, so its value is determined by chance. Random variables are typically denoted using letters such as X. When experiments are conducted in a way such that the outcome is a numerical result, it is said that the outcome is a random variable. A numerical variable is a numerical measure of the outcome of a probability experiment, so its value is determined by chance.

In the mathematical language of measurement theory, the difference between variables is defined as the distance from the measurement point (called the sampling point) to the measurement point. This allows the forward measurement, called the distribution of the variable, to be determined; so the distribution is a measure of the probability of a set of all values ​​of a random variable. The two variables may differ but differ in important respects; so they will be free.

Consider the special cases of random variables and non-continuous random variables, corresponding to how random variables take values ​​in non-continuous problems (for example, as finite sets) or in the range of real numbers. There is another important point, especially in the theory of random processes, where it is important to determine arbitrary sequences or arbitrary functions.

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By answering the presented question, we may conclude that The expressions measurement "32-gallon" fish tank means that the fish tank can hold 32 gallons of water.

In mathematics, you can multiply, divide, add, or subtract. An expression is constructed as follows: Number, expression, and mathematical operator A mathematical expression is made up of numbers, variables, and functions (such as addition, subtraction, multiplication or division etc.) It is possible to contrast expressions and phrases. An expression or algebraic expression is any mathematical statement that has variables, integers, and an arithmetic operation between them. For example, the phrase 4m + 5 has the terms 4m and 5, as well as the provided expression's variable m, all separated by the arithmetic sign +.

The measurement "32-gallon" fish tank means that the fish tank can hold 32 gallons of water.

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The system of equations has infinite solutions.

Here we have the system of equations:

x + 2y = 0

4y = -2x

And we want to solve it graphically. To do so, we need to graph both of the equations in the same coordinate axis, and find the point where the graphs intercept.

In the graph below you can see that there is only one line, that happens because both of the equations represent the same line.

Then the system of equations has infinite solutions.

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Answer:

0.654 + 0.45 = 1.104

0.9 + 0.3 = 1.2

0.004 + 0.8342 = 0.8382

0.98 + 0.245 = 1.225

0.04 + 0.8532 = 0.8932

0.57 + 0.436 + 0.4321 = 1.4381

1.54 + 45.6 + 32.78 = 79.92

35.78 + 0.6 + 4.3 + 4.893 = 45.573

Step-by-step explanation:

If I missed one, please let me know. <3 Hope this helps

The x-intercepts of the parabola are approximately (0.8, 0) and (2.2, 0).

A parabola is a symmetrical curve with a U- or smile-shaped shape. It is a particular kind of conic section, a shape created by cutting a cone with a plane.

The set of all points in a plane that are equally distant from a fixed point (referred to as the focus) and a fixed line is known as a parabola (called the directrix).

To find the y-intercept of a parabola, we can set x=0 and solve for y.

y = x² - 3x + 7/4

When x=0:

y = (0)² - 3(0) + 7/4 = 7/4

So the y-intercept of the parabola is (0, 7/4).

To find the x-intercepts, we can set y=0 and solve for x:

0 = x² - 3x + 7/4

We can solve this quadratic equation using the quadratic formula:

[tex]x = \dfrac{[ -(-3) \pm\sqrt{( (-3^2- 4(1)(\dfrac{7}{4})} ) ]} { (2x_1)}[/tex]

[tex]x = \dfrac{[ 3 \pm \sqrt{( 9 - 7 )} ]} {2}[/tex]

[tex]x =\dfrac{ [ 3 \pm \sqrt{(2)} ]} { 2}[/tex]

So the x-intercepts of the parabola are approximately (0.8, 0) and (2.2, 0).

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The differential equation describing the cell phone sales is: dy/dt = (8/475)y(100 - y/4)

The logistic growth model is typically described by the following differential equation:

dy/dt = ry(1 - y/K)

where y is the population size, t is time, r is the intrinsic growth rate, and K is the carrying capacity (maximum population size).

In this case, we know that the carrying capacity is K = 100 (thousand cell phones). We also know that when t = 0 (i.e., at the beginning of sales), y = 0. Furthermore, we know that sales are increasing by 8 thousand phones per month when 38 thousand phones have been sold. This means that the growth rate, r, is also a function of y:

r(y) = 8*(1 - y/38)

Putting this all together, we can write the differential equation for the cell phone sales as:

dy/dt = r(y)y(1 - y/K)

Substituting the expression for r(y) yields:

dy/dt = 8*(1 - y/38)y(1 - y/100)

Simplifying further, we get:

dy/dt = (8/475)y(100 - y/4)

So the differential equation describing the cell phone sales is:

dy/dt = (8/475)y(100 - y/4)

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The range of (X,Y) is the triangular region in the first quadrant of the x-y plane bounded by the x-axis, y-axis, and the line x + y = 1.

The value of constant c is 2.

The value of P(Y < 2X^2) = 2/3.

To determine the range of (X,Y) in the x-y plane, we need to find the region of the x-y plane where the joint PDF is non-zero. From the given PDF, we have that the joint PDF is non-zero for x, y > 0 and x + y < 1.

Therefore, the range of (X,Y) is the triangular region in the first quadrant of the x-y plane bounded by the x-axis, y-axis, and the line x + y = 1.

To find the constant c, we can use the fact that the total probability over the entire range of (X,Y) must be equal to 1:

∫∫ f(x,y) dA = ∫∫ cx + 1 dA = 1

where A is the region of integration, which in this case is the triangular region as described above.

Evaluating the double integral over this region, we get:

∫0^1 ∫0^(1-x) cx + 1 dy dx = 1

Simplifying and solving for c, we get:

c = 2

To find P(Y < 2X^2), we need to integrate the joint PDF over the region where Y < 2X^2. Since this region is not a simple geometric shape, we need to break it up into two parts based on the value of X:

P(Y < 2X^2) = ∫∫ f(x,y) dA, where A is the region where Y < 2X^2

= ∫0^1 ∫0^(2x^2) 2x + 1 dy dx + ∫0^(√2/2) ∫0^(1-x) 2x + 1 dy dx

Evaluating these integrals, we get:

P(Y < 2X^2) = 2/3

Therefore, the probability that Y is less than 2 times X squared is 2/3. This tells us about the relationship between X and Y and how likely it is for Y to be smaller than a multiple of X squared.

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Complete question is:

Let X and Y be jointly continuous random variables with joint PDF

cx +1                  x, y > 0, x+y<1

0                           otherwise

Show the range of (X,Y), in the x−y plane.

Find the constant c.

Find P(Y<2x^2).