Solve the differential equation. dy + 6ydx=9e -⁶x dx y=

Answer:

18x^5 - 7x^2y - 2xy^2 + 17y^4

Step-by-step explanation:

Standard form means that the variables are in alphabetical order, and the degree of the variable closer to the start of the alphabet decreases while the degree of the variable closer to the end of the alphabet increases.

The chances of success in cycling across the United States is higher on the central route because more than half of the people who started on this route finished it (52%), while on the other routes the chances of successful completion is lower (route north (0%) south (0.02%).

According to the above, to find the probability of success in crossing the United States from coast to coast, we must divide the number of successful attempts by the total number of attempts as shown below:

North Route

0 ÷ 2 = 0

Center Route

10 ÷ 19 = 0.52

South Route

1 ÷ 39 = 0.025

According to the above, the highest probability of success is found in the central route with 0.52, that is to say that more than half of those who tried this route finished.

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The range of the relation is ( 0,2)

The set which contains all the second elements, on the other hand, is known as the range of the relation.

For example, the two terms a and b have the following values

a: 1, 2 ,4, 8, 10

b: 1, 4 , 12, 14.

The range of relation between the two values is ( 1 ,4) because the two values are common to both term.

Similarly, looking at the term x and y , we can see that only 0 and 2 are common to both terms.

Therefore the range of relation is (0,2)

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The expected value of Lucas going fishing is $26.56. This is calculated by multiplying the probability of each outcome (catching 0, 1, 2, or 3 walleye) by its corresponding payoff ($0, $0, $20, or $100) and adding the results.

To calculate the expected value of Lucas going fishing, we need to consider all possible outcomes and their respective probabilities

Lucas catches all 3 walleye Probability = (3/4) * (2/3) * (1/2) = 1/4 (since he has to catch each walleye in succession, with decreasing probabilities)

Value = $100

Lucas catches 2 walleye Probability = (3/4) * (2/3) * (1/2) * (1/4) * 3 = 9/32 (he has to catch 2 walleye in any order and then not catch the catfish in the remaining attempt)

Value = $20

Lucas catches 1 walleye Probability = (3/4) * (2/3) * (1/2) * (1/4) * (1/4) * 3 = 3/32 (he has to catch 1 walleye and then not catch the other two walleye and the catfish)

Value = $0

Lucas catches no walleye and no catfish Probability = (1/4) = 1/4 (since he has to catch the catfish)

Value = $0

Therefore, the expected value of Lucas going fishing is

E(X) = (1/4)$100 + (9/32)$20 + (3/32)$0 + (1/4)$0 = $26.56

So, on average, Lucas can expect to make $26.56 each time he goes fishing.

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You can use one of the following properties to prove that a triangle is isosceles without knowing the measure of each side.

Angle-Angle (AA): A triangle is equal if two angles of one triangle meet (equal in measure) two angles of another triangle. For an isosceles triangle, if you can prove that two angles are equal, then the triangle must be an isosceles triangle. This is because two right angles are the defining feature of an isosceles angle, as opposed to two congruent sides.

Base Angles: If the base angles of a triangle are equal (equal in measure), then the triangle is isosceles. This is because in an isosceles triangle, the base angles are always congruent.

Vertex angle bisection: If the vertex angle bisector of a triangle is also a perpendicular bisector of the opposite side (base), then the triangle is isosceles. This is because in an isosceles triangle, the two sides of the vertex angle always bisect the base and are perpendicular.

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Using Rule of thumb,

1) A common rule of thumb is to spend no more than 30% of your income on rent. Therefore, the maximum amount you should spend on rent is:

$2,800 x 0.3 = $840

2) Using the same rule of thumb, the maximum amount you should spend on rent is:

$1,900 x 0.3 = $570

3) To afford an $820/month apartment, you should aim to spend no more than 30% of your income on rent:

$820 = 0.3x

x = $2,733.33

So your monthly income must be at least $2,733.33 to afford the apartment.

4) Following the same reasoning, your monthly income must be at least:

$900 = 0.3x

x = $3,000

5) To start renting, you need to pay first and last month's rent, plus a $650 security deposit, for a total of:

$665 x 2 + $650 = $1,980

So you would need $1,980 to start renting the apartment.

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The value of the expression 65,536 x 4h for h = -4 is 256. The value represents the number of bacteria in the culture 4 hours before 8:00 A.M.

To find the value of the expression for h = -4, we substitute -4 for h in the expression [tex]65536*4^{h}[/tex]. This gives us

65536*4⁻⁴

To simplify this, we need to evaluate the exponent first. Remember that a negative exponent means we take the reciprocal of the base raised to the positive version of the exponent. So 4⁻⁴ is the same as 1/(4⁴), or 1/256.

Substituting that back into the expression, we get

65536*(1/256) = 256

So the value of the expression for h = -4 is 256.

The value of the expression in part (a) represents the number of bacteria in the culture 4 hours before 8:00 A.M. Since h is negative, we are looking at a time before 8:00 A.M. Specifically, h = -4 means we are looking at 4 hours before 8:00 A.M., or 4:00 A.M.

So the expression 65536*4⁻⁴ tells us how many bacteria were in the culture at 4:00 A.M., assuming the bacteria quadrupled every hour from that point until 8:00 A.M.

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Answer:

Step-by-step explanation:

To use a number line to find the number that is 7 more than -9, we start by finding -9 on the number line. Then, we move 7 units to the right (because we want to find a number that is 7 more), staying on the number line.

Starting from -9 on a number line, we can count 7 units to the right, which brings us to:

-9 + 7 = -2

Therefore, the number that is 7 more than -9 is -2.

The ratios that have a unit rate of 3 include the following:

A. 15/2 cups: 2 1/2 cups

C. 2 cups: 2/3 cups

F. 2 1/2 cups: 5/6 cups

In Mathematics, the unit rate is sometimes referred to as unit ratio and it can be defined as the quantity of material that is equivalent to a single unit of product or quantity.

15/2 cups : 5/2 cups

15/2 ÷ 5/2 : 5/2 ÷ 5/2

15/2 × 2/5  : 1

3 : 1  (True)

1 cup: 1/4 cups

1 × 4 : 1/4 × 1

4 : 1   (False)

2/3 cups: 1 cup

2/3 × 3/2  : 1 × 3/2

1 : 3/2   (False)

3 3/4 cups: 2 cups

(4 × 3 + 3)/4 : 2

15/4 : 2

15/8  : 2/2

15/8 : 1    (False).

2 cups: 2/3 cups

2 × 3/2 : 2/3 × 3/2

3  : 1   (True).

2 1/2 cups: 5/6 cups

(2 × 2 + 1)/2 : 5/6

5/2  : 5/6

5/2 × 6/5 : 5/6 × 6/5

3 : 1  (True).

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Complete Question:

Which ratios have a unit rate of 3? Choose all that apply.

15/2 cups: 2 1/2 cups

1 cup: 1/4 cups

2/3 cups: 1 cup

3 3/4 cups: 2 cups

2 cups: 2/3 cups

2 1/2 cups: 5/6 cups

10³ times 8 bacteria will Harriet have in the dish at the end of 6 hours, if she has 10³ bacteria now and they double every 2 hours, option D.


Starting with the initial number of bacteria: 10³

Since the bacteria double every 2 hours, after 2 hours, there will be 10³ × 2 bacteria.

After another 2 hours (total of 4 hours), the bacteria will double again: (10³ × 2) × 2 = 1³ × 2²

After the final 2 hours (total of 6 hours), the bacteria will double once more: (10³ × 2²) × 2 = 10³ × 2³

So, at the end of 6 hours, Harriet will have 10³ × 2³ bacteria in the dish. The correct answer is D. 10³ times 8.

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The algebraic expression is equivalent to the expression is 125/4x + 34. Option B

Algebraic expressions are simply defined as expressions that are composed of terms, variables, constants, coefficients and factors.

These algebraic expressions are also composed of mathematical operations, such as;

From the information given, we have;

25/4(5x-4/5)+29

expand the bracket, we get;

125x - 100/5 + 29

Find the LCM, we get;

125x - 100 + 145/5

Divide the values

125/4x + 34

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The probability of selecting a red bead is 3/20 when the probability of selecting a white bead, replacing it, and then selecting a red bead.

We need to find the probability of selecting a red bead when first a white bead is selected and then it is replaced and then selected a red bead. The formula to find the probability is,

P(A) = f / N

Where,

f = number of outcomes

N = total number of outcomes

Given data:

Total number of beads =  10

Number of blacks beads = 2

Number of white beads = 3

Number of red beads = 5

The probability of selecting a white bead is given as,

P(A) = f / N

P(W) = 3/10

When the bead is replaced, the probability of selecting a red bead is  P(R) = 5/10

The probability of selecting a white bead and then a red bead is the product of the probabilities of each event:

P(white and red) = P(white) × P(red)

= (3/10) × (5/10)

= 3/20

Therefore, the probability of selecting a white bead, replacing it, and then selecting a red bead is 3/20.

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Answer: B, C and E

Step-by-step explanation: the term growing means the value is increasing, and a number multiplied by 1 gives the same number; therefore in order for that number to increase, it need to be multiplied by a bigger number which is greater than 1, in this case it can not be a number less than 0 like 0.85, so all the number which are greater than 1 in the bracket are correct.

the number before the bracket is the population before increasing, so you can substitute any value for t and try out different outcomes, you will get only B, C and E as number greater than the original number

The equation of the straight line connecting these points in the form of y = mx + c is given as y = (1/2)x + 1.

The line in the slope-intercept equation is given as,

y = mx + c

where:

m = the slope of the line

c = the y-intercept

The slope m of the line connecting two points (x1, y1) and (x2, y2) is given by the formula

= (y₂ - y₁) / (x₂ - x₁)

Substituting the coordinates values of P and Q into this formula, we get

[tex]m = (3 - 0) / (4 - (-2))[/tex]

= 3/6

= 1/2

Therefore, the value of the m is 1/2

We can find the value of c by substituting the m and x values in the equation. using the point P and  Substituting x and y values in the equation we get

x = - 2

y = 0  

[tex]0 = (1/2) × (-2) + c[/tex]

c = 1

Therefore, the value of c is 1.

By substuting the m and c values in the standard slope-intercept formula we get y = (1/2)x + 1.

Therefore, the equation of the line connecting points P and Q is y = (1/2)x + 1.

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Step-by-step explanation:

If u want to find the volume

V= width × length × height

V= 8.5 cm × 12 cm × 29 cm

V= 2958 cm3

It would be more efficient to take the new road from Crimson to Sunshine to get home.

To determine the shortest route home, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In this case, the two sides are the distances you traveled west (6 miles) and south (8 miles).

Using the Pythagorean theorem, we have:

Hypotenuse² = 6² + 8²
Hypotenuse² = 36 + 64
Hypotenuse² = 100
Hypotenuse = √100 = 10 miles

So, the shortest route home (the new road) is 10 miles. If you go back the way you came, you would travel 6 miles west and then 8 miles north, for a total of 14 miles. The longer route (14 miles) is 4 miles longer than the shorter route (10 miles). Therefore, you should take the new road to get home more quickly.

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The circle has horizontal tangent lines at the points (1, 2 + √(14)) and (1, 2 - √(14)) and the circle has vertical tangent lines at the points (1 + √(8), -2) and (1 - √(8), -2).

To find the points where the circle defined by x^2 + y^2 - 2x - 4y = 11 has horizontal and vertical tangent lines, we need to use implicit differentiation to find the derivatives of x and y with respect to each other.

Taking the derivative of both sides of the equation with respect to x, we get:

2x + 2y(dy/dx) - 2 - 4(dy/dx) = 0

Simplifying, we get:

(dy/dx) = (x-1) / (y+2)

To find the points where the circle has horizontal tangent lines, we need to find where the derivative dy/dx is equal to zero. This occurs when x-1 = 0, or x = 1. Substituting this value of x back into the original equation, we get:

1 + y^2 - 4y = 11

Simplifying, we get:

y^2 - 4y - 10 = 0

Using the quadratic formula, we get:

y = 2 ± √(14)

To find the points where the circle has vertical tangent lines, we need to find where the derivative dy/dx is undefined, which occurs when y+2 = 0, or y = -2. Substituting this value of y back into the original equation, we get:

x^2 - 2x + 4 = 11

Simplifying, we get:

x^2 - 2x - 7 = 0

Using the quadratic formula, we get:

x = 1 ± √(8)

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The critical value z a/2 that corresponds to a 97% level of confidence is 1.96, and the critical value z a/2 that corresponds to an 80% level of confidence is 1.28.

In statistical hypothesis testing, the critical value is the point beyond which we reject the null hypothesis. To find the critical values, we need to use the standard normal distribution table or calculator.

For a 97% level of confidence, the significance level (α) is 0.03, and the critical value can be found by dividing α by 2 to get 0.015 (since it's a two-tailed test), and then finding the corresponding z-value using the standard normal distribution table or calculator. The z-value is 1.96.

Similarly, for an 80% level of confidence, the significance level (α) is 0.20, and the critical value can be found by dividing α by 2 to get 0.10 (since it's a two-tailed test), and then finding the corresponding z-value using the standard normal distribution table or calculator. The z-value is 1.28.

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The value of the expression when x=8 is 56/9.

To find the value of an expression, follow these steps:

To find the value of the expression when x=8, we substitute 8 for x in the expression:

(8^2 - 8) / (8+1)

= (64 - 8) / 9

= 56/9

Therefore, the value of the expression when x=8 is 56/9.

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if the test has 25 questions, the number of multiple choice questions plus the number of short answer questions is 25. The correct answer is c.

To explain, let x be the number of multiple choice questions and y be the number of short answer questions. We know that there are 25 questions in total, so x + y = 25.

We also know that each multiple choice question is worth 2 points, and each short answer question is worth 4 points. If we let M be the total number of points from the multiple choice questions, and S be the total number of points from the short answer questions, we can set up the equation:

M + S = 80

We can also express M and S in terms of x and y:

M = 2x
S = 4y

Substituting these equations into the first equation, we get:

2x + 4y = 80

Dividing both sides by 2:

x + 2y = 40

Now we have two equations with two variables:

x + y = 25
x + 2y = 40

Subtracting the first equation from the second, we get:

y = 15

Substituting this into the first equation, we get:

x + 15 = 25

x = 10

Therefore, there are 10 multiple choice questions on the test.

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"The median student height in Class A is equal to the median student height in Class B."

Based on the given information, we can conclude that the median student height in Class A is e.

qual to the median student height in Class B. However, we cannot make any definitive conclusions about the range or mean heights of the two classes based on this limited information.

The range is a measure of the spread of the data and is calculated by subtracting the minimum value from the maximum value. Without knowing the actual height values for each student in both classes, we cannot compare the ranges and determine which class has a greater range.

The mean height is a measure of the central tendency of the data and is calculated by adding up all the heights and dividing by the total number of students. Again, without knowing the actual height values, we cannot calculate the mean heights for each class and compare them.

Therefore, the only conclusion that can be made based on the given information is that the median student height in Class A is equal to the median student height in Class B.

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Tim and Bella working together can paint the room in 2.4 hours.

We can use the formula:

1/Total Work = 1/Time1 + 1/Time2

where Total Work is the work to be done (in this case, painting the room), Time1 is the time taken by Tim to complete the work, and Time2 is the time taken by Bella to complete the work.

Let x be the time taken by Tim and Bella working together to complete the work. Then we can write:

1/1x = 1/6 + 1/4

Simplifying this equation, we get:

1/1x = 5/12

Multiplying both sides by x, we get:

x = 12/5 = 2.4 hours.

Therefore, Tim and Bella working together can paint the room in 2.4 hours.

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Paul has 18 interest-free days for the $300 stereo purchase.

He will need to pay the full balance of his June billing statement.

If Paul's credit card closes on the 6th of the month and his payment is due on the 24th, then he has 18 days between the close of the billing cycle and the due date of his payment.

If Paul purchases a stereo for $300 on June 8th, then the transaction will be included in his billing cycle for the month of June. Since his billing cycle closes on the 6th, the $300 charge will appear on his June billing statement.

If Paul pays off his credit card in full each month, then he will need to pay the full balance of his June billing statement by the due date of June 24th to avoid finance charges. This means he will need to pay $300 for the stereo, plus any other charges that may have been included on his billing statement for the month of June.

Therefore, the number of interest-free days that Paul will have for the $300 stereo purchase is 18 days, which is the number of days between the billing cycle close date (June 6th) and the payment due date (June 24th).

To summarize:

Paul has 18 interest-free days for the $300 stereo purchase.

Paul will need to pay the full balance of his June billing statement, including the $300 stereo charge, by June 24th to avoid finance charges.

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The probability of picking a number greater than 1 or a divisor of 2 is 3/4.

The numbers greater than 1 or divisor of 2 in the set {1, 2, 3, 4} are {2, 3, 4}.

Therefore, the probability of picking a number greater than 1 or a divisor of 2 is:

P(greater than 1 or divisor of 2) = P(2 or 3 or 4) = P(2) + P(3) + P(4)

Since there are four equally likely outcomes, the probability of each individual outcome is 1/4. Thus, we have:

P(greater than 1 or divisor of 2) = 1/4 + 1/4 + 1/4 = 3/4

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Alexi to consider these factors before deciding how many apples to produce depends on the demand for apples in her area, the size of her garden, and her ability to produce apples efficiently.

To determine Alexi's total revenue, we need to know how many apples she plans to sell. Let's assume that Alexi plans to sell X apples.

If Alexi sells each apple for $3, her total revenue will be:

Total revenue = Price per apple x Number of apples sold

Total revenue = $3 X X

Total revenue = $3X

To determine the cost of producing the apples, we need more information about Alexi's production costs. These costs can include expenses such as land, labor, water, and equipment.

Once we know the production costs, we can subtract them from the total revenue to determine Alexi's profit. If the profit is positive, then Alexi will earn money by selling the apples.

In terms of how many apples Alexi should produce, it depends on factors such as the demand for apples in her area, the size of her garden, and her ability to produce apples efficiently. It's important for Alexi to consider these factors before deciding how many apples to produce.

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Let's assume that the cylindrical can has a height of "h" and a radius of "r". We want to find the values of "h" and "r" that minimize the amount of material needed to manufacture the can.

The amount of material needed to manufacture the can can be represented by the surface area of the can, which is the sum of the area of the top and bottom circles and the lateral area of the cylinder.

The area of the top and bottom circles can be calculated using the formula for the area of a circle:

A_top = A_bottom = πr^2

The lateral area of the cylinder can be calculated using the formula for the lateral surface area of a cylinder:

A_lateral = 2πrh

Therefore, the total surface area of the cylindrical can can be calculated as:

A_total = A_top + A_bottom + A_lateral

= 2πr^2 + 2πrh

Now, we need to express "h" in terms of "r" and the volume of the can, which is given as 180 cm^3. The formula for the volume of a cylinder is:

V = πr^2h

Substituting the given value of the volume and solving for "h", we get:

h = 180/(πr^2)

Substituting this expression for "h" in the equation for the total surface area, we get:

A_total = 2πr^2 + 2πr(180/(πr^2))

= 2πr^2 + 360/r

To find the values of "r" and "h" that minimize the surface area, we need to take the derivative of "A_total" with respect to "r", set it equal to zero, and solve for "r".

dA_total/dr = 4πr - 360/r^2 = 0

Solving for "r", we get:

r = (360/(4π))^(1/3) ≈ 4.35 cm

Substituting this value of "r" in the expression for "h", we get:

h = 180/(π(4.35)^2) ≈ 3.9 cm

Therefore, the height and radius that minimize the amount of material needed to manufacture the can are approximately 3.9 cm and 4.35 cm, respectively.

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Since the limit of the ratio is less than 1 (1/3 < 1), the series is absolutely convergent (Option A). To determine the convergence of the series, we can use the Ratio Test. The series in question is:

Σ (3n² + 3) / (3n³ + 1)
First, we find the ratio between consecutive terms:
R = (a_(n+1)) / a_n = ((3(n+1)² + 3) / (3(n+1)³ + 1)) / ((3n² + 3) / (3n³ + 1))
R = (3(n+1)² + 3)(3n³ + 1) / ((3n² + 3)(3(n+1)³ + 1))
Now, as n goes to infinity:
lim (n -> ∞) R = lim (n -> ∞) (9n² + 6n + 3)(3n³ + 1) / (9n² + 3)(9n³ + 3n² + 3n + 1)
In this case, the highest power of n in the numerator and denominator is n⁵. To simplify, we can divide both the numerator and denominator by n⁵:
lim (n -> ∞) (9 + 6/n + 3/n²)(3 + 1/n²) / (9 + 3/n)(9 + 3/n² + 3/n + 1/n³)
As n approaches infinity, the terms with n in the denominator approaches zero:
lim (n -> ∞) (9)(3) / (9)(9) = 27 / 81 = 1/3
Since the limit of the ratio is less than 1 (1/3 < 1), the series is absolutely convergent (Option A).

To determine whether the series 3n² +  3n³ +1 is absolutely convergent, conditionally convergent, or divergent, we can use the ratio test.
Using the ratio test, we have:
lim n→∞ |(3(n+1)² + 3)/(3n² + 3n³ +1)
= lim n→∞ |(3n² + 6n + 3)/(3n³ + 3n² +1)
= lim n→∞ |(n² + 2n + 1)/(n³ + n²)
= 0
Since the limit is less than 1, the series is absolutely convergent. Therefore, the answer is A. Absolutely convergent.

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The height of the pile of books in terms of a and b is 4a cm + 2b cm.

To find the height of the pile of books in terms of a and b, we need to multiply the number of books by the thickness of each book. Since there are (2a+b) books in the pile and the thickness of each book is 2cm, the height of the pile can be expressed as:

(2a+b) x 2cm

Expanding this expression, we get:

4a cm + 2b cm

Therefore, the height of the pile of books in terms of a and b is 4a cm + 2b cm. This means that for every additional 'a' book, the height of the pile will increase by 4cm and for every additional 'b' book, the height of the pile will increase by 2cm.

It's important to note that this formula assumes that all the books are the same size and have the same thickness. If there are any variations in size or thickness, the formula may not accurately represent the height of the pile. Additionally, it's important to ensure that all units of measurement are consistent (in this case, cm for both the thickness of the book and the height of the pile).

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The measure of the smallest angle is 37.1 degrees

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

The legs of a right triangle measure 11.4 meters and 15.1 meters

So, the measure of one of the acute angles is

tan(x) = 11.4/15.1

Evaluate

tan(x) = 0.7550

Take the arc tan of both sides

So, we have

x = 37.1

This means that the measure of the smallest angle is 37.1 degrees

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The probability that Sam will get a parking ticket given that he has to park in a no-parking zone on Monday is approximately 0.1308 or 13.08%.

We can use Bayes' theorem to solve this problem. Let A be the event that Sam gets a parking ticket and B be the event that Sam parks in a no-parking zone. Then, we want to find P(A|B), which is the conditional probability of A given B.

Bayes' theorem states that P(A|B) = P(B|A)*P(A)/P(B), where P(B|A) is the probability of B given A, P(A) is the prior probability of A, and P(B) is the prior probability of B.

From the problem, we know that P(A) = 0.07, P(B|A) = 1 (since if Sam parks in a no-parking zone, he will definitely get a parking ticket), and P(B|not A) = 0.50 (since if Sam finds a legal parking space, he has a 0.50 probability of parking in a no-parking zone).

To find P(B), we can use the law of total probability, which states that P(B) = P(B|A)*P(A) + P(B|not A)*P(not A), where P(not A) = 1 - P(A).

Therefore, P(B) = 10.07 + 0.50(1-0.07) = 0.5351.

Finally, we can use Bayes' theorem to find P(A|B):

P(A|B) = P(B|A)P(A)/P(B) = 10.07/0.5351 ≈ 0.1308.

Therefore, the probability that Sam will get a parking ticket given that he has to park in a no-parking zone on Monday is approximately 0.1308 or 13.08%.

To know more about probability, refer here:

https://brainly.com/question/30034780s#

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