TRUE or FALSE:


1. Each exterior angle of a regular hexagon is acute


2. The sum of the interior angles of a polygon is not necessarily a multiple of 180


3. In any polygon, the larger the number of vertices, the smaller the measure of an exterior angle

Answer: The probability of getting heads on a fair coin is 1/2, and the probability of rolling an even number on an eight-sided fair die is 4/8 = 1/2. Therefore, the probability of getting both heads and an even number is:

P(heads and even number) = P(heads) x P(even number) = (1/2) x (1/2) = 1/4

The probability of getting tails on a fair coin is also 1/2, and the prime numbers less than 4 are 2 and 3. The probability of rolling a 2 or a 3 on an eight-sided fair die is 2/8 = 1/4. Therefore, the probability of getting tails and a prime number less than 4 is:

P(tails and prime number less than 4) = P(tails) x P(prime number less than 4) = (1/2) x (1/4) = 1/8

So the probability that Oscar gets heads and an even number is 1/4, and the probability that Oscar gets tails and a prime number less than 4 is 1/8.

Step-by-step explanation: can i get brianliest :D

The probability would be 7/12.

Here, we have,

Consider A is the event that there is snowing in first three days and B is the event that there is snowing in next four days.

According to the question,

P(A) = 1/3

P(B)= 1/4

Thus, the probability that it snows at least once during the first week of January

= snow in first three days or snow in next four days

= P(A∪B)

=P(A) + P(B) - P(A∩B)

 ( ∵ A and B are independent ⇒P(A∩B) = 0  )

=1/3 + 1/4

=7/12

Hence, The probability would be 7/12.

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The area of the square is 16 square units.

The area of the rhombus is 24 square units.

Part 1: Using the distance formula, we can find the lengths of the diagonals:

Diagonal AC = √[tex][(7-1)^2 + (2-10)^2[/tex]] = √(36 + 64) = √100 = 10

Diagonal BD = √[[tex](12-(-4))^2 + (12-0)^2[/tex]] = √(256 + 144) = √400 = 20

Part 2: The area of a rhombus can be calculated using the formula: Area = (diagonal 1 x diagonal 2)/2.

So, for this rhombus, the area = (10 x 20)/2 = 100 square units.

Part 3: The area formulas for a square and rhombus are not interchangeable because they have different ways of computing their areas. A square has all four sides equal in length, while a rhombus has opposite sides equal in length. The diagonals of a square bisect each other at right angles, and each diagonal divides the square into two congruent triangles, so the area of a square is simply side length squared (Area =[tex]s^2[/tex]). However, the diagonals of a rhombus bisect each other at right angles, but they do not necessarily divide the rhombus into congruent triangles. Therefore, the area of a rhombus is calculated using the lengths of the diagonals (Area = (diagonal 1 x diagonal 2)/2).

For example, let's consider a square with side length 4 units and a rhombus with diagonals 6 units and 8 units.

The area of the square = [tex]4^2 = 16[/tex]square units.

The area of the rhombus = (6 x 8)/2 = 24 square units.

As we can see, the formulas are not interchangeable.

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The function f(x) = 11x - 11e^(-x) is concave upward on the interval (-∞, ln(11)) and concave downward on the interval (ln(11), ∞). The inflection point is at x = ln(11).

Find the first and second derivatives of the function f(x):

f'(x) = 11 + 11e^(-x)

f''(x) = 11e^(-x)

Set the second derivative equal to zero to find any potential inflection points:

11e^(-x) = 0

e^(-x) = 0

There are no solutions to this equation, so there are no inflection points in the function.

Determine the sign of the second derivative on either side of the potential inflection point(s) to identify the intervals of concavity:

For x < ln(11), e^(-x) > 0, so f''(x) > 0, meaning the function is concave upward on the interval (-∞, ln(11)).

For x > ln(11), e^(-x) < 0, so f''(x) < 0, meaning the function is concave downward on the interval (ln(11), ∞).

Therefore, the final answer is: The function f(x) = 11x - 11e^(-x) is concave upward on the interval (-∞, ln(11)) and concave downward on the interval (ln(11), ∞). The inflection point is at x = ln(11).

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Light travels at a constant speed of approximately 3 x 10⁸ meters per second in a vacuum, which is also known as the speed of light.

The speed of light is a fundamental constant in physics and is denoted by the symbol "c". In a vacuum, such as outer space, light travels at a constant speed of approximately 299,792,458 meters per second, which is equivalent to 3 x 10⁸ meters per second (to three significant figures).

In the question, we were given the distance that light travels in one year (9.45 x 10¹⁵ meters) and the number of seconds in one year (3.15 x 10⁷ seconds). To find how far light travels per second, we simply divided the distance per year by the time per year.

To find how far light travels per second, we need to divide the distance it travels in a year by the number of seconds in a year:

Distance per second = Distance per year / Time per year

Distance per second = 9.45 x 10¹⁵ meters / 3.15 x 10⁷ seconds

Distance per second = 3 x 10⁸ meters per second (approx.)

Therefore, light travels approximately 3 x 10⁸ meters per second, which is also known as the speed of light.

It is worth noting that the speed of light is an extremely important quantity in physics and has many implications for our understanding of the universe. For example, the fact that the speed of light is constant in all reference frames is a key component of Einstein's theory of relativity. Additionally, the speed of light plays a crucial role in astronomy and cosmology, as it allows us to measure the distances between celestial objects and study the behavior of light over vast distances.

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Answer: 5/6

Step-by-step explanation:

probabilities = possibilities/outcome

To get an even you can possible roll 2, 4, 6 with 6 total outcomes

P(even) = probability of getting an even = 3/6

To get a divisor of 3: only 3 and 6 can be divided by 3

P(divisor of 3)=2/6

Because of the or in the statement, you add the 2 probabilities

P(even or divisor of 3) =  3/6+2/6 = 5/6

Answer:

Step-by-step explanation:

False.

Solids with curves can be laid out as a net. In fact, there are many solids with curves that can be represented by a net. For example, a cylinder can be represented by a net consisting of two circles for the top and bottom faces and a rectangle wrapping around the circumference of the circles for the side face. Similarly, a cone can be represented by a net consisting of a circle for the base and a sector of a circle for the lateral face, which can be unfolded and attached to the circular base.

While it may be more difficult to visualize and create a net for a solid with curves compared to a solid with flat faces, it is still possible to do so in many cases.

The variance in the number of rainy days in Marina's city in two weeks during the month of August is approximately 3.96.

We know that the probability of rain on any given day is0.43, and we want to find the  friction in the number of  stormy days in two weeks, which is 14 days.  

The  friction of a binomial distribution is given by the formula  

Var( X) =  npq  

where n is the number of trials,

p is the probability of success on each trial,

and q is the probability of failure on each trial( q =  1- p).

 In this case, n =  14,

p = 0.43, and

q = 0.57.

Substituting these values, we get  

Var( X) =  npq  

Var( X) =  14 x0.43 x0.57  

Var( X) = 3.9642  

Rounding to three significant  numbers, we get   Var( X) = 3.96

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There is a vertical asymptote at x = 0.

A vertical asymptote is a vertical line that the graph of a function approaches but never touches or crosses. In the case of a rational function such as f(x) = x^3/(x^2-4), vertical asymptotes occur where the denominator of the function is equal to zero.

In this case, the denominator is x^2 - 4, which is equal to zero when x = ±2. However, we need to check whether these values are in the domain of the function. Since the interval of interest is (–18, 19), we see that only x = 2 is in the domain of the function.Therefore, the only vertical asymptote of the function f(x) = x^3/(x^2-4) on the interval (–18, 19) is at x = 0, which is the value of x where the denominator is closest to zero.

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The volume of the can is 36π cubic inches, the surface area is 44π square inches, and the ratio of surface area to volume is 11/9.

To find the volume and surface area of a can with a diameter of 4 inches and a height of 9 inches, you can follow these steps:

1. Calculate the radius, Since the diameter is 4 inches, the radius (r) is half of that, which is 2 inches.

2. Find the volume, The formula for the volume (V) of a cylinder is V = πr^2h, where r is the radius and h is the height. In this case, V = π(2^2)(9) = 36π cubic inches.

3. Calculate the surface area, The formula for the surface area (A) of a cylinder is A = 2πrh + 2πr^2. Here, A = 2π(2)(9) + 2π(2^2) = 36π + 8π = 44π square inches.

4. Determine the ratio of surface area to volume, To find this ratio, divide the surface area by the volume. In this case, the ratio is (44π)/(36π). The π's cancel out, and the ratio simplifies to 44/36, which further simplifies to 11/9.

So, the volume of the can is 36π cubic inches, the surface area is 44π square inches, and the ratio of surface area to volume is 11/9.

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Answer:

X = 2² * 3² * 5³ * 7.

Step-by-step explanation:

140 = 2*2*5*7

15^2 = 3*5*3*5

So X = 2² * 3² * 5³ * 7

To find the prime factorization of X, we first need to simplify the expression.

Now, we can factor 140 and 225 into their prime factors:

So,

We can now use the distributive property to multiply these factors together:

Therefore, the prime factorization of X is:

The first four nonzero terms of the Taylor series for f(y) about 0 are:

[tex]-2y^2 + 48y^4/2![/tex] + ... = -[tex]2y^2 + 24y^4[/tex] + ...

To find the Taylor series for the function f(y) = ln(1 - 2y^4) about 0, we need to compute its derivatives at 0 and evaluate them at each term. Let's start by finding the first four derivatives:

f(y) = ln(1 - 2[tex]y^4)[/tex]

f'(y) = [tex]-8y^3 / (1 - 2y^4)[/tex]

f''(y) =[tex](24y^6 - 32y^2) / (1 - 2y^4)^2[/tex]

f'''(y) =[tex](-144y^9 + 384y^5) / (1 - 2y^4)^3[/tex]

f''''(y) =[tex](1920y^12 - 7680y^8 + 3456y^4) / (1 - 2y^4)^4[/tex]

Now we can evaluate each derivative at 0 to get the first four nonzero terms of the Taylor series:

f(0) = ln(1) = 0

f'(0) = 0

f''(0) = -2

f'''(0) = 0

f''''(0) = 48

Therefore, the first four nonzero terms of the Taylor series for f(y) about 0 are: -2y^2 + 48y^4/2! + ... = -2y^2 + 24y^4 + ...

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The simple interest rate charged on the purchase per monthly payment is approximately 0.5735%.

To determine the simple interest rate charged per monthly payment on the flat screen television, please follow these steps:

1. Calculate the total amount paid in monthly payments: 24 payments * $80 = $1920.
2. Subtract the down payment: $1920 - $100 = $1820.
3. Subtract the original cost from the total amount paid: $1820 - $1600 = $220. This is the total interest paid.
4. Divide the total interest by the number of monthly payments: $220 / 24 = $9.1667 interest per month.
5. Calculate the interest rate per monthly payment: ($9.1667 / $1600) * 100 = 0.5735% per month.

The simple interest rate charged on the purchase per monthly payment is approximately 0.5735%.

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The runner would take a total of 93.48 seconds to complete the sprint 19 times.

To find the total time the runner takes to complete the sprint 19 times, we can multiply the time it takes for one sprint by the number of sprints:

Total time = 4.92 seconds/sprint * 19 sprints

Total time = 93.48 seconds

Therefore, the runner would take a total of 93.48 seconds to complete the sprint 19 time.

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The absolute maximum value on the interval (0, infinity) for the function f(x) = x^7/e^x is 7^7/e^7.

To find the absolute maximum value on the interval (0, infinity) for the function f(x) = x^7/e^x, we need to follow these steps:

1. Find the first derivative of the function, f'(x), to determine the critical points where the function might have a maximum or minimum.
2. Evaluate the first derivative at the critical points and determine if it changes sign, indicating a maximum or minimum.
3. Verify if the function has an absolute maximum on the given interval.

Step 1: Find the first derivative f'(x) using the quotient rule.
f'(x) = (e^x * 7x^6 - x^7 * e^x) / (e^x)^2

Step 2: Simplify f'(x) and find the critical points.
f'(x) = x^6(7 - x) / e^x
f'(x) = 0 when x = 0 (not included in the interval) or x = 7

Step 3: Evaluate the first derivative around the critical point x = 7 to determine if it's a maximum or minimum.
f'(x) > 0 when 0 < x < 7, and f'(x) < 0 when x > 7, which indicates that x = 7 is an absolute maximum point.

Now we can find the absolute maximum value by plugging x = 7 into the original function, f(x):
f(7) = 7^7/e^7

Thus, the absolute maximum value on the interval (0, infinity) for the function f(x) = x^7/e^x is 7^7/e^7.

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a) The quantity (in hundreds of units) that will give maximum profit: [tex]x^{2}[/tex] + 6x - 3022 = 0

b) The maximum profit is approximately $202,573.42.

An equation is a mathematical statement that shows that two expressions are equal. It typically contains variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division.

(a) To find the quantity that will give maximum profit, we need to find the level of output at which marginal revenue equals marginal cost.

The total revenue function for the monopoly is TR = px, where p is the price and x is the quantity sold. The marginal revenue is the derivative of the total revenue with respect to x, which is MR = d(TR)÷dx = p + x(dp÷dx).

To find the marginal revenue function, we differentiate the demand function with respect to x:

dp÷dx = -(2÷3)x

So, the marginal revenue function is:

MR = (2104 - (1÷3)[tex]x^{2}[/tex]) - (2÷3)[tex]x^{2}[/tex]

To find the marginal cost function, we differentiate the average cost function with respect to x:

dC÷dx = 22 + 2x

So, the marginal cost function is:

MC = 22 + 2x

To find the level of output at which MR = MC, we set the two functions equal to each other:

(2104 - (1÷3)[tex]x^{2}[/tex]) - (2÷3)[tex]x^{2}[/tex] = 22 + 2x

Simplifying this equation, we get:

[tex]x^{2}[/tex] + 6x - 3022 = 0

(b) Using the quadratic formula, we find that:

x = (-6 ± √( 36- 4(1)(-3022))) / 2(1)

x = (-6 ± √(36444)) / 2

x = (-6 ± 190.81) ÷ 2

x ≈ -98.4 or x ≈ 92.4

Since we can't produce a negative quantity, we choose x ≈ 92.4 as the quantity that will give maximum profit.

Therefore, the level of output that will maximize profit is 924 units (in hundreds of units).

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No, the values in the table do not represent a proportional relationship because the y-values are not a constant multiple of the corresponding x-values.

Based on the given table:

x | 0 | 1 | 2 | 3
y | 0 | 3 | 5 | 6

The values in the table do not represent a proportional relationship. To be proportional, all of the y-values should be a constant multiple of the corresponding x-values. However, in this case, the ratio of y/x is not constant (3/1, 5/2, 6/3). Therefore, the relationship is not proportional.

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The side lengths of the three squares are 8, 9, and 10, and the total perimeter is 27 units.

Let's call the side length and perimeter of the first square "s", the second square "s+1", and the third square "s+2".

The area of a square is given by the formula A =[tex]s^2[/tex], where A is the area and s is the side length.

So the total area of the three squares is:

A_total =[tex]s^2[/tex] + (s+1[tex])^2[/tex] + (s+2[tex])^2[/tex]

We are given that the total area is 245 square units:

A_total = 245

Substituting this into our expression for A_total, we get:

245 =[tex]s^2[/tex] + (s+1[tex])^2[/tex] + (s+2[tex])^2[/tex]

Expanding the squares, we get:

245 = 3[tex]s^2[/tex]+ 6s + 5

Simplifying, we get a quadratic equation:

3[tex]s^2[/tex]+ 6s - 240 = 0

Dividing by 3, we get:

[tex]s^2[/tex] + 2s - 80 = 0

We can factor this quadratic as:

(s+10)(s-8) = 0

So s = -10 or s = 8. Since s must be positive (it represents a side length), we have:

s = 8

Therefore, the side lengths of the three squares are 8, 9, and 10.

The total perimeter is the sum of the side lengths of the three squares:

P_total = s + (s+1) + (s+2)

P_total = 8 + 9 + 10

P_total = 27

Therefore, the total perimeter is 27 units.

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Therefore, the value of x to the nearest hundredth is approximately 72.73 units.

A triangle is a geometrical shape that consists of three straight sides and three angles. It is a polygon with three vertices, and the sum of its interior angles always adds up to 180 degrees. Triangles are classified based on the length of their sides and the measure of their angles. The most common types of triangles are equilateral, isosceles, and scalene, based on the length of their sides, and acute, right, and obtuse, based on the measure of their angles. Triangles have a wide range of applications in mathematics, physics, engineering, and other fields.

Here,

Therefore, in this triangle, sin(67°) = 67/x, where x is the length of the hypotenuse.

We can rearrange this equation to solve for x:

x = 67 / sin(67°)

Using a calculator, we find that sin(67°) is approximately 0.921, so:

x = 67 / 0.921

x ≈ 72.73

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Using circle theorems  the angles in triangle ΔABM are

Circle theorems are theorems that govern circle peoperties

To find the angles in triangle ABM, we notice that in ΔABO, since OB = OA (radius of the circle), then ΔABO is an isoceles triangle.

So, ∠OAB = ∠ABO  (Base angles of an isoceles triangle)

Now, ∠OAB = ∠ABO = ∠A = 30°

Also ∠OAB + ∠ABO = ∠BOM (sum of opposite interior angles)

30° +  30° = ∠BOM

∠BOM = 60°

Now since OB is the radius and touches MV at B, and thus perpendicular to it. so, ∠OBM = 90°

Now, ∠OAB + ∠OBM = ∠ABM

30° + 90° = ∠ABM

∠ABM = 120°

Now in triangle ΔABM

∠BAM + ∠ABM + ∠BMA = 180° (sum of angles in a triangle)

Now

So, making ∠BMA subject of the formula, we have that

∠BMA = 180° - ∠BAM + ∠ABM

So, substituting the values of the variables into the equation, we have that

∠BMA = 180° - ∠BAM + ∠ABM

∠BMA = 180° - 30° - 120°

∠BMA = 180° - 150°

∠BMA = 30°

So, the angles are

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To find the volume and surface area of the composite figure, we need to find the volume and surface area of the right cylinder and the hemisphere separately, and then add them together.

First, let's find the volume of the right cylinder.

The formula for the volume of a right cylinder is V = πr^2h, where r is the radius and h is the height. In this case, the radius is half of the diameter, which is 4/2 = 2 inches. So, the volume of the right cylinder is:

V1 = π(2)^2(12) = 96π cubic inches

Next, let's find the volume of the hemisphere. The formula for the volume of a hemisphere is V = (2/3)πr^3, where r is the radius. In this case, the radius is also 2 inches. So, the volume of the hemisphere is:

V2 = (2/3)π(2)^3 = 16/3π cubic inches

To find the total volume of the composite figure, we just need to add V1 and V2:

Vtotal = V1 + V2 = 96π + 16/3π = 320/3π cubic inches

Now, let's find the surface area of the composite figure.

To do this, we need to find the surface area of the curved surface of the cylinder, the top of the cylinder, and the curved surface of the hemisphere separately, and then add them together.

The formula for the surface area of the curved surface of a cylinder is A = 2πrh, where r is the radius and h is the height. In this case, the radius is still 2 inches, and the height is 12 inches. So, the surface area of the curved surface of the cylinder is:

A1 = 2π(2)(12) = 48π square inches

The formula for the surface area of the top of a cylinder is A = πr^2, where r is the radius. In this case, the radius is still 2 inches. So, the surface area of the top of the cylinder is:

A2 = π(2)^2 = 4π square inches

Finally, the formula for the surface area of the curved surface of a hemisphere is A = 2πr^2, where r is the radius. In this case, the radius is still 2 inches. So, the surface area of the curved surface of the hemisphere is:

A3 = 2π(2)^2 = 8π square inches

To find the total surface area of the composite figure, we just need to add A1, A2, and A3:

Atotal = A1 + A2 + A3 = 48π + 4π + 8π = 60π square inches

So, the volume of the composite figure is 320/3π cubic inches, and the surface area of the composite figure is 60π

square inches.

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The original price of the television was $240.

Let's denote the original price of the television by "x".

From the first sentence, the website applies a 4.5% processing fee and charges $6.00 for shipping. Therefore, the cost of the television with these fees is:

x + 0.045x + 6.00 = 1.045x + 6.00

From the second sentence, the customer uses a coupon that takes 15% off of the final price. Therefore, the price after the discount is:

0.85(1.045x + 6.00) = 0.88825x + 5.10

The problem states that the customer paid $218.28 for the order. Therefore, we can set up the following equation:

0.88825x + 5.10 = 218.28

Solving for x, we get:

0.88825x = 213.18

x = 240

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If it has a 15% yearly interest rate and the interest is compounded monthly and the card has $0 minimum payments for the first four years it is active. Therefore, Emma will owe approximately $529.27 in four years.

To calculate how much Emma will owe in four years, we need to use the compound interest formula: A = P (1 + r/n)^(n*t)

where:

A = the amount of money at the end of the investment period

P = the principal amount (the initial amount of money borrowed)

r = the annual interest rate (as a decimal)

n = the number of times the interest is compounded per year

t = the time period in years

In this case, the principal amount is $300, the annual interest rate is 15%, and the interest is compounded monthly (so n = 12). The time period is four years.

Plugging in the values, we get:

A = 300(1 + 0.15/12)^(12*4)

A ≈ $529.27

Therefore, Emma will owe approximately $529.27 in four years if she doesn't pay off her credit card until after college.

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The solutions of the equation algebraically are x = 5 and x = -7

From the information given, we have the quadratic equation;

|x| = x2 + x − 35

The sign , '| |' represents the modulus sign and says that the value must be a positive value.

Then, we have;

x² + x + x - 35

add the like terms

x² + 2x - 35

Find the pair factors of -35 that add up to 2, we have;

x² + 7x - 5x - 35

Group the expression in pairs

(x² + 7x) - (5x - 35)

factorize the expression

x(x + 7) - 5(x + 7)

Then, we have that;

x - 5 = 0

x = 5

x + 7 = 0

x = -7

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The dimensions of the rectangular sticker can be found by using the area and perimeter formulas. After solving the resulting system of equations, it can be concluded that the sticker has dimensions of 15 cm by 7 cm.

Let the length and width of the rectangle be l and w respectively.

Given that the area of the rectangle is 105 square centimeters.

So, lw = 105 --- equation (1)

Also, given that the perimeter is 44 centimeters.

Perimeter = 2(l + w) = 44

l + w = 22 --- equation (2)

Solving equations (1) and (2) simultaneously, we get:

l = 15 and w = 7

Therefore, the dimensions of the rectangular sticker are 15 cm by 7 cm.

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Based on the equation,  the expected average number of leaves on a tree 8 weeks after spring has started would be approximately 1966.64.

To write a linear equation that can be used to approximate the data distribution, we need to use the two given points on the scatter plot. Let's assume the first point is (0, 1700) and the second point is (6, 1900).

The slope of the line passing through these points can be calculated as:

slope = (1900 - 1700) / (6 - 0) = 200 / 6 = 33.33 (approx)

Using the point-slope form of a linear equation, we can write:

y - 1700 = 33.33(x - 0)

Simplifying, we get:

y = 33.33x + 1700

Therefore, the linear equation that can be used to approximate the data distribution is: Y = 33.33x + 1700 (Option C)

To find the expected average number of leaves on a tree 8 weeks after spring has started, we need to substitute x = 8 in the above equation and solve for Y:

Y = 33.33(8) + 1700 = 1966.64 (approx)

Therefore, the expected average number of leaves on a tree 8 weeks after spring has started would be approximately 1966.64.

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The critical point (180, 120) will yield a maximum revenue of 23,700. So, the correct answer is the (180, 120).

To determine which critical point will yield a maximum revenue, we need to evaluate the objective function at each point and compare the results.

Objective Function: Revenue = 75x + 85y

1. Point (0,0):
Revenue = 75(0) + 85(0) = 0

2. Point (180, 120):
Revenue = 75(180) + 85(120) = 13500 + 10200 = 23700

3. Point (300,0):
Revenue = 75(300) + 85(0) = 22500 + 0 = 22500

Comparing the revenues at each point:
(0,0) - Revenue = 0
(180, 120) - Revenue = 23700
(300,0) - Revenue = 22500

The critical point (180, 120) will yield a maximum revenue of 23,700. So, the correct answer is the (180, 120).

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Answer:

  (x, y) = (5, 3)

Step-by-step explanation:

You want the solution to the system of equations ...

It is convenient to subtract the second equation from 3 times the first:

  3(4x -3y) -(5x -9y) = 3(11) -(-2)

  12x -9y -5x +9y = 33 +2

  7x = 35 . . . . . . . simplify

  x = 5 . . . . . . . . . . divide by 7

  4(5) -3y = 11 . . . . . substitute into the first equation

  9 = 3y . . . . . . . . add 3y-11 to both sides

  y = 3 . . . . . . . . divide by 3

The solution is (x, y) = (5, 3).

__

Additional comment

We find using a graphing calculator to be the easiest way to solve a pair of simultaneous equations. The attachment shows the solution is the one we found above.

The approach of "substitution" is straightforward, if error-prone. Basically, you solve one equation for either variable, then use that expression in the other equation. Here, for example, you can solve the first for x:

  x = (11 +3y)/4

Then use that in the second equation:

  5(11 +3y)/4 -9y = -2

  5(11 +3y) -36y = -8 . . . . eliminate the denominator

  55 +15y -36y = -8 . . . . . eliminate the parentheses

  -21y = -63 . . . . . . . . . . . simplify, subtract 55

  y = 3 . . . . . . . . . . . . divide by -21

  x = (11 +3(3))/4 = 20/4 = 5 . . . . find x

In the above, we used "elimination." We took advantage of the fact that the y-coefficients were related by a factor of 3. To cancel y-terms, we need to have the equations we "add" have opposite signs for the y-terms. Here, we do that by multiplying the first by 3 (to make -9y), then subtracting the second equation (which has -9y, so will cancel). We could have subtracted 3 times the first from the second, but that would make all the resulting coefficients be negative, which we like to avoid.

Or, we could have multiplied the first equation by -3 to make the y-coefficients opposite, then added the results. (Again, that would give negative coefficients in the sum.) Planning ahead can help avoid mistakes due to minus signs.

You should choose Account 1 to obtain the greater amount in 10 years. Account 1 will have a higher balance after 10 years.

Account 1 pays 6% annual interest compounded quarterly, which means the interest is calculated and added to the principal four times a year. You can use the formula A = P(1 + r/n)^(nt) to calculate the future value, where A is the final amount, P is the principal ($2500), r is the annual interest rate (0.06), n is the number of times compounded per year (4), and t is the number of years (10).

Account 2 pays 4% annual interest compounded continuously. You can use the formula A = Pe^(rt) to calculate the future value, where A is the final amount, P is the principal ($2500), e is the base of the natural logarithm (approximately 2.71828), r is the annual interest rate (0.04), and t is the number of years (10).

When you compare the final amounts for both accounts, Account 1 will have a higher balance after 10 years.

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The required, Taub paid a tax of 0.055 or 5.5%.

To find the tax percentage Taub paid, we need to subtract 1 from the total multiplier and convert the result to a percentage.

1.055 - 1 = 0.055

So Taub paid a tax of 0.055 or 5.5%.

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