Of 10,000 grocery store transactions, 895 have been identified as having coffee, ice cream, and chips as part of the same transaction. Calculate the support of the association rule.Multiple Choice11.1730.08958.950.895

Answer:

  x ∈ {-4, 0, 2}

Step-by-step explanation:

You want the solutions to the cubic 2x³ +4x² -16x = 0.

We observe that x and 2 are factors of all terms, so this can be written ...

  2x(x² +2x -8) = 0

The quadratic will have binomial factors with constants that are factors of -8 that have a sum of 2.

  2x(x +4)(x -2) = 0

Solutions are the values of x that make the factors zero:

  x = 0

  x +4 = 0   ⇒   x = -4

  x -2 = 0   ⇒   x = 2

The solutions are x = -4, 0, 2.

To obtain a feasible solution, you would need to revise the assignment limits or add additional constraints that do not violate the given constraint.

Based on the constraint you provided, x21 + x22 + x23 + x24 ≤ 3, it means that the sum of variables x21, x22, x23, and x24, representing the number of tasks assigned to agents 1, 2, 3, and 4 respectively, cannot exceed 3.

This constraint implies that agent 3 can be assigned to a maximum of 2 tasks (since x23 ≤ 2), and agent 2 can be assigned to a maximum of 3 tasks (since x22 ≤ 3).

However, there seems to be a contradiction with the statement that "agent 3 can be assigned to 2 tasks" and "agent 2 can be assigned to 3 tasks" because the sum of these maximum assignments would already exceed 3, which is not feasible according to the constraint.

Therefore, To obtain a feasible solution, you would need to revise the assignment limits or add additional constraints that do not violate the given constraint, such as reducing the maximum number of tasks that can be assigned to agent 2 or agent 3, or adjusting the total number of tasks available for assignment.

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Answer:

3344 cubic yards

Step-by-step explanation:

The volume of a rectangular prism is length x width x height.

If the area of the base is 152, that means the length x width = 152

So, 152 x 22 = 3344.

In the two functions as the value of V(x) increases, the value of W(x) also increases.

The value of functions, V(x) and W(x) is determined as follows;

for h(-2, 1/4); the value of the functions is calculated as follows;

v(x) = 2ˣ ⁺ ³ = 2⁻²⁺³ = 2¹ = 2

w(x) = 2ˣ ⁻ ³ = 2⁻²⁻³ = 2⁻⁵ = 1/32

for h(-1, 1/2); the value of the functions is calculated as follows;

v(x) = 2ˣ ⁺ ³ = 2² = 4

w(x) = 2ˣ ⁻ ³ = 2⁻⁴ = 1/16

for h(0, 1); the value of the functions is calculated as follows;

v(x) = 2ˣ ⁺ ³ = 2³ = 8

w(x) = 2ˣ ⁻ ³ = 2⁻³ = 1/8

for h(1, 2); the value of the functions is calculated as follows;

v(x) = 2ˣ ⁺ ³ = 2⁴ = 16

w(x) = 2ˣ ⁻ ³ = 2⁻² = 1/4

for h(2, 4); the value of the functions is calculated as follows;

v(x) = 2ˣ ⁺ ³ = 2⁵ = 32

w(x) = 2ˣ ⁻ ³ = 2⁻¹ = 1/2

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When measured in straight line, the distance of the cars apart would be = 10 miles.

To calculate the distance of the cars apart in a straight line, the Pythagorean formula should be used. That is;

C² = a²+b²

c² = 8²+6²

= 64+36

c² = 100

c = √100

= 10 miles

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The correct answer is option C i.e. 0.15 - 0.02 to 0.15 + 0.02

The correct calculation for a 95% confidence interval for the true proportion of all US adults who feel a gym membership is a necessity is:

Margin of error = z√(p(1-p)/n)

where z is the z-score corresponding to the desired level of confidence (95% in this case), p is the sample proportion (0.15), and n is the sample size (1522).

From a standard normal distribution table, the z-score for a 95% confidence level is approximately 1.96.

Substituting these values into the formula, we get:

Margin of error = 1.96 * √(0.15*(1-0.15)/1522) ≈ 0.02

Therefore, the 95% confidence interval is:

0.15 - 0.02 to 0.15 + 0.02

which simplifies to: [0.13, 0.17]

So, the correct answer is option C.

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The indefinite integral of 4t(1-16t^4) dt is: 2t^2 - (4/5)t^6 + c, Here, C is the constant of integration, which can be written as C = C1 + C2.

To find the indefinite integral of the given function, we'll integrate term by term. The given function is:

∫(4t - 16t^4) dt

Now we'll integrate each term:

∫4t dt - ∫16t^4 dt

For the first term, the power rule for integration states that ∫t^n dt = (t^(n+1))/(n+1) + C, where n is a constant:

∫4t dt = 4∫t^1 dt = 4(t^(1+1))/(1+1) + C1 = 4t^2/2 + C1 = 2t^2 + C1

For the second term, we'll apply the same rule:

∫16t^4 dt = 16∫t^4 dt = 16(t^(4+1))/(4+1) + C2 = 16t^5/5 + C2 = (16/5)t^5 + C2

Now combine the results:

∫(4t - 16t^4) dt = 2t^2 + (16/5)t^5 + C

Here, C is the constant of integration, which can be written as C = C1 + C2.

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The nine fundamental solutions to the differential equation are:

[tex]e^{(-3+i)t}, e^{(-3-i)t}, e^t, te^t,[/tex] 1, t, t²/2!, t³/3!, [tex]t^4[/tex]/4!, [tex]e^{(-5+i)t}, ~and ~e^{(-5-i)t}[/tex]

We have,

The characteristic equation of the given differential equation is:

[tex](r^2 + 6r + 10)^2 \times r^2 (r - 1)^3 = 0[/tex]

We can find the fundamental solutions by looking at the roots of the characteristic equation.

The roots can be categorized as follows:

Roots of multiplicity 2 = -3 + i and -3 - i

Roots of multiplicity 2 =  1

Root of multiplicity 1 =  0

Root of multiplicity 2 = -5 + i and -5 - i

For each of these roots, we need to find the corresponding fundamental solution.

For the roots (-3 + i) and (-3 - i), the corresponding fundamental solutions are:

[tex]e^{(-3+i)t}~ and~ e^{(-3-i)t}[/tex]

For root 1, the corresponding fundamental solutions are:

[tex]e^t~and~te^t[/tex]

For the root 0, the corresponding fundamental solutions are:

1, t, t²/2!, t³/3!, ..., [tex]t^8[/tex]/8!

For the roots (-5 + i) and (-5 - i), the corresponding fundamental solutions are:

[tex]e^{(-5+i)t} ~and~e^{(-5-i)t}[/tex]

Therefore,

The nine fundamental solutions to the differential equation are:

[tex]e^{(-3+i)t}, e^{(-3-i)t}, e^t, te^t,[/tex] 1, t, t²/2!, t³/3!, [tex]t^4[/tex]/4!, [tex]e^{(-5+i)t}, ~and ~e^{(-5-i)t}[/tex]

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The surface described by the equation [tex]r^2 + z^2 = 4[/tex]is a right circular cylinder with a radius of 2 units, centered along the z-axis.

The surface whose equation is given is a cylinder with a radius of 2 units and a height of 4 units, centered on the z-axis.
Hi! I'd be happy to help you identify the surface with the given equation. The equation provided is:

[tex]r^2 + z^2 = 4[/tex]

This equation represents a right circular cylinder with a radius of 2 units, centered along the z-axis. Here's why:

1. Notice that the equation contains r^2 and [tex]z^2[/tex] terms, which suggests a cylindrical coordinate system.
2. The equation does not contain the θ term, which implies that the surface is symmetric about the z-axis.
3. The equation is in the form [tex]r^2 + z^2[/tex] = constant, which is the equation of a right circular cylinder in cylindrical coordinates.

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The stress (σ) for n = 100 is approximately 333.33 MPa.

The stress (σ) for n = 103 is also approximately 333.33 MPa.

To calculate the stress (σ) for n = 100 and 103 using the given Sut (ultimate tensile strength) and Fsut (factor of safety for ultimate tensile strength), we can use the formula:

σ = Sut / Fsut

Let's assume the given values of Sut and Fsut are as follows:

Sut = 500 MPa (megapascals)

Fsut = 1.5 (dimensionless)

For n = 100:

Plugging in the values into the formula, we get:

σ = Sut / Fsut

= 500 MPa / 1.5

≈ 333.33 MPa

So, the stress (σ) for n = 100 is approximately 333.33 MPa.

Similarly, for n = 103:

Using the same formula with the given values of Sut and Fsut:

σ = Sut / Fsut

= 500 MPa / 1.5

≈ 333.33 MPa

So, the stress (σ) for n = 103 is also approximately 333.33 MPa.

Please note that these calculations are based on the given values of Sut and Fsut, and the units are assumed to be in megapascals (MPa) as per the given formula.

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By answering the presented question, we may conclude that As a result, exponential about 739 banana plants in the plantation will be affected with the blight after 9 years.

The word "exponential growth" refers to the process of increasing quantity through time. When the instantaneous rate of change of a quantity with respect to time is proportional to the quantity, this is said to be proportional to the quantity. Exponential growth is a statistical pattern in which bigger gains are seen with time. Compound interest delivers exponential rewards in the world of finance. Savings accounts with compound interest can occasionally experience exponential growth. characterised by a rapid increase in the exponential growth rate (in size or extent). exponentially. The exponential function formula is f(x)=abx, where a and b are positive real values. Draw exponential functions for various values of a and b using the tools provided below.

To tackle this problem, we may apply the exponential growth formula:

[tex]N = N0 * (1 + r)^t[/tex]

Where N0 is the initial number of infected plants (476)

r = rate of increase (5% = 0.05)

t = time span (9 years)

When we plug in the values, we get:

[tex]N = 476 * (1 + 0.05)^9 \sN = 476 * 1.55128 \sN = 738.94[/tex]

When we round to the next full number, we get:

N ≈ 739

As a result, about 739 banana plants in the plantation will be affected with the blight after 9 years.

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Check the picture below.

[tex]\cfrac{2^3}{8^3}=\cfrac{V}{96\pi }\implies \cfrac{1}{64}=\cfrac{V}{96\pi }\implies \cfrac{96\pi }{64}=V\implies 4.71\approx V[/tex]

a. Using backwards finite difference approximations for both partial derivatives, we get:

∂Q/∂x ≈ (Q(i,j) - Q(i-1,j))/a

∂Q/∂y ≈ (Q(i,j) - Q(i,j-1))/a

Therefore,

∂Q/∂x + ∂Q/∂y ≈ (Q(i,j) - Q(i-1,j))/a + (Q(i,j) - Q(i,j-1))/a

≈ Q(i,j)/a - [Q(i-1,j) + Q(i,j-1)]/a

b. Using backwards finite difference approximations for both partial derivatives twice, we get:

∂^2Q/∂x^2 ≈ (Q(i,j) - 2Q(i-1,j) + Q(i-2,j))/a^2

∂^2Q/∂y^2 ≈ (Q(i,j) - 2Q(i,j-1) + Q(i,j-2))/a^2

Therefore,

∂^2Q/∂x^2 + ∂^2Q/∂y^2 ≈ (Q(i,j) - 2Q(i-1,j) + Q(i-2,j))/a^2 + (Q(i,j) - 2Q(i,j-1) + Q(i,j-2))/a^2

≈ 2Q(i,j)/a^2 - [Q(i-1,j) + Q(i-2,j) + Q(i,j-1) + Q(i,j-2)]/a^2

c. Using backwards finite difference approximations for both partial derivatives thrice, we get:

∂^3Q/∂x^3 ≈ (Q(i,j) - 3Q(i-1,j) + 3Q(i-2,j) - Q(i-3,j))/a^3

∂^3Q/∂y^3 ≈ (Q(i,j) - 3Q(i,j-1) + 3Q(i,j-2) - Q(i,j-3))/a^3

Therefore,

∂^3Q/∂x^3 + ∂^3Q/∂y^3 ≈ (Q(i,j) - 3Q(i-1,j) + 3Q(i-2,j) - Q(i-3,j))/a^3 + (Q(i,j) - 3Q(i,j-1) + 3Q(i,j-2) - Q(i,j-3))/a^3

≈ 3Q(i,j)/a^3 - [Q(i-1,j) + Q(i-2,j) + Q(i-3,j) + Q(i,j-1) + Q(i,j-2) + Q(i,j-3)]/a^3

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The value of the area is πab which is enclosed by the ellipse with the equation (x²/a²) + (y²/b²) = 1.

To find the area enclosed by the ellipse with the equation (x²/a²) + (y²/b²) = 1.

To find the area of this ellipse, use the formula A = πab, where A is the area, a is the semi-major axis, and b is the semi-minor axis.

First, identify the values of a and b from the given equation.
In the equation (x²/a²) + (y²/b²) = 1, a² is the coefficient of x², and b² is the coefficient of y².

Now, calculate the area using the formula A = πab.
Plug the values of a and b into the formula and multiply them with π to find the area.

So, the area enclosed by the ellipse (x²/a²) + (y²/b²) = 1 is A = πab.

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Answer:

The correct answer is: Yes; it is a reflection over the y-axis.

To see why, imagine folding the graph along the y-axis. Points on the right-hand side of the y-axis remain on the right-hand side, while points on the left-hand side move to the right. This transformation is equivalent to reflecting the original figure across the y-axis.

Step-by-step explanation:

There do not exist positive integers a and n such that a^2+3=3^n.

The given proof is not complete. The statement to be proven is that there do not exist positive integers a and n such that a^2+3=3.

The proof starts by assuming the opposite, i.e., assuming that there exist positive integers a and n such that a^2+3=3. However, the proof then only considers the case where n=1, which is not the most general case.

The proof correctly shows that if we put n=1, we get a^2+3=3, which simplifies to a^2=0. However, the conclusion that this is impossible is not explained. The reason this is impossible is that a is a positive integer, so a^2 must also be a positive integer. But a^2=0 implies that a=0, which contradicts the assumption that a is a positive integer.

To complete the proof, we need to consider the case where n>=2. In this case, we have:

a^2 + 3 = 3^n

Subtracting 3 from both sides, we get:

a^2 = 3^n - 3

We can factor the right-hand side as:

a^2 = 3(3^(n-1) - 1)

Since a is a positive integer, a^2 must be a multiple of 3. But 3^(n-1) - 1 is never a multiple of 3 for n>=2, so a^2 cannot be equal to 3(3^(n-1) - 1). Therefore, there do not exist positive integers a and n such that a^2+3=3^n.

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Answer:

(the answer is y ≤ X + 1).....

(a) To find the price the company should charge for each unit to sell 6,500 units, we need to substitute q with 6.5 (since q is measured in thousands of units) in the demand function p = 26e^(-0.6q): p = 26e^(-0.6 * 6.5)

After calculating, we get: p ≈ $2.98

So, the company should charge approximately $2.98 per unit to sell 6,500 units.

(b) To find how many units will sell if the company prices the products at $6.50 each, we need to solve for q in the demand function p = 26e^(-0.6q) with p = $6.50: 6.50 = 26e^(-0.6q)

Now, we need to solve for q: q = ln(6.50/26) / -0.6 ≈ 1.884

Since q is measured in thousands of units, the company will sell approximately 1,884 units when the price is $6.50 each.

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Answer:

sorry can't understand what your trying to say but i can help a lil if you translate it into english

All the inequalities that represent the graph include the following:

B. y > -5/4(x) + 5

E. y + 5 > -1.25(x - 8)

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

y - y₁ = m(x - x₁)

Where:

First of all, we would determine the slope of this line;

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Slope (m) = (5 - 0)/(0 - 4)

Slope (m) = 5/-4

Slope (m) = -5/4

At data point (0, 5) and a slope of -5/4, a linear equation for this line can be calculated by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y - 5 = -5/4(x - 0)  

y - 5 = -1.25(x - 0)

y = -5x/4 + 5

y > -5x/4 + 5 (shaded above the dashed line).

At data point (8, -5) and a slope of -5/4, a linear equation for this line can be calculated by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y - (-5) = -5/4(x - 8)  

y + 5 = -1.25(x - 8)

y + 5 > -1.25(x - 8)

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The evaluation of -2/3+1/6-5/12 is -11/12

A fraction has two parts, the numerator and the denominator.

In a simple fraction, both are integers. Examples are; 2/5 , 3/5. A complex fraction has a fraction in the numerator or denominator. In a proper fraction, the numerator is less than the denominator.

Solving, -2/3 +1/6 -5/12

1/6 -2/3 -5/12

= (2-8-5)/12

= (2-13)/12

= -11/12

therefore the evaluation of -2/3+1/6-5/12 is -11/12

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The solution of the differential equation is  U=x+2y. (A)

To solve the differential equation (x + 2y)dx + ydy = 0 using substitution, you can use the substitution U = x + 2y.


1. Substitute U for x+2y: dU = (dx + 2dy)

2. Replace (x + 2y)dx + ydy with dU - 2ydy + ydy: dU - ydy = 0

3. Factor out dy: dU - ydy = dy(U - y) = 0

4. Separate variables: (1/dU) dU = dy/y

5. Integrate both sides: ∫(1/dU) dU = ∫(dy/y)

6. Obtain the solution: ln|U| = ln|y| + C

7. Replace U with x+2y: ln|x+2y| = ln|y| + C

8. Exponentiate both sides: x+2y = k*y, where k = e^C

Thus, the differential equation (x + 2y)dx + ydy = 0 can be solved using the substitution U = x + 2y.(A)

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Answer:

3x10^3

Step-by-step explanation:

Move the decimal so there is one non-zero digit to the left of the decimal point. The number of decimal places you move will be the exponent on the

10

The sign of the exponent will depend on the direction you are moving the decimal.

Answer:

3x10^3

Step-by-step explanation:

Move the decimal so there is one non-zero digit to the left of the decimal point. The number of decimal places you move will be the exponent on the

10

The sign of the exponent will depend on the direction you are moving the decimal.

The absolute maximum value of f(t) is approximately -0.1213 at t = 9/4, and the absolute minimum value is -2 at t = 4.

To find the absolute maximum and absolute minimum values of f on the given interval [−1, 4], we first need to find the critical points of the function f(t) = t − 3√t.
Taking the derivative of f(t) with respect to t, we get:
f'(t) = 1 - (3/2)t^(-1/2)
Setting f'(t) = 0 to find critical points, we get:
0 = 1 - (3/2)t^(-1/2)
(3/2)t^(-1/2) = 1
t^(-1/2) = 2/3
t = (2/3)^(-2) = 2.25
So the only critical point of f(t) on the interval [−1, 4] is t = 2.25.
Now we need to evaluate f(t) at the endpoints of the interval and at the critical point to determine the absolute maximum and minimum values of f on the interval:
f(-1) = -1 - 3√(-1) = -1 - 3i
f(4) = 4 - 3√4 = 4 - 6 = -2
f(2.25) = 2.25 - 3√2.25 = 2.25 - 3(1.5) = -2.25
Therefore, the absolute maximum value of f on the interval [−1, 4] is f(-1) = -1 - 3i, and the absolute minimum value of f on the interval is f(4) = -2.
To find the absolute maximum and minimum values of f(t) = t - 3√t on the interval [-1, 4], we need to evaluate the function at its critical points and endpoints.
First, we find the critical points by taking the derivative of the function and setting it to zero:
f'(t) = 1 - (3/2)t^(-1/2)
To solve for critical points, set f'(t) = 0:
0 = 1 - (3/2)t^(-1/2)
(3/2)t^(-1/2) = 1
t^(-1/2) = 2/3
t = (2/3)^(-2) = 9/4
Since 9/4 is within the interval [-1, 4], it is a valid critical point.
Now, evaluate the function at the critical point and the endpoints:
f(-1) = -1 - 3√(-1)

(Note: This value is complex, and we're looking for absolute max/min in the real domain, so we'll ignore this endpoint)
f(9/4) = (9/4) - 3√(9/4) ≈ -0.1213
f(4) = 4 - 3√4 = -2
So, the absolute maximum value of f(t) is approximately -0.1213 at t = 9/4, and the absolute minimum value is -2 at t = 4.

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The average value of P=f(t)=2.04(1.03)^t for 0≤t≤30 is approximately 3.236. The average value of P=f(t)=2.04(1.03) for 0≤t≤30 is 2.10.

To find the average value of the function P=f(t)=2.04(1.03)^t for 0≤t≤30, you'll need to evaluate the appropriate integral and use the formula for the average value of a function.
The formula for the average value of a function is:
Average value = (1/(b-a)) * ∫[f(t) dt] from a to b
In this case, a = 0, b = 30, and f(t) = 2.04(1.03)^t.
Step 1: Evaluate the integral.
∫[2.04(1.03)^t dt] from 0 to 30
Step 2: Use a calculator to find the definite integral value.
We should find that the integral value is approximately 97.091.
Step 3: Substitute the integral value, a, and b into the average value formula.
Average value = (1/(30-0)) * 97.091
Step 4: Calculate the average value.
Average value ≈ (1/30) * 97.091 ≈ 3.236
So, the average value of P=f(t)=2.04(1.03)^t for 0≤t≤30 is approximately 3.236.

To find the average value of P=f(t)=2.04(1.03) for 0≤t≤30, we need to first evaluate the integral of the function over the given interval.
∫(0 to 30) 2.04(1.03) dt
Using a calculator, we can simplify and solve this integral as follows:
2.04(1.03)∫(0 to 30) dt
= 2.10t |(0 to 30)
= 2.10(30) - 2.10(0)
= 63.00
So, the integral of P=f(t) over the interval 0≤t≤30 is 63.00.
To find the average value of P over this interval, we divide this integral by the length of the interval:
Average value of P = (1/30-0) * 63.00
= 2.10
Therefore, the average value of P=f(t)=2.04(1.03) for 0≤t≤30 is 2.10.

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The following parts can be answered by the concept of combination of functions.

20) fg: Since f is even and g is odd, the product (fg) will be an odd function.

21) fg: The answer is the same as #20. The product (fg) will be an odd function.

22) g∘f: For a composition of functions, the even/odd properties depend on the functions themselves. Since g is odd and f is even, the composition g∘f will also be an odd function.

23) f∘f: Since both functions are even, the composition of two even functions, f∘f, will result in an even function.

24) g∘g: Similarly, since both functions are odd, the composition of two odd functions, g∘g, will result in an even function.
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The value of x is 8.37

What is the area of the shaded region?

The area of the shaded area is the difference between the total area of the polygon and the area of the portion of the polygon that is not shaded. In polygons, the area of the shaded component might appear in two different ways. A polygon's sides or its center are both potential locations for the shaded area.

Here, we have

Given: The area of the shaded region is 20cm².

we have to find the value of x.

x in this case is the radius. In fact, both the height and base are the radius.

To find the radius, we need to form an equation. The only info given is with the area of the shaded area which is 20cm².

The area of the sector - an area of the triangle = the shaded area.

Area of the sector = πr²/4

Area of triangle = (1/2)bh

Area of the triangle = x²/2

The area of the triangle is in that way, as the height and base are x ( and x is the radius here!)

=  πr²/4 - x²/2

Multiply 4 with the whole equation as it is the LCM.

= 4(πr²/4 - x²/2) = 80

= πx² - 2x² = 80

= 1.142x² = 80

x² = 70.1

x = 8.37

Hence, the value of x is 8.37

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The probability that the average price of 45 randomly selected combo meals around town will be less than $6.50 can be calculated using the central limit theorem.

According to the central limit theorem, the sampling distribution of the sample mean becomes approximately normal, regardless of the distribution of the population, if the sample size is large enough (n > 30).

Therefore, we can assume that the sample mean of the 45 combo meals follows a normal distribution with a mean of $6.75 and a standard deviation of $1.08/sqrt(45) = $0.161.

To find the probability that the sample mean is less than $6.50, we need to standardize the distribution using the z-score formula:

z = (x - μ) / (σ / sqrt(n))

where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.

Plugging in the values, we get:

z = (6.50 - 6.75) / (1.08 / sqrt(45)) = -1.73

Looking up the z-score in the standard normal distribution table, we find that the probability of a z-score less than -1.73 is approximately 0.04.

Therefore, the probability that the average price of 45 randomly selected combo meals around town will be less than $6.50 is 0.04 or 4%.

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The equation of the osculating circle to the function [tex]f(x) = \frac{1}{16} x^4 - \frac{1}{4} x^2[/tex]at the origin is [tex]x^2 + (y - 4/3)^2 = 16/9[/tex].

The radius of the circle is 4/3, and its center is at (0, 4/3).

To find the equation of the osculating circle to the function [tex]f(x) = \frac{1}{16} x^4 - \frac{1}{4} x^2[/tex] at the origin, we need to find the radius and center of the circle.

The osculating circle at a point (a, f(a)) has the same curvature as the graph of the function at that point, so we can use the formula for curvature:

[tex]k = |f''(a)| / [1 + (f'(a))^2]^{(3/2)[/tex]

where f''(a) and f'(a) are the second and first derivatives of f(x) evaluated at x = a.

At the origin (a = 0), we have:

f(a) = f(0) = -0.0625

f'(a) = f'(0) = 0

f''(a) = f''(0) = 3/4

Substituting these values into the formula for curvature, we get:

[tex]k = |f''(0)| / [1 + (f'(0))^2]^{(3/2)}\\= (3/4) / [1 + 0^2]^{(3/2)[/tex]

= 3/4

Since the radius of the osculating circle is 1/k, the radius of the circle at the origin is:

r = 1 / (3/4) = 4/3

To find the center of the circle, we note that it must lie on the normal line to the graph of f(x) at the origin.

Since the slope of the tangent line at the origin is f'(0) = 0, the slope of the normal line is undefined (i.e., it is vertical).

Therefore, the center of the osculating circle is at (0, r), or (0, 4/3).

The equation of the osculating circle is therefore:

[tex](x - 0)^2 + (y - 4/3)^2 = (4/3)^2\\x^2 + (y - 4/3)^2 = 16/9[/tex]

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