5^a - 3^a = 2b a and b are less then 5 and positive integers work out all the possible values of a and b

Answer:

a) To find the probability that X1 is less than or equal to 0.5 and X2 is greater than or equal to 0.25, we need to integrate the given density function over the region where X1 ≤ 0.5 and X2 ≥ 0.25.

P(X1 ≤ 0.5, X2 ≥ 0.25) = ∫∫(x1,x2) f(x1,x2) dxdy

where the limits of integration are:

0.25 ≤ x2 ≤ 1

0 ≤ x1 ≤ 0.5

Substituting the given density function:

P(X1 ≤ 0.5, X2 ≥ 0.25) = ∫0.25^1 ∫0^0.5 (x1 + x2) dx1 dx2

Evaluating the inner integral:

P(X1 ≤ 0.5, X2 ≥ 0.25) = ∫0.25^1 [(x1^2/2) + x1x2] |0 to 0.5 dx2

Simplifying the expression:

P(X1 ≤ 0.5, X2 ≥ 0.25) = ∫0.25^1 [(0.125 + 0.25x2)] dx2

Evaluating the upper and lower limits:

P(X1 ≤ 0.5, X2 ≥ 0.25) = [0.125x2 + 0.125x2^2] |0.25 to 1

Substituting the limits:

P(X1 ≤ 0.5, X2 ≥ 0.25) = [(0.125 + 0.125) - (0.03125 + 0.015625)]

Solving for the final answer:

P(X1 ≤ 0.5, X2 ≥ 0.25) = 21/64

Therefore, the probability that X1 is less than or equal to 0.5 and X2 is greater than or equal to 0.25 is 21/64.

b) To find the probability that X1 + X2 is less than or equal to 1, we need to integrate the given density function over the region where X1 + X2 ≤ 1.

P(X1 + X2 ≤ 1) = ∫∫(x1,x2) f(x1,x2) dxdy

where the limits of integration are:

0 ≤ x1 ≤ 1

0 ≤ x2 ≤ 1-x1

Substituting the given density function:

P(X1 + X2 ≤ 1) = ∫0^1 ∫0^(1-x1) (x1 + x2) dx2 dx1

Evaluating the inner integral:

P(X1 + X2 ≤ 1) = ∫0^1 [(x1x2 + 0.5x2^2)] |0 to (1-x1) dx1

Simplifying the expression:

P(X1 + X2 ≤ 1) = ∫0^1 [(x1 - x1^2)/2 + (1-x1)^3/6] dx1

Evaluating the integral:

P(X1 + X2 ≤ 1) = [x1^2/4 - x1^3/6 - (1-x1)^4/24] |0 to 1

Substituting the limits:

P(X1 + X2 ≤ 1) = (1/4 - 1/6 - 1/24) - (0/4 - 0/6 - 1/24)

Solving for the final answer:

P(X1 + X2 ≤ 1) = 1/8

Therefore, the probability that X1 + X2 is less than or equal to 1 is 1/8.

Answer:

[tex]tan(x)=\frac{4}{3}[/tex]

Step-by-step explanation:

In the unit circle,

- [tex]cos(a)=\frac{x}{r}[/tex] where [tex]a[/tex] is the degree measure, [tex]x[/tex] is the x-coordinate of the triangle, and [tex]r[/tex] is the radius of the circle

- [tex]sin(a)=\frac{y}{r}[/tex] where [tex]a[/tex] is the degree measure, [tex]y[/tex] is the y-coordinate of the triangle, and [tex]r[/tex] is the radius of the circle

Thus, since tangent is equal to sine over cosine, we can simplify our knowledge to:  [tex]tan(a)=\frac{sin(a)}{cos(a)}=\frac{y}{x}[/tex]

In this problem, [tex]sin(x)=\frac{4}{5}[/tex]. We can conclude from our previous knowledge that [tex]y=4[/tex] and the radius is 5.

Similarly, [tex]cos(x)=\frac{3}{5}[/tex], which means [tex]x=3[/tex] and the radius is the same, at 5.

Since we know that [tex]x=3[/tex] and [tex]y=4[/tex], we can find the value of  [tex]tan(x)[/tex] by using the formula [tex]tan(x)=\frac{y}{x}[/tex] and plug in the numbers.

Therefore, [tex]tan(x)=\frac{4}{3}[/tex].

Learn more about tangents at https://brainly.com/question/12465859

First, 6.7 % can be written in decimal form as 0.067  (6.7 / 100 = 0.067).

Let's use the variable x to represent the number of copies sold to date.

Then we can write and solve the following equation to represent 6.7% of the total sold to date:

0.067 • x = 33600

You can solve this equation by dividing both sides of the equation by 0.067:

0.067 • x  =  33600

0.067             0.067

x = 501493

To date, 500000 copies would have been sold rounded to the nearest whole.

The smallest that will qualify the window opening per the code is 0.71

What is rectangular?

A quadrilateral with four right angles is a rectangle. It can alternatively be described as a parallelogram with a right angle or an equiangular quadrilateral, where equiangular denotes that all of its angles are equal. A square is a rectangle with four equally long sides.

Here, we have

Given: a basement bedroom must have a window with an opening area of at least 5.7 square feet per the international residential code. A rectangular basement window opening 0.75 meters wide.

First, we convert square feet into square meters.

5.7 square feet = 0.53 square meters

Now,

0.53 / 0.75 = 0.71

Hence, the smallest that will qualify the window opening per the code is 0.71

To learn more about the rectangular from the given link

https://brainly.com/question/31186475

#SPJ1

Answer:

5/12

Step-by-step explanation:

We are only looking at people who have a dog, which is 12

Now we need to determine if they have a cat

P( cat given that they have a dog)

            = number of people with a cat/ they have a dog

           = 5/ (5+7)

          = 5/12

[tex] \: [/tex]

((x/y)^(n+1)) > ((x/y)^n) for all n ≥ 1 and x > y > 0.

To prove this statement by induction, we will use the principle of mathematical induction.

Base case: When n = 1, we have:

((x/y)^(1+1)) = ((x/y)^2) = (x^2)/(y^2)

((x/y)^1) = (x/y)

Since x > y > 0, we have x/y > 1. Therefore, (x^2)/(y^2) > (x/y), which means that the base case is true.

Inductive step: Assume that ((x/y)^(k+1)) > ((x/y)^k) for some arbitrary positive integer k. We want to prove that this implies that ((x/y)^(k+2)) > ((x/y)^(k+1)).

Starting with ((x/y)^(k+2)), we can write:

((x/y)^(k+2)) = ((x/y)^(k+1)) * ((x/y)^1)

Using the induction hypothesis, we know that ((x/y)^(k+1)) > ((x/y)^k), and we also know that x/y > 1. Therefore, we have:

((x/y)^(k+2)) > ((x/y)^k) * (x/y)

Simplifying this expression, we get:

((x/y)^(k+2)) > ((x/y)^k) * (x/y)

((x/y)^(k+2)) > ((x^k)/(y^k)) * (x/y)

((x/y)^(k+2)) > ((x^(k+1))/(y^(k+1)))

Therefore, we have shown that ((x/y)^(k+2)) > ((x/y)^(k+1)) for all positive integers k, which completes the inductive step.

By the principle of mathematical induction, we have proven that ((x/y)^(n+1)) > ((x/y)^n) for all n ≥ 1 and x > y > 0.

Check the picture below.

[tex]\cfrac{3^2}{4^2}=\cfrac{27}{A}\implies \cfrac{9}{16}=\cfrac{27}{A}\implies 9A=432\implies A=\cfrac{432}{9}\implies A=48[/tex]

The probability that a randomly selected light bulb will last between 750 and 900 hours is given as follows:

P = 47.5%.

The Empirical Rule states that, for a normally distributed random variable, the symmetric distribution of scores is presented as follows:

In the context of this problem, we have that:

The normal distribution is symmetric, hence the probability of an observation between the mean and two standard deviations above the mean is given as follows:

0.5 x 95 = 0.475 = 47.5%.

More can be learned about the Empirical Rule at https://brainly.com/question/10093236

#SPJ1

Answer:

Let's call the number we're trying to find "x".

According to the problem, when this number is decreased by 40% of itself, we get 54.

In other words,

x - 0.4x = 54

Simplifying the left side, we get:

0.6x = 54

Dividing both sides by 0.6, we get:

x = 90

Therefore, the number we're looking for is 90.

The perimeter is 27 feet and the area is 35 square feet

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

The figure

The perimeter is the sum of tthe side lengths

So, we have

Perimeter = 7.5 + 6 + (6 - 2.5) + 4 + 2.5 + 3.5

Evaluate

Perimeter = 27

The area is calculated as

Area = 6 * 3.5 + 4 * (6 - 2.5)

Evaluate

Area = 35

Hence, teh area is 35 square feet

Read more about area at

https://brainly.com/question/24487155

#SPJ1

Answer:

To solve the inequality 3x - 14 ≥ 11 - x, we need to isolate the variable x on one side of the inequality symbol. We can do this by adding x to both sides and adding 14 to both sides:

3x - 14 + x ≥ 11

Combining like terms, we get:

4x - 14 ≥ 11

Adding 14 to both sides, we get:

4x ≥ 25

Dividing both sides by 4, we get:

x ≥ 6.25

Therefore, the solution to the inequality is x ≥ 6.25, which can be represented as the interval [6.25, ∞) on the number line

The f(x) of the linear function is:

f(x) = -4x - 7

Since f is a linear function. The general form of a linear function is:

y = mx + b

where y = f(x), m is the slope and b is the y-intercept

Since f (−3) = 5, we have:

x = -3 and y = 5

Substitute into y = mx + b:

y = mx + b

5 = -3m + b ---- (1)

Also, f(5) = −27, we have:

x = 5 and y = -27

Substitute into y = mx + b:

y = mx + b

-27 = 5m + b ---- (2)

Solving (1) and (2) simultaneously by elimination method:

-3m + b = 5

5m + b = -27

-8m = 32

m = 32/(-8)

m = -4

Put m = -4 in (1) and solve for b:

-3x + b = 5

-3(-4) + b = 5

12 + b = 5

b = 5 - 12

b = -7

Put m and b into f(x) = mx + b to get f(x). That is:

f(x) = mx + b

f(x) = -4x - 7

Learn more about linear function on:

https://brainly.com/question/4025726

#SPJ1

Step-by-step explanation:

all triangles here are right-angled.

ABC, ADB, ADC, AED.

let's say CD = x, AD = height

just by using Pythagoras :

8² = height² + (10-x)² = height² + 100 -20x + x²

64 = height² + 100 -20x + x²

6² = height² + x²

36 = height² + x²

64-36 = 100 - 20x

28 = 100 - 20x

-72 = -20x

x = 72/20 = 3.6 ft

6² = height² + 3.6²

36 = height² + 12.96

height² = 23.04

height = 4.8 ft

height² = ED² + 3²

23.04 = ED² + 9

ED² = 14.04

ED = 3.746998799... ft ≈ 4 ft

1. The equation for the line, which is tangent to the lemniscate at the point P is y = -√3x + (5/4 + √3/8). The equation for the line which is tangent to the lemniscate at Q is y = (-5/3)x + 11/3.

The pace at which a function is changing at a specific point is known as its derivative. It shows the angle at which the tangent line to the curve at that location slopes. A key idea in calculus, the derivative can be utilised to tackle a range of issues, such as curve analysis, rates of change, and optimisation.

The tangent line to the lemniscate at point P, is determined using the derivative of the function.

(x² + y²)² = 4(x² - y²)

Taking the derivative on both sides we have:

2(x² + y²)(2x + 2y(dy/dx)) = 8x - 8y(dy/dx)

dy/dx = (x² + y²)/(y - x)

Substituting  P= (√5/8, √3/8) for the x and y we have:

dy/dx = (√5/8)² + (√3/8)²) / (√3/8 - √5/8) = -√3

Thus, the slope of the tangent line at point P is -√3.

Using the point slope form:

y - y1 = m (x - x1)

Substituting the values we have:

y - (√3/8) = -√3(x - √5/8)

y = -√3x + (5/4 + √3/8)

Hence, equation for the line, which is tangent to the lemniscate at the point P is y = -√3x + (5/4 + √3/8).

Bonus question:

The equation of tangent for the lemniscate at point Q = (2,1) is:

dy/dx = (2² + 1²)/(1 - 2) = -5/3

Using the point slope form:

y - 1 = (-5/3)(x - 2)

y = (-5/3)x + 11/3

Hence, equation for the line which is tangent to the lemniscate at Q is y = (-5/3)x + 11/3.

Learn more about derivative here:

https://brainly.com/question/25752367

#SPJ1

option C is correct. The range of penguin heights is greater at Cityview Zoo than at Countryside Zoo.

How to calculate data?

Based on the given data, we can see that the penguin heights at Countryside Zoo are distributed more evenly across the range of heights, with heights ranging from 35cm to 45cm, and a fairly consistent distribution across this range. In contrast, the penguin heights at Cityview Zoo are more concentrated around the height of 38cm, with fewer penguins at the extremes of the range.

Therefore, option C is correct. The range of penguin heights is greater at Cityview Zoo than at Countryside Zoo.

Option A is incorrect because the range of penguin heights is greater at Countryside Zoo than at Cityview Zoo.

Option B is incorrect because the ranges of penguin heights are not the same.

Option D is also incorrect because there is no mode of penguin heights at either zoo. A mode is defined as the value that appears most frequently in a distribution. In this case, no height appears more frequently than any other.

In conclusion, we can say that the penguin heights are distributed differently at the two zoos, with Countryside Zoo having a more evenly distributed range of heights, while Cityview Zoo has a more concentrated distribution around a particular height.

To know more about Data visit :-

https://brainly.com/question/26711803

#SPJ1

The cumulative probabilities for the given probability distribution were calculated, and the median of the discrete random variable was found to be 20.

To find the median, we need to find the smallest value of the random variable for which the cumulative probability equals or exceeds 0.5.

The cumulative probabilities are:

(≤0) = 0.02

(≤5) = 0.07

(≤10) = 0.15

(≤15) = 0.31

(≤20) = 0.58

(≤25) = 1

The cumulative probability is the sum of the probabilities of all events that have an outcome less than or equal to a given value. For example, the cumulative probability for the event of collecting 5 dollars or less is the sum of the probabilities for collecting 0 dollars and 5 dollars, which is 0.02 + 0.05 = 0.07. Similarly, the cumulative probability for the event of collecting 10 dollars or less is the sum of the probabilities for collecting 0 dollars, 5 dollars, and 10 dollars, which is 0.02 + 0.05 + 0.08 = 0.15. The same process is used to calculate the cumulative probabilities for all other values. The median is the smallest value of the random variable for which the cumulative probability is greater than or equal to 0.5.

To learn more about probability please click on below link        

https://brainly.com/question/16484393

#SPJ1

Answer:

[tex]3x-3y=31[/tex]

Step-by-step explanation:

Adding the equations yields [tex]3x-3y=20+11=31[/tex].

The correct statement that compares the distributions is:

D On average, nonsmokers weighed 13 lbs less than smokers.

What is the variability?

Variability refers to the amount of spread or dispersion in a set of data. It is a measure of how much the data values in a sample or population differ from each other.

One commonly used measure of variability is the standard deviation, which is the square root of the variance. The variance is the average of the squared differences from the mean.

Looking at the graph, we can see that the center of the distribution of smokers is around 178 lbs, while the center of the distribution of nonsmokers is around 165 lbs. This means that, on average, nonsmokers weigh less than smokers.

Option A is incorrect because the range is not a good measure of variability, and it does not necessarily mean that there is more variability in the weights of nonsmokers.

Option B is incorrect because the graph clearly shows that nonsmokers weigh less on average than smokers.

Option C is incorrect because we cannot make any conclusion about half of the smokers weighing more than all of the nonsmokers from the graph.

Option E is incorrect because the graph shows that the variability in the weights of smokers is greater than that of nonsmokers.

Hence, The correct statement that compares the distributions is:

D On average, nonsmokers weighed 13 lbs less than smokers.

To learn more about the variability visit:

https://brainly.com/question/14052211

#SPJ1

Answer:

The travel bag that can hold exactly 3,150 cubic inches is the bag that is 21 inches long.

Step-by-step explanation:

The travel bags can be modelled as rectangular prisms.

The formula for the volume of a rectangular prism is:

[tex]\boxed{\textsf{Volume} = l \times w\times d}[/tex]

where:

Given that all three bags have a depth of 10 inches and a width of 15 inches, we can create an equation for the volume of any of the bags by substituting d = 10 and w = 15 into the formula:

[tex]\begin{aligned}\sf Volume &=l \times w \times d\\&= l \times 15\times10\\&=l \times 150\\&=150\;l\end{aligned}[/tex]

To determine which travel bag can hold exactly 3,150 cubic inches, substitute volume = 3150 into the formula and solve for length, l:

[tex]\begin{aligned}\sf Volume &=150\;l\\\\ \implies 3150&=150\;l\\\\\dfrac{3150}{150}&=\dfrac{150\;l}{150}\\\\21&=l\\\\l&=21\;\sf in\end{aligned}[/tex]

Therefore, the travel bag that can hold exactly 3,150 cubic inches is the bag that is 21 inches long.

After factoring the given expression -2xy³ - 8xy² + xy, the resultant answer is -2xy²(xy-4).

The concept of algebraic expressions is the use of letters or alphabets to represent numbers without providing their precise values.

A group of terms coupled with the operations +, -, x, or form an expression, such as 4 x 3 or 5 x 2 3 x y + 17.

A statement with an equal sign, such as 4 b 2 = 6, asserts that two expressions are equal in value and is known as an equation.

So, we have the expression:

-2xy³ - 8xy² + xy

Now, factor out as follows:

=  -2xy³ - 8xy² + xy

= xy(-2xy² - 8xy)

= -2xy²(xy-4)

Therefore, after factoring the given expression -2xy³ - 8xy² + xy, the resultant answer is -2xy²(xy-4).

Know more about expressions here:

https://brainly.com/question/1859113

#SPJ1

Answer:

you just add a 0 at the end of the answer of what 32 divided by 4 is, so in this case 320 divided by 4 is 80

Step-by-step explanation:

32 divided by 4 is 8.

320 divided by 4 is 80.

To get from 32 to 320 all you need is a 0 at the end, so you can just add the 0 the end of the answer. This means you're going from an 8, to an 80.

OR

Another way you can look at it is 32 multiplied by 10 to get 320. So you need to mutiple your answer by 10 to get the right answer.

32*10=320

8*10=80

Hope this helps!

Answer: 16.5 gallons of water.

Step-by-step explanation:

If it was me. I would be setting up as a table to keep my work organized.

So first we find how much 1 minute is.

15g : 10m

15/10 : 10m/10

1.5g : 1m

Then I multiply how many minutes there are.

1.5g x 11 : 1m x 1

16.5g : 11m

And there we find the answer of 16.5 gallons.

Happy Solving

Answer:16.5

Step-by-step explanation:

Answer:

The given line has a slope of -3/2, since it is in the form y = mx + b, where m is the slope. Any line that is parallel to this line will also have a slope of -3/2.

To find the equation of the line that passes through the point (-2,3) and has a slope of -3/2, we can use the point-slope form of the equation of a line:

y - y1 = m(x - x1)

where (x1, y1) is the given point and m is the slope. Substituting in the values we have:

y - 3 = (-3/2)(x - (-2))

y - 3 = (-3/2)x - 3

y = (-3/2)x - 3 + 3

y = (-3/2)x

Therefore, the equation of the line that is parallel to y = -3/2x - 8 and passes through the point (-2,3) is y = (-3/2)x.

Note that this line does not have a y-intercept, since it passes through the point (-2,3) and has a slope of -3/2.

Answer:

O = { -4,-3,-2,-1,0,-1 }

The values of function f(x) positive, and negatives are (-infinity, infinity )

A function is a procedure or link that connects every element of one non-empty set A to at least one element of another non-empty set B. The phrases "domain" and "co-domain" are used in mathematics to define a function f between two sets, A and B. The constraint F = (a,b)| is satisfied by all values of a and b.

In the case of the question,

f (x) = x²

f (x) will be positive for all x values. As a result of the function:

x² = x × x

That is, when any number or integer is multiplied by itself, the result is positive. (For example, - - = + and + + = +)

As a result, f (x) = x2 will be positive for (,).

To know more about function visit:

https://brainly.com/question/11624077

#SPJ1

Answer:

P=2(l+w)=2·(9+10)=38ft

Answer:

yes.

Step-by-step explanation:

cause yes.

Answer:

The average temperature on the Equator is 137°F warmer than the average temperature at the South Pole.

76°c

Step-by-step explanation: