Quadratic Inequalities

The length of the legs are 20 cm and 21 cm

The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

We know that from the Pythagoras theorem;

[tex]c^2 = a^2 + b^2[/tex]

Let the hypotenuse be c and the other two sides be a and b

We have that;

[tex]29^2 = x^2 + (x -1)^2\\841 = x^2 + x^2 - 2x + 1\\841 = 2x^2 - 2x + 1\\2x^2 - 2x + 1 - 841 = 0\\2x^2 - 2x - 840 = 0\\x = -20 or 21[/tex]

Since length can not be negative, x = 21 cm

Thus the other leg is 20 cm

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There is not sufficient evidence at the α=0.05 level to conclude that the proportion of people in Antonia's city that can speak more than one language is greater than 26%.

Here's a step-by-step explanation:

1. Identify the null hypothesis (H₀) and the alternative hypothesis (Hₐ): H₀: p = 0.26, Hₐ: p > 0.26.

2. Determine the significance level (α): α = 0.05.

3. Calculate the test statistic (z): In this case, z ≈ 2.25.

4. Determine the P-value: The P-value is given as approximately 0.1.

5. Compare the P-value to the significance level: If the P-value is less than or equal to the significance level (α), reject the null hypothesis. In this case, 0.1 > 0.05, so we do not reject the null hypothesis.

Based on the information provided, there is not sufficient evidence at the α=0.05 level to conclude that the proportion of people in Antonia's city that can speak more than one language is greater than 26%.

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Dominic will need to save a minimum of $121 per month to pay for his portion of the first year's tuition.

Dominic's parents will pay 80% of the annual tuition cost, which is:

0.8 x $7,260 = $5,808

So, Dominic will need to pay the remaining 20% of the tuition cost, which is:

0.2 x $7,260 = $1,452

Since Dominic has one year to save his portion of the tuition, he will need to save:

$1,452 / 12 months = $121 per month

Therefore, Dominic will need to save a minimum of $121 per month to pay for his portion of the first year's tuition.

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The expression 0.85(1.4s) explains what happened to the price of the jacket through the two changes made by the store owner.

So, the expression 0.85(1.4s) explains what happened to the price of the jacket through the two changes made by the store owner.

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The area of the triangle to the nearest tenth is 5.8cm²

A triangle is a closed, 2-dimensional shape with 3 sides, 3 angles, and 3 vertices. A triangle is also a polygon. Examples of triangle include, isosceles, equilateral , scalene e.t.c

The area of a triangle is expressed as;

A = 1/2 b h for right angle triangle and A = absinC for others.

Here; x = 4.7, y = 7.9, W = 162

area = 1/2× 4.7 × 7.9 sin162

= 1/2 × 4.7 × 7.9 × 0.31

= 5.8 cm²( nearest tenth)

Therefore the area of the triangle is 5.8cm²

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With the surface area of the square pyramid 84 square inches and side length of the base is 6, the value of x is 4 inches, by assuming x as the slant height of the square pyramid.

Assuming that x refers to the slant height of the square pyramid, we can use the formula for the surface area of a square pyramid to solve for x:

Surface area of a square pyramid = base area + (0.5 x perimeter of base x slant height)

Since the base of the square pyramid is a square with side length 6,

the base area is 6² = 36 square inches.

The perimeter of the base is 4 times the side length, so it is 4 x 6 = 24 inches.

Substituting these values into the formula and simplifying, we get:

84 = 36 + (0.5 x 24 x x)

84 - 36 = 12x

48 = 12x

x = 4

Therefore, the value of x, the slant height of the square pyramid, is 4 inches.

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The pet boarder can accommodate 70 dogs.

Let's calculate how many dogs and cats the pet boarding facility can hold using the ratio of dogs to cats, which is 5:2.

First, we can figure out how many parts there are in the ratio: 5 + 2 = 7.

This indicates that there are 7 equal components in the ratio, 5 of which are dogs and 2 of which are cats.

We need to multiply the result by the number of dog components (5) after dividing the total number of parts (7) into the 98 available locations to get the number of dogs:

Number of dogs = (5 / 7) * 98

Number of dogs = 70

Therefore, the pet boarder can accommodate 70 dogs.

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Answer:

a. 5.74%.

b. 7.29%

c. 20.18%

d. 34.87%

e. 65.13%

Step-by-step explanation:

a. This problem can be solved using the binomial distribution formula: P(X=k) = (n choose k) * p^k * (1-p)^(n-k), where n is the sample size, k is the number of successes, p is the probability of success, and (n choose k) is the binomial coefficient.

For this problem, n=7, p=0.1, and we want to find P(X=3). Therefore, we have:

P(X=3) = (7 choose 3) * 0.1^3 * (0.9)^4 = 0.0574, or 5.74%.

b. We have n=5, p=0.1, and we want to find P(X=2). Therefore, we have:

P(X=2) = (5 choose 2) * 0.1^2 * (0.9)^3 = 0.0729, or 7.29%.

c. To find the probability that 4 or 5 out of 10 bulbs are defective, we can use the binomial distribution to find the probabilities of each outcome separately and add them together. We have n=10 and p=0.1.

P(4 out of 10 are defective) = (10 choose 4) * 0.1^4 * (0.9)^6 = 0.1937, or 19.37%.

P(5 out of 10 are defective) = (10 choose 5) * 0.1^5 * (0.9)^5 = 0.0081, or 0.81%.

P(4 or 5 out of 10 are defective) = P(4 out of 10 are defective) + P(5 out of 10 are defective) = 0.1937 + 0.0081 = 0.2018, or 20.18%.

d. To find the probability that no bulbs out of 10 are defective, we can use the binomial distribution with n=10 and p=0.1, and find P(X=0). Therefore, we have:

P(X=0) = (10 choose 0) * 0.1^0 * (0.9)^10 = 0.3487, or 34.87%.

e. To find the probability that one or more bulbs out of 10 are defective, we can use the complement rule and subtract the probability of no bulbs being defective from 1. Therefore, we have:

P(one or more bulbs out of 10 are defective) = 1 - P(X=0) = 1 - 0.3487 = 0.6513, or 65.13%.

Answer:

k = - 0.36 or k = 30.36

Step-by-step explanation:

k² - 30k = 11

to complete the square

add ( half the coefficient of the k- term )² to both sides

k² + 2(- 15)k + 225 = 11 + 225

(k - 15)² = 236 ( take square root of both sides )

k - 15 = ± [tex]\sqrt{236}[/tex] ≈ ± 15.36 ( to the nearest hundredth )

add 15 to both sides

k = 15 ± 15.36

Then

k = 15 - 15.36 = - 0.36

or

k = 15 + 15.36 = 30.36

The Taylor series centered at 4 for f(x) = ln(1+x) is: f(x) = ln(5) + (x-4)/5 - (x-4)^2/50 + (2/125)*(x-4)^3 - (6/625)*(x-4)^4 + ... The interval of convergence for this series is (-∞, ∞).

Let's find the Taylor series centered at 4 for the function f(x) = ln(1+x).

We can use the formula for the Taylor series coefficients:

f^(n)(x) = (-1)^(n-1) * (n-1)! / (1+x)^n

where f^(n)(x) denotes the nth derivative of f(x).

Using this formula, we can find the Taylor series centered at 4: f(4) = ln(1+4) = ln(5) f'(x) = 1/(1+x), so f'(4) = 1/5 f''(x) = -1/(1+x)^2, so f''(4) = -1/25 f'''(x) = 2/(1+x)^3, so f'''(4) = 2/125 f''''(x) = -6/(1+x)^4, so f''''(4) = -6/625 and so on.

Putting it all together, the Taylor series centered at 4 for f(x) is:

f(x) = ln(5) + (x-4)/5 - (x-4)^2/50 + (2/125)*(x-4)^3 - (6/625)*(x-4)^4 + ...

To find the interval of convergence, we can use the ratio test:

lim |(f^(n+1)(x) / f^(n)(x)) * (x-4)/(x-4)| = lim |(-1) * (n+1) * (1+x)^2 / (1+x)^n| * |x-4| = lim (n+1) * (1+x)^2 / (1+x)^n * |x-4| = lim (n+1) / (1+x)^(n-2) * |x-4|

Since this limit is zero for all values of x, the interval of convergence is the entire real line, (-∞, ∞).

So the final answer is: The Taylor series centered at 4 for f(x) = ln(1+x) is: f(x) = ln(5) + (x-4)/5 - (x-4)^2/50 + (2/125)*(x-4)^3 - (6/625)*(x-4)^4 + ... The interval of convergence for this series is (-∞, ∞).

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The capacity of 1 thumb drive is 4 gigabytes and the capacity of 1 compact disk is 2 gigabytes.

The first step is to form the system of equations that represent the information in the question:

4x + y = 18 equation 1

4x + 3y = 22 equation 2

The elimination method would be used to determined the required values.

Subtract equation 1 from equation 2

2y = 4

y = 4/2

y = 2

Substitute for y in equation 1: 4x + 2 = 18

4x = 18 - 2

4x = 16

x = 16/4

x = 4

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The domain of the function f(x) is {x | 0 ≤ x ≤ 3650} and the range is {f(x) | 20 ≤ f(x) ≤ 9145}, where f(x) represents the amount Billy would spend to rent x films.

The domain D of the function f(x) is the collection of all possible values for x. In this scenario, because Billy is not permitted to rent more than ten films every day, the maximum number of films he may rent in a year is ten times the number of days in a year, or 10 x 365 = 3650 films.

Furthermore, because he must pay a membership fee of $20 regardless of how many movies he rents, the minimum number of movies he could rent in a year is zero. As a result, the domain of the function f(x) is as follows:

D = {x | 0 ≤ x ≤ 3650}

The range R of the function f(x) is the collection of all possible values for f(x). In this scenario, the function: returns the amount Billy would spend to rent x films.

f(x) = 2.5x + 20

where 2.5x is the rental charge for x films and 20 is the membership fee. Because x can be any value between 0 and 3650, the smallest value that f(x) can be is:

f(0) = 2.5(0) + 20 = 20

and f(x) has the following maximum value:

f(3650) = 2.5(3650) + 20 = 9145

As a result, the function f(x) has the following range:

R = {f(x) | 20 ≤ f(x) ≤ 9145}

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The tomatoes start to get mushy after more than 6 days.

Note: The calculation assumes that the number of tomatoes harvested is large enough for the continuous division process to approach zero.

Let's assume that Marcos initially harvested X cherry tomatoes.

According to the given information, each day Marcos keeps one-third of the tomatoes and brings the remaining amount into the office. This means that after the first day, Marcos would have 2/3 (or 2/3X) of the original harvest left.

On the second day, he would keep one-third of the remaining tomatoes and bring 2/3 × 1/3 (or 2/9) of the original harvest into the office. This would leave him with 2/3 × 2/3 (or 4/9) of the original harvest.

In general, after N days, the amount of tomatoes left can be calculated using the following formula:

Amount of tomatoes left = (2/3)^N × X

The question states that Marcos has less than one eighty-first of the original harvest left when the tomatoes start to get mushy. In other words, when the amount of tomatoes left is less than 1/81 of the original harvest, we can write the following inequality:

[tex](2/3)^N × X < 1/81[/tex]

To find the number of days (N) when the tomatoes start to get mushy, we need to solve for N in the inequality above.

Taking the logarithm (base 2/3) of both sides of the inequality:

[tex]N > log2/3(1/81) / log2/3(2/3)[/tex]

N > 4 / (2/3)

N > 4 × 3/2

N > 12/2

N > 6

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The expression that fits perfect for the given requirement is 5z + 7.50, under the condition that a food truck owner charges z dollars per burrito combo and $1.50 for a side of guacamole.

Here we have to apply the principles of solving algebraic equations, due to the expression provided.

From the  given information, here the food truck owner charges z dollars per burrito combo along with $1.50 for a side of guacamole.

According to the information it is given that the  expression is  5(z+1.50)  which helps to state the total cost of 5 burrito combos and 5 sides of guacamole.

Lets now formulate the expression for the given required equation

= 5(z + 1.50)

= 5z + 7.50.

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A food truck owner charges z dollars per burrito combo and $1.50 for a side of guacamole. The expression 5(z+1.50) represents the total cost of 5 burrito combos and 5 sides of guacamole. What expression also represent the cost of 5 burrito combos and 5 sides if guacamole that cost 1.50 each.

The center is located at (2, −4), and the radius is 4.

To find the center and radius of this equation, we need to complete the square for both the x and y terms.

Starting with the x terms:

x^2 - 4x

To complete the square, we need to add and subtract (4/2)^2 = 4:

x^2 - 4x + 4 - 4

Now we can simplify:

(x - 2)^2 - 4

And for the y terms:

y^2 + 8y

We need to add and subtract (8/2)^2 = 16:

y^2 + 8y + 16 - 16

Simplifying:

(y + 4)^2 - 16

Now we can rewrite the original equation:

(x - 2)^2 - 4 + (y + 4)^2 - 16 = -4

Combining like terms:

(x - 2)^2 + (y + 4)^2 = 4

This is in the form of a circle:

(x - h)^2 + (y - k)^2 = r^2

Where the center is (h, k) and the radius is r.

So the center is located at (2, -4) and the radius is 4.

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The secant of a °45 angle is defined as the ratio of the length of the hypotenuse to the length of the adjacent side. In this case, the adjacent side is also a leg of length 5, so:

secant(45) = hypotenuse/adjacent = 5√2/5 = √2

The secant function, denoted as sec(x), is a trigonometric function that is defined as the reciprocal of the cosine function, cos(x).

In other words,

sec(x) = 1 / cos(x)

The secant function is defined for all values of x except for those where the cosine function is equal to zero, which corresponds to the values x = (2n+1)π/2 where n is an integer. At these points, the secant function is undefined.

In a °45-°45-°90 triangle, the two legs are congruent, so if the length of the hypotenuse is 5√2, then each leg has a length of:

leg = hypotenuse/√2 = (5√2)/√2 = 5

The sine of a °45 angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In this case, the side opposite the angle is a leg of length 5, so:

sine(45) = opposite/hypotenuse = 5/5√2 = 1/√2 = √2/2

The secant of a °45 angle is defined as the ratio of the length of the hypotenuse to the length of the adjacent side. In this case, the adjacent side is also a leg of length 5, so:

secant(45) = hypotenuse/adjacent = 5√2/5 = √2

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1. x-3y -3=0, 3x-9y-2=0 has no solution

2. 2x+y=5,3x+2y=8 has a unique solution

3. 3x-5y=20,6x-10y=40 has infinitely many solution

4.  x-3y-7 =0,3x-3y-15=0 has No solution

5. 8x+5y=9,3x+2y=4 has a unique solution

(i) To solve using substitution method, we can rearrange the first equation to x=3y+3 and substitute it into the second equation:

3(3y+3) - 9y - 2 = 0

9y + 9 - 9y - 2 = 0

7 = 0

This is a contradiction, so the pair of equations has no solution.

(ii) To solve using elimination method, we can multiply the first equation by 2 and subtract it from the second equation:

3x + 2y = 8

(4x + 2y = 10)

-x = -2

So, x = 2. Substituting this value into the first equation, we get:

2x + y = 5

2(2) + y = 5

y = 1

Therefore, the unique solution is (x,y) = (2,1).

(iii) To solve using elimination method, we can multiply the first equation by 2 and subtract it from the second equation:

6x - 10y = 40

(6x - 10y = 40)

0 = 0

This equation is true for any value of x and y, so the pair of equations has infinitely many solutions.

(iv) To solve using elimination method, we can subtract the first equation from the second equation:

3x - 3y - 15 - (x - 3y - 7) = 0

2x - 22 = 0

x = 11

Substituting this value into the first equation, we get:

11 - 3y - 7 = 0

-3y = -4

y = 4/3

Therefore, the unique solution is (x,y) = (11,4/3).

(v) To solve using elimination method, we can multiply the first equation by 2 and subtract it from the second equation:

3x + 2y = 4

(16x + 10y = 18)

-29x - 18y = -14

Solving for y, we get:

y = (29/18)x + (7/9)

Substituting this expression for y into the first equation, we get:

8x + 5((29/18)x + (7/9)) = 9

(143/18)x = 2/9

x = 2/13

Substituting this value into the expression for y, we get:

y = (29/18)(2/13) + (7/9) = 41/117

Therefore, the unique solution is (x,y) = (2/13,41/117).

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The only option that represents the universal set from the subsets is:

B: A set of cricket players of the school.

The universal set is defined as a set that is said to consist of all the elements or objects, which includes its own elements. This universal set is represented by just a symbol 'U'

Now, a subset is also defined as a part of the universal set which is basically a group under the universal set.

Now, from the given options, the only one that could possible represent a universal set of all the options given is "a set of cricket players of the school."

Therefore we conclude that Option B provides the correct answer to the universal set definition.

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The selling price of the necklace at the store is $32.64.

To calculate the selling price of the necklace, we will first find the markup amount and then add it to the original cost.

Markup = (Original Cost) x (Markup Percentage)
Markup = $24 x 36% = $24 x 0.36 = $8.64

Now, add the markup amount to the original cost to find the selling price:

Selling Price = Original Cost + Markup
Selling Price = $24 + $8.64 = $32.64

So, the selling price of the necklace at the store is $32.64.

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The expression to represent the situation is 2.85g + 3.15c.

Joe bought g gallons of gasoline for $2.85 per gallon and c cans of oil for $3.15 per can.

An algebraic expression is made up of variables and constants, along with algebraic operations such as addition, subtraction, division, multiplication etc.

Therefore, the expression that can be used to determine the total amount Joe spent on gasoline and oil can be calculated as follows:

Therefore,

total cost = 2.85g + 3.15c

where

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Ordered pairs must be placed in the following order: (6, 1), (8, 8), (12, 22). (20, 50). The right answer is D.

According to the assertions, it deals with a number of operations in mathematics.

For the first series, let the number be x, and for the second, let it be y.

The first rule is to multiply by 2, then, beginning with 6, remove 4.

The sequence can be written as 6 is the first number: = 2x - 4

next number = 2(6) - 4

                    = 12-4

                    =  8

The following phrase is thus 8. The following two terms can be written as 12 and 20.

Values of x = 6, 8, 12, 20

The second rule is to add three and then multiply by two, counting backward from one.

= (y + 3)2

First term = 1

Next term = (1 + 3)2

               =  8

The next two words can be acquired in a similar manner.

y = 1, 8, 22, 50

Therefore, ordered pairs must be placed in the following order: (6, 1), (8, 8), and (12, 22). (20, 50). The right answer is D.

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Correct question:

The first rule is to multiply by 2, then subtract 4 starting from 6. The second rule is to add 3, then multiply by 2 starting from 1. What are the first four ordered pairs using the two sequences?

A. (0, 0) (6, 1), (8, 2), (12, 22)

B. (6, 1), (10, 8), (14,22), (22, 50)

C. (6, 1), (8, 8), (12, 20), (20, 45)

D. (6, 1), (8, 8), (12, 22), (20, 50)

discount amount= CP-Disount percentage

or,480=CP-25/100

or480×100+25 =CP

48025 is the original price of shirt

The area of the playground include the following: 900 square yards.

In Mathematics and Geometry, the area of a regular polygon can be calculated by using this formula:

Area = (n × s × a)/2

Where:

In Mathematics and Geometry, the area of a parallelogram can be calculated by using the following formula:

Area of a parallelogram, A = base area × height

Area of playground, A = 45 × 20

Area of a playground, A = 900 square yards.

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The volume of the larger cone will be 48 cm³.

This is because when a shape is enlarged by a scale factor of 2, the volume is increased by a factor of 2³ (or 8). So, 6 x 8 = 48.


When we enlarge a shape by a scale factor, we are multiplying all of its dimensions by that factor. In the case of a cone, this means that we are increasing the radius and the height of the cone by a factor of 2.

We can use the formula for the volume of a cone to find out how the volume changes when we enlarge it. The formula for the volume of a cone is V = (1/3)πr²h, where r is the radius and h is the height.

If we multiply the radius and the height of the cone by 2, we get a new cone with a radius of 2r and a height of 2h. Plugging these new values into the formula for the volume of a cone, we get:

V' = (1/3)π(2r)²(2h) = (1/3)π(4r²)(2h) = (8/3)πr²h

We can simplify this expression by multiplying the original volume by 8/3:

V' = (8/3) x 6 = 48

So the volume of the larger cone is 48 cubic centimeters.

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The evaluated probability of randomly choosing a  participant dreams in black and white or color is 0.13, under the condition that the given Students are passing from psychology surveyed 200 of their fellow students regarding their dreams.

Probability means the possible chances of an event occurring in a particular time frame. It is a considered a branch of mathematics that deals with the occurrence of a random event.

The value is presented from zero to one. Probability has been induced in math to predict how prone are the events going to occur. The meaning of probability is basically the express  something that is likely to happen.

Black Probability = 4/200=0.02

White Probability = 12/200=0.06

Probability of all other color = 10/200=0.05

So, probability of randomly choosing a participant dreams in black and white or color =0.02+0.06+0.05=0.13

Therefore, the probability of randomly selecting participant dreams in black and white or color = 0.13

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The complete question is

Students majoring in psychology surveyed 200 of their fellow students about their dreams. The results of the survey are shown in the Venn diagram. Let B be the event that the participant dreams in black and white and let C be the event that the participant dreams in color.

What is the probability that a randomly selected participant dreams in black and white or color?

Mark originally had 28 shoes.

Now we can use the equation for Grace's number of shoes to find her final count:

Grace = x + 16 = 28 + 16 = 44

So Grace ends up with 44 shoes.

Let's start by setting up an equation to represent the given information:

Let the number of shoes Mark has be represented by "x"

Grace has 16 more shoes than Mark:

Grace = x + 16

Mark gives Grace 12 shoes, so Grace now has:

Grace = x + 16 + 12 = x + 28

Mark realized he has half as many shoes as Grace:

x = 0.5(x + 28)

Simplifying this equation:

x = 0.5x + 14

0.5x = 14

x = 28

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To investigate whether there is an association between cell phone brand and age-group, the researcher can conduct a chi-squared test of independence.

This test compares the observed frequencies in the contingency table to the expected frequencies if there were no association between the variables. If the test results in a p-value less than the chosen significance level (usually 0.05), then the researcher can reject the null hypothesis of no association and conclude that there is evidence of an association between cell phone brand and age-group. The degrees of freedom for this test would be (3-1) * (4-1) = 6.

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Answer:1000

Step-by-step explanation:what about 1000 people

The general solution to the given differential equation:

[tex]y = (9x + C)e^(-6x)[/tex]

To solve the given differential equation, dy + 6ydx = 9e^(-6x)dx, we can first rewrite it as a first-order linear differential equation:

[tex]dy/dx + 6y = 9e^(-6x)[/tex]

Now, we will find the integrating factor, which is e^(∫P(x)dx), where P(x) is the coefficient of y in the equation:

Integrating factor =[tex]e^(∫6dx) = e^(6x)[/tex]

Next, we multiply the entire differential equation by the integrating factor:

[tex]e^(6x)(dy/dx + 6y) = 9e^(-6x)e^(6x)[/tex]

This simplifies to:

[tex]e^(6x)(dy/dx) + 6e^(6x)y = 9[/tex]

Now, the left side of the equation is the derivative of the product of y and the integrating factor:
[tex]d/dx(y * e^(6x)) = 9[/tex]

To solve for y, integrate both sides with respect to x:

[tex]∫[d/dx(y * e^(6x))]dx = ∫9dx[/tex]

This results in:

[tex]y * e^(6x) = 9x + C[/tex]

Finally, we solve for y by dividing by e^(6x):

[tex]y = (9x + C)e^(-6x)[/tex]

And that is the general solution to the given differential equation.

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The probability is approximately 45.4% (rounded to the nearest tenth of a percent).

To find the probability that a randomly selected point within the circle falls in the white area, we need to find the area of the white region and divide it by the total area of the circle.

The total area of the circle is:

A = πr² = π(4 cm)² = 16π cm²

The area of the white region can be found by subtracting the area of the two semicircles from the area of the circle:

White area = A - 2(1/2π(2.5 cm)²) = 16π - 2(4.375π) = 7.25π cm²

So, the probability that a randomly selected point within the circle falls in the white area is:

P(white area) = (white area)/(total area) = (7.25π)/(16π) ≈ 45.4%

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