Theorem 9.6.4: when is the function V(x, y) = ax² + bxy + cy² positive definite? When is it negative definite?

The probability that a senator is under 70 years old given that he or she is at least 50 years old is 0.75.

To answer this question, we need to use conditional probability. Let A be the event that a senator is under 70 years old, and let B be the event that a senator is at least 50 years old. We want to find the probability of A given B, denoted as P(A|B).

Using Bayes' theorem, we have:

P(A|B) = P(B|A) * P(A) / P(B)

where P(B|A) is the probability that a senator is at least 50 years old given that they are under 70 years old (which is 1), P(A) is the probability that a senator is under 70 years old (which we do not know yet), and P(B) is the probability that a senator is at least 50 years old (which we also do not know yet).

To find P(A), we need more information. Let's assume that we know the following:

The total number of senators is 100.

The number of senators who are under 70 years old is 60.

The number of senators who are at least 50 years old is 80.

Using this information, we can calculate P(A) and P(B) as follows:

P(A) = number of senators under 70 / total number of senators = 60/100 = 0.6

P(B) = number of senators at least 50 / total number of senators = 80/100 = 0.8

Now we can plug these values into Bayes' theorem:

P(A|B) = P(B|A) * P(A) / P(B)

P(A|B) = 1 * 0.6 / 0.8

P(A|B) = 0.75

Therefore, the probability that a senator is under 70 years old given that he or she is at least 50 years old is 0.75.

Learn more about probability

brainly.com/question/11234923

#SPJ11

So the vector form of the quadratic function y = x² + 4x + 3 is: y = (x + 2)² - 1.

To change from standard form to vertex form, we need to complete the square.

First, we group the x-terms together and factor out any common coefficient of x², giving:

y = x² + 4x + 3

y = 1(x² + 4x) + 3

Next, we need to add and subtract a constant inside the parentheses to complete the square. To determine this constant, we take half of the coefficient of x (4) and square it:

(4/2)² = 4

So we add and subtract 4 inside the parentheses:

y = 1(x² + 4x + 4 - 4) + 3

Now we can factor the quadratic expression inside the parentheses as a perfect square:

y = 1[(x + 2)² - 4] + 3

Simplifying and rearranging terms, we get:

y = (x + 2)² - 1

To know more about vector,

https://brainly.com/question/20426452

#SPJ11

Answer:

I'm pretty sure the answer is 80.

Step-by-step explanation:

5 × 8 × 2 = 80

An equation of the line in point-slope form, using the coordinates of the labeled point is y + 3 = 0.8(x + 2).

In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical equation (formula):

y - y₁ = m(x - x₁)

Where:

At data point (-2, -3) and a slope of 0.8, a linear equation for this line can be calculated by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y - (-3) = 0.8(x - (-2))  

y + 3 = 0.8(x + 2)

Read more on point-slope here: brainly.com/question/24907633

#SPJ1

The correct answer is c. Yes, because the evidence was not strong enough to suggest that average reading and writing scores differ.

In hypothesis testing, we use a significance level to determine whether to reject or fail to reject the null hypothesis. If the p-value is less than the significance level, we reject the null hypothesis and conclude that there is evidence to support the alternative hypothesis. If the p-value is greater than the significance level, we fail to reject the null hypothesis and conclude that there is not enough evidence to support the alternative hypothesis.

Based on the results of the hypothesis test, if we failed to reject the null hypothesis (which is the case in this question), then we cannot conclude that the average reading and writing scores differ. Therefore, it is reasonable to expect a confidence interval for the average difference between the reading and writing scores to include 0. A confidence interval is a range of values that is likely to contain the true population parameter with a certain degree of confidence. Since we cannot conclude that the average scores differ, it is possible that the true population parameter is 0, and thus 0 could be included in the confidence interval.

Learn more about "hypothesis test": https://brainly.com/question/15980493

#SPJ11

The relation for t as a function of v is:

t = 200km/v

Here we need to remember the relation:

distance = speed*time.

Here we know that the quantities are:

speed = v

time = t

distance = 200km

Then we can write:

200km = v*t

Now we can isolate the variable t, so we have t as a function of v:

t = 200km/v

That is what we wanted.

learn more about speed at:

https://brainly.com/question/4931057

#SPJ1

Conditional probability is calculated by multiplying the probability of the preceding event by the updated probability of the succeeding, or conditional, event.

True. The notation P(a | b) represents the conditional probability of event a given that event b has occurred. It is read as "the probability of a given b."

learn about conditional probability,

https://brainly.com/question/10567654

#SPJ11

The functions categorized as decay or growth are

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

y = 2250(1-0.9)^x

y=10.(1+0.04)^x

An exponential function is represented as

y = ab^x

If b > 0, then it is a growth function

Otherwise, it a decay function

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

y = 2250(1-0.9)^x -- decay

y=10.(1+0.04)^x -- growth

Read more about exponential functions at

https://brainly.com/question/2456547

#SPJ1

Using mathematical operators, the value of x in the equations are -1/3, (2.39 or 0.71) and no solution respectively.

To determine the value of x in the equations, we need to use mathematical functions or operators;

1. 8 + 9x(x + 4) = (3x - 2)(3x + 2)

Open the brackets

8 + 9x² + 36x = 9x² + 6x - 6x - 4

Collect like terms

8 + 9x² + 36x - 9x² + 4 = 0

8 + 4 + 36x = 0

12 + 36x = 0

36x = -12

x = -12/36

x = -1/3

2. 2x(2x - 2) - 17 = (5x + 2x)(2x - 5)

Open the brackets;

4x² - 4x - 17 = 10x² - 25x + 4x² - 10x

collect like terms

14x² - 35x - 4x² - 4x + 17 = 0

10x² - 31x + 17 = 0

solving the quadratic equation;

x = 2.39 or x = 0.71

3. (2x + 1)(4x² - 2x + 1) = 4x(2x² - 5)

Open brackets;

8x³ - 4x² + 2x + 4x² - 2x + 1 = 8x³ - 20x

collect like terms

8x³ + 1 - 8x³ + 20 = 0

21 = 0

The equation has no solution

Learn more on mathematical operations here;

https://brainly.com/question/4721701

#SPJ1

The area of the diagram ABDC is 625cm2.

We are given that;

RQ is parallel to ST, which implies that ∆PRQ and ∆PST are similar by the AA criterion (angle-angle). We are also given that ST = 4 cm, RP = 10 cm, and PT = 5 cm;

Now,

we can use the fact that RP = 10 cm and PT = 5 cm to find PQ by using the Pythagorean theorem. PQ is the hypotenuse of the right triangle PRT, so we get:

PQ2 = RP2 + PT2 PQ2 = 102 + 52 PQ2 = 100 + 25 PQ2 = 125 PQ = √125 PQ ≈ 11.18 cm

Now we can use PQ and ST as corresponding sides to find the scale factor. We get:

Scale factor = PQ / ST Scale factor ≈ 11.18 / 4 Scale factor ≈ 2.795

ii. To find |RQ|, we can use the scale factor and multiply it by |ST|. We get:

|RQ| = Scale factor * |ST| |RQ| ≈ 2.795 * 4 |RQ| ≈ 11.18 cm

b. Find the area of ∆PST if the area of ∆PRQ is 80 cm2.

To find the area of ∆PST, we can use the fact that the ratio of the areas of similar triangles is equal to the square of the scale factor. We get:

Area of ∆PST / Area of ∆PRQ = (Scale factor)2 Area of ∆PST / 80 = (2.795)2 Area of ∆PST / 80 = 7.812 Area of ∆PST = 7.812 * 80 Area of ∆PST ≈ 625 cm2

Therefore, by the area the answer will be 625cm2.

Learn more about the area;

https://brainly.com/question/1658516

#SPJ1

Therefore, there are 8820 groups of 9 that contain 5 women and 4 men using combination formula.

To solve part (b)(i), we need to use the combination formula. The number of ways to choose 5 women out of 9 is given by the combination C(9,5), which is:

C(9,5) = 9! / (5! * 4!)

= 126

Similarly, the number of ways to choose 4 men out of 8 is given by the combination C(8,4), which is:

C(8,4) = 8! / (4! * 4!)

= 70

To get the total number of groups of 9 that contain 5 women and 4 men, we need to multiply these two numbers:

126 * 70 = 8820

To know more about combination,

https://brainly.com/question/20211959

#SPJ11

The number of moles for the first equation is 1.8 and the number of moles for the second equation is 0.0240.

The balanced chemical equation is 4Al + 3O² → 2Al₂O₃. The molar ratio of Al to O₂ is 4:3. Therefore, if 1.35 moles of O₂

4 moles Al/₃ moles O₂ = x moles Al/1.35 moles O₂

x = 1.8 moles of Al

The balanced chemical equation is N + 2H₂O → NaOH + H₂. The molar ratio of N to H₂ is 1:1. Therefore, if 0.0240 moles of N is used, then the number of moles of H₂ produced would be:

1 mole N/1 mole H₂ = 0.0240 moles N/x moles H₂

To know more about chemical reactions follow

https://brainly.com/question/31681528

#SPJ1

Answer:

68 1/4 lbs

Step-by-step explanation:

its just the answer

Answer:

x = -2

Step-by-step explanation:

5x + 2x = -8 - 6

7x = -14

x = -2

The height of the ball will be 16.94 feet. Then the baseball travels over the fence.

Given that:

Distance at time t, x = 140t

Height at time t, y = 3.1 + 40t − 16t²

Height, h = 10 feet

Distance, x = 320 feet

The time is calculated as,

320 = 140t

t = 320/140

t = 16/7

The height at t = 16/7 is calculated as,

y = 3.1 + 40(16/7) − 16(16/7)²

y = 3.1 + 97.43 - 83.59

y = 16.94 feet

y > 10 feet

The height of the ball will be 16.94 feet. Then the baseball travels over the fence.

More about the function link is given below.

https://brainly.com/question/5245372

#SPJ1

Manufacturers change the size of compact disks so that they are made of the same material and have the same thickness as a current disk but have one third of the diameter, the moment of inertia will change by a factor of 1/9 or approximately 0.111.

The moment of inertia of a disk is proportional to the square of its radius (I = (1/2)mr^2). If the diameter of the new compact disk is one-third of the diameter of the current disk, then its radius will be one-sixth of the radius of the current disk (r_new = r_current/3). Therefore, the moment of inertia of the new compact disk will decrease by a factor of (1/6)^2 = 1/36, or 19.

Given that the new compact disk has one-third of the diameter of the current disk, the moment of inertia (I) will change as a function of the radius (r) squared. Since the diameter is halved, the radius is also one-third, and the moment of inertia is given by the formula I = k * r^2, where k is a constant.

When we reduce the radius to one-third (r/3), the new moment of inertia (I') can be calculated as:

I' = k * (r/3)^2
I' = k * (r^2/9)

To find the factor by which the moment of inertia has changed, we need to divide the new moment of inertia (I') by the original moment of inertia (I):

Factor = I'/I = (k * r^2/9) / (k * r^2) = 1/9

So, the moment of inertia will change by a factor of 1/9 or approximately 0.111.

To learn more about diameter, click here:

brainly.com/question/5501950

#SPJ11

The correct answer would be

B) half of her classmates have a sibling and half do not

Add totals for classmates who have chores and those who do not have chores for each group.

7 + 8 = 15

9 + 6 = 15

Therefore, the correct answer would be

B) half of her classmates have a sibling and half do not

Based on the fact that the total for the classmates that have chores and those that don't have are equal which is 15, we can conclude that option B is correct.

Read more about algebra here:

https://brainly.com/question/22399890

#SPJ1

Answer: The exponential function that models the population growth of the city over time is:

y = 350,000 * (1 + 0.044)^t

where t is the number of years since the start of the study, y is the city's population, and 0.044 is the annual growth rate expressed as a decimal.

The function is an example of exponential growth, where the population is increasing at a constant rate over time.

Step-by-step explanation:

Answer: Average speed is given by the formula:

speed = distance ÷ time

Given that Darlene rode her bike 8 2/5 miles in 3/4 of an hour. We need to convert the mixed number 8 2/5 to an improper fraction.

8 2/5 = (8 x 5 + 2) / 5 = 42/5

So, Darlene rode her bike a distance of 42/5 miles in a time of 3/4 hours.

Substitute speed = distance ÷ time

speed = (42/5) ÷ (3/4)

To divide by a fraction, we invert and multiply:

speed = (42/5) x (4/3)

speed = 56/5

So, Darlene's average speed was 56/5 miles per hour (mph).

To simplify this fraction, we can divide both the numerator and denominator by 1:

speed = 11/1

To find the probability that someone who tests negative for opium use does not use opium, we need to use Bayes' theorem.

Let's assume that the prevalence of opium use in the population is 5%. This means that out of 100 people, 5 people use opium.

Now, let's say that the test for opium use has a sensitivity of 95% and a specificity of 98%. This means that out of 100 people who use opium, 95 will test positive, and out of 100 people who do not use opium, 98 will test negative.

Using Bayes' theorem, we can calculate the probability that someone who tests negative for opium use does not use opium as follows:

P(Not using opium | Negative test) = P(Negative test | Not using opium) * P(Not using opium) / P(Negative test)

P(Negative test | Not using opium) = 0.98 (specificity)
P(Not using opium) = 0.95 (complement of the prevalence)
P(Negative test) = P(Negative test | Not using opium) * P(Not using opium) + P(Negative test | Using opium) * P(Using opium)
P(Negative test | Using opium) = 1 - 0.95 = 0.05 (false negative rate)
P(Using opium) = 0.05 (prevalence)

P(Negative test) = 0.98 * 0.95 + 0.05 * 0.05 = 0.0935

P(Not using opium | Negative test) = 0.98 * 0.95 / 0.0935 ≈ 0.994

Therefore, the probability that someone who tests negative for opium use does not use opium is approximately 0.994, or 0.994 in decimal format rounded to three decimal places.
In this case, we want to find the probability of not using opium, given that someone tests negative.

Let's define the events as follows:
- A: Person does not use opium
- B: Person tests negative for opium

We want to find P(A|B), which is the probability of event A occurring given that event B has occurred. Using the conditional probability formula:

P(A|B) = P(A ∩ B) / P(B)

Unfortunately, without specific data on the probabilities of A, B, and A ∩ B (not using opium and testing negative), we cannot provide a numeric answer. However, if you have this information, you can simply plug the values into the formula and divide P(A ∩ B) by P(B) to obtain the probability in decimal format. Finally, round the result to three decimal places.

To learn more about probability : brainly.com/question/11234923

#SPJ11

Answer: 87.5

Step-by-step explanation:

sum of values/number of values
85+93+91+81/4

350/4

87.5

The radius of a sphere with volume 288π cm³ is equal to 6 centimeter.

In Mathematics and Geometry, the volume of a sphere can be calculated by using the following mathematical equation (formula):

Volume of a sphere = 4/3 × πr³

Where:

r represents the radius.

By substituting the given parameters into the formula for the volume of a sphere, we have the following;

288π = 4/3 × π × r³

r³ = 864/4

Radius, r = ∛216

Radius, r = 6 centimeter.

Read more on volume of a sphere here: brainly.com/question/20394302

#SPJ1

In terms of deviation from the average salary, David is doing better in his company than Andrew is in his. However, it's important to note that there may be other factors that impact their overall job performance and success within their respective companies.

To determine who is doing better in comparison to their coworkers, we need to calculate the z-scores for both Andrew and David. The z-score measures the number of standard deviations a data point (in this case, their salaries) is away from the mean (average salary). The formula for the z-score is:

z = (X - μ) / σ

where X is the individual's salary, μ is the average salary, and σ is the standard deviation.

For Andrew:
X = $45,000 EC
μ = $41,000 EC
σ = $7,000 EC

z = ($45,000 - $41,000) / $7,000
z = $4,000 / $7,000
z = 0.5714

For David:
X = $38,000 EC
μ = $35,000 EC
σ = $1,700 EC

z = ($38,000 - $35,000) / $1,700
z = $3,000 / $1,700
z = 1.7647

Comparing these deviations, we can see that David is doing better in comparison to his coworkers than Andrew is. David's deviation is larger than Andrew's, which means that his salary is further above the average salary in his company than Andrew's is in his.

A higher z-score indicates a better position compared to coworkers. In this case, David has a higher z-score (1.7647) compared to Andrew (0.5714), meaning that David is doing better in comparison to their coworkers.

Learn more about Average:

brainly.com/question/24057012

#SPJ11

The relative fitness of red birds is 1, not 0.5.

To determine the relative fitness of each bird type, we need to compare the average number of offspring each type produces in the next generation.

For red birds, the average number of offspring is 10.

For orange birds, the average number of offspring is 8.

For yellow birds, the average number of offspring is 2.

To calculate the relative fitness of each bird type, we divide each average number of offspring by the highest average number of offspring (which is 10):

- Redbirds: 10/10 = 1
- Orange birds: 8/10 = 0.8
- Yellow birds: 2/10 = 0.2

Find out more about the population

brainly.com/question/28450930

#SPJ11

There will be $530.68 in the account at the end of 2 years, if Arianna deposits $500 in an account that pays 3% interest, compounded semiannually.

The formula accrued amount in a compounded interest is expressed as;

A = P( 1 + r/n )^( n × t )

Where A is accrued amount, P is principal, r is interest rate and t is time.

Given the data in the question;

Principal P = $500

Compounded semi annually n = 2

Time t = 2 years

Interest rate r = 3%

Accrued amount A = ?

First, convert R as a percent to r as a decimal

r = R/100

r = 3/100

r = 0.03

Plug the values into the above formula:

A = P( 1 + r/n )^( n × t )

A = $500( 1 + 0.03/2 )^( 2 × 2 )

A = $500( 1 + 0.015 )^( 4 )

A = $530.68

Therefore, the accrued amount is $530.68.

Learn more about compound interest here: brainly.com/question/27128740

#SPJ1

The gradient of the line is 2

[tex]Distance AB = \sqrt{[(5-(-3))^2 + (2-4)^2]}= \sqrt{68} \\Distance AC = \sqrt{[(-3-0)^2 + (4-(-3))^2]} = \sqrt{58} \\Distance BC = \sqrt{[(5-0)^2 + (2-(-3))^2]}= \sqrt{50}[/tex]

mAB = (2 - 4) / (5 - (-3)) = -1/2

This is the slope of AB

-1 / (-1/2) = 2

we have to find the point that is perpendicular to AB

(-3 + 5) / 2 = 1

(4 + 2) / 2 = 3

1 , 3 are  perpendicular to AB

y - 3 = 2(x - 1)

y - 3 = 2x - 2

y = 2x + 1

There fore the gradient of the line is 2

Read more on gradient here:https://brainly.com/question/23016580

#SPJ1

The given representation is a quadratic function.

The given table can be represented in the form of equation as,

y = x²

When x = 0, y = 0

When x = 1, y = 1

When x = 2, y = 2² = 4

When x = 3, y = 3² = 9

When x = 4, y = 4² = 16

This can be written as,

y = x² + 0x + 0

This is a quadratic function.

Hence the given representation is a quadratic function.

Learn more about Quadratic Functions here :

https://brainly.com/question/18958913

#SPJ1