The length of time for one individual to be served at a restaurant is a random variable having an exponential distribution with an expected weight time of 4minutes.f(y)={λe−λy,for 0≤y≤[infinity]0,otherwiseFind the probability that an individual would wait longer than 10minutes to be served?(a) 0.00.(b) 0.08.(c) 0.94.(d) None of the above.

Answer:

7.1 in

Step-by-step explanation:

We know that this is an isosceles right triangle because the right triangle's legs are congruent.

The ratio of side lengths in an isosceles right triangle is:

1 : 1 : √2

Therefore, the length of the hypotenuse (the missing side) in the diagrammed triangle is:

5√2 in

This can be approximated as 7.1 in.

The decision would depend on the specific cost of the cousin's ticket and the cost of the closer seats relative to the available budget.

Part A: In order to determine if Jonah's claim is correct, we need to know the cost of the closer seats. If the cost of the closer seats is less than or equal to Jonah's savings of $103, then it would be possible for Jonah's dad to move the family to those seats by using Jonah's savings. However, if the cost of the closer seats exceeds $103, then Jonah's claim would not be correct, as his savings alone would not be sufficient to cover the cost. Without information about the cost of the closer seats, we cannot definitively determine if Jonah is correct.

Part B: If Jonah's cousin wants to come to the game and they have a total of $350, the family's seating options would be influenced by this additional expense. The cost of the cousin's ticket would need to be deducted from the $350 budget. If there is enough remaining after purchasing the cousin's ticket, Jonah's dad could consider using the combined savings of $103 from Jonah and the remaining budget to move the family to closer seats, provided the cost of those seats is within the remaining budget. However, if the cost of the closer seats, including the cousin's ticket, exceeds the remaining budget, then it would not be possible to move the family to closer seats. The decision would depend on the specific cost of the cousin's ticket and the cost of the closer seats relative to the available budget.

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Answer:

kindly mark brainlist if helped

Step-by-step explanation:

To compute the value of k, we can use the given information that after 98 seconds (t = 98), the temperature of the engine is 233°F (T = 233) with an outside temperature of 71°F (T₀ = 71). Plugging these values into the equation:

In (T - T₀) = -kt + 4.718

We have:

In (233 - 71) = -k(98) + 4.718

In (162) = -98k + 4.718

Taking the natural logarithm (ln) of both sides:

ln(162) = ln(-98k + 4.718)

Now, solve for k by rearranging the equation:

-98k + 4.718 = e^(ln(162))

-98k + 4.718 ≈ 5.2428 (rounded to four decimal places)

-98k ≈ 5.2428 - 4.718

-98k ≈ 0.5248

k ≈ 0.00535 (rounded to five decimal places)

A. The value of k, rounded to the nearest hundredth, is approximately 0.01.

To find the temperature T of the engine after resting for 212 seconds (t = 212), we can plug the values into the equation:

In (T - T₀) = -kt + 4.718

In (T - 71) = -(0.01)(212) + 4.718

In (T - 71) ≈ -2.12 + 4.718

In (T - 71) ≈ 2.598

Exponentiating both sides:

T - 71 ≈ e^(2.598)

T - 71 ≈ 13.4464

T ≈ 13.4464 + 71

T ≈ 84.4464

B. The temperature of the engine, after resting for a total of 212 seconds, is approximately 84°F.

To determine if the car is safe to drive after 3 minutes (t = 3 minutes = 180 seconds) of waiting, we can find the temperature T using the value of k:

In (T - 71) = -(0.01)(180) + 4.718

In (T - 71) ≈ -1.8 + 4.718

In (T - 71) ≈ 2.918

Exponentiating both sides:

T - 71 ≈ e^(2.918)

T - 71 ≈ 18.5277

T ≈ 18.5277 + 71

T ≈ 89.5277

The temperature of the engine after 3 minutes of waiting is approximately 90°F.

C. Since the temperature of the engine after 3 minutes of waiting is within the safe range of 190°F to 220°F, Prolific's car is safe to drive after 3 minutes of waiting.

A:

B:

C:

The measure of the amount of random sampling error in a survey's result is known as margin of error.

The margin of error is a statistical concept that quantifies the degree of uncertainty or sampling error associated with survey results. It provides an estimate of the range within which the true population parameter is likely to fall. The margin of error is typically expressed as a percentage and is based on the sample size and the level of confidence desired.

In survey research, random sampling error refers to the natural variability that occurs when a subset of individuals, known as the sample, is selected to represent a larger population. It arises because the sample is not an exact replica of the entire population. The margin of error takes into account this inherent variability and provides a measure of how much the survey results might deviate from the true population values.

A larger sample size generally leads to a smaller margin of error, as it reduces the random variability associated with sampling. Similarly, a higher level of confidence, such as 95% confidence level, results in a larger margin of error to account for a wider range of potential values.

By considering the margin of error, survey researchers can assess the reliability and precision of their findings, providing a range of values within which the true population parameter is likely to reside.

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There are exists a point ϕ ∈ (a, b) such that f(ϕ) = (1/n) * [f(p1) + f(p2) + ... + f(pn)].

To prove that there exists a point ϕ ∈ (a, b) such that f(ϕ) = (1/n) * [f(p1) + f(p2) + ... + f(pn)], we can utilize the Mean Value Theorem for Integrals.

Let F(x) be the antiderivative of f(x) on the interval [a, b]. By the Mean Value Theorem for Integrals, there exists a point c ∈ (a, b) such that the average value of F(x) on [a, b] is equal to F(c):

1/(b - a) * ∫[a to b] F(x) dx = F(c)

Since F(x) is the antiderivative of f(x), we can rewrite the equation as:

1/(b - a) * ∫[a to b] f(x) dx = F(c)

Taking the definite integral of f(x) from a to b, we have:

1/(b - a) * ∫[a to b] f(x) dx = F(b) - F(a)

Since f(x) is continuous on [a, b], it is also continuous on the closed interval [a, b]. Therefore, by the Extreme Value Theorem, f(x) attains its maximum and minimum values on [a, b]. Let M be the maximum value of f(x) and m be the minimum value of f(x) on [a, b].

Since f(x) is continuous, it satisfies the Intermediate Value Property. Therefore, for any y ∈ [m, M], there exists a point d ∈ [a, b] such that f(d) = y.

Now, consider the points p1, p2, ..., pn ∈ (a, b). Let A = f(p1) + f(p2) + ... + f(pn). Since f(x) satisfies the Intermediate Value Property, there exists a point ϕ ∈ (a, b) such that f(ϕ) = A/n.

Hence, we have proven that there exists a point ϕ ∈ (a, b) such that f(ϕ) = (1/n) * [f(p1) + f(p2) + ... + f(pn)].

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The surface area of the given pyramids and cone would be listed below as follows:

1.)576.4in²

2.)71.4yd²

To calculate the surface area the following steps should be taken.

For question 1.)

The formula for surface area of square based pyramid;

= a²+2al

where;

a² = base area = 11² = 121in

l = 20.7in

a = 11

SA = 121+2(11×20.7)

= 121+455.4

= 576.4in²

For question 2.)

The formula for surface area of triangular pyramid ;

SA= B+1/2Ps

B = base area = 15.6yd

P = 18yd

Slant height = 6.2 yd

SA = 15.6+1/2×18×6.2

= 71.4yd²

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To rewrite the expression 18a^3b + 9ab^2 using a common factor, we can factor out the common factor from both terms. In this case, the common factor is 9ab.

Taking out the common factor, we have:

18a^3b + 9ab^2 = 9ab(2a^2 + b)

So, the expression 18a^3b + 9ab^2 can be simplified as 9ab(2a^2 + b) by factoring out the common factor 9ab.

This process is known as factoring out the greatest common factor (GCF). By factoring out the GCF, we simplify the expression and make it more manageable and easier to work with.

Factoring out the GCF is a useful technique in algebra to simplify expressions and solve equations. It helps in identifying common factors and allows us to rearrange terms more easily.

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Answer:

(a) Please refer to explanation (in part 1)

(b) [tex]V=\frac{2}{3}x(22-x^{2})[/tex]

(c) [tex]\frac{176}{9}\sqrt{\frac{22}{3}} \text{cm}^{3}[/tex]

Step-by-step explanation:

The explanation is attached below.

At the hot wings restaurant, a fraction of 5/9 of the patrons ordered the inferno hot wings, and 1/8 of those patrons passed out from the intensity of the sauce. The fraction of patrons who passed out are 5/72.

Given that 5/9 of the patrons ordered the inferno hot wings, this fraction represents the portion of patrons who were exposed to the intense sauce. Out of this group, 1/8 passed out due to the sauce's intensity.

To find the fraction of patrons who passed out, we multiply the fractions 5/9 and 1/8:

(5/9) * (1/8) = 5/72.

Therefore, the fraction 5/72 represents the proportion of patrons who passed out from the intensity of the inferno hot wing sauce.

This information is important for understanding the effects of the spicy sauce and can be used by the restaurant to gauge the intensity of the dish and potentially make adjustments to cater to different preferences. Additionally, it provides insights into the customer experience and can influence future menu decisions or considerations regarding the heat levels of their offerings.

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Answer:98

Step-by-step explanation:3/7=42/y

                                            3y=294

                                            3y/3=294/3

                                             y=98

                                             42/98 simplified equals 3/7

The value of x in the equation is 19.

We have,

To find the value of x, we need to set the expressions bd and ac equal to each other and solve for x.

Given:

bd = 8x - 7

ac = 6x + 31

Setting bd = ac:

8x - 7 = 6x + 31

Now, solve this equation for x:

8x - 6x = 31 + 7

2x = 38

x = 38/2

x = 19

Therefore,

The value of x in the equation is 19.

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The smallest positive degree, β, that satisfies the equation cos(β) / 3 + 0.482 = 0.16 is approximately 203.53 degrees.

To find the smallest positive degree, β, that satisfies the equation cos(β) / 3 + 0.482 = 0.16, we need to isolate the cosine term and solve for β.

First, let's rearrange the equation:

cos(β) / 3 = 0.16 - 0.482

cos(β) / 3 = -0.322

Next, multiply both sides of the equation by 3 to eliminate the fraction:

cos(β) = -0.322 * 3

cos(β) = -0.966

To find the smallest positive degree, we can use the inverse cosine (cos⁻¹) function:

β = cos⁻¹(-0.966)

Using a calculator, we can evaluate the inverse cosine to find the corresponding angle. The result is approximately 156.47 degrees.

However, we need to find the smallest positive degree, so we subtract this angle from 360 degrees:

Smallest positive degree = 360 - 156.47

Smallest positive degree ≈ 203.53 degrees.

Therefore, the smallest positive degree, β, that satisfies the equation cos(β) / 3 + 0.482 = 0.16 is approximately 203.53 degrees.

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The equality of partial and Pearson correlations provides some insights into the relationship between X and Y, however, it is not sufficient to determine statistical significance, causality, range of scores.

The fact that the partial correlation is equal to the Pearson correlation does not automatically imply statistical significance. Statistical significance is determined by conducting hypothesis tests or calculating p-values, which require additional information such as sample size and significance level.

The statement (b) does not provide evidence of a causal relationship between X and Y. Correlation alone does not establish causality, as there may be other confounding factors or alternative explanations for the observed relationship.

The range of scores on X and Y cannot be inferred solely from the equality of partial and Pearson correlations. The range of scores depends on the actual data and variability within X and Y, which is not addressed in the statement.

The statement (d) suggests that the variable that was partialed out does not fully account for the correlation between X and Y.

However, it does not specify the nature of the variable or the method used for partial correlation. Further analysis and context are needed to draw conclusions about the role of the partialed-out variable.

In summary, while the equality of partial and Pearson correlations provides some insights into the relationship between X and Y, it is not sufficient to determine statistical significance, causality, range of scores, or the full explanation for the correlation observed. Additional analysis and considerations are necessary to make conclusions in these areas.

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0.006336 g of this sample is left after 6 half-lives.

Amount remaining = initial amount x (1/2)^number of half-lives

In this case, the initial amount is 0.4055 g and the number of half-lives is 6. So:

Amount remaining = 0.4055 g x (1/2)⁶ = 0.006336 g

if a sample of thulium-171 has a mass of 0.4055 g and undergoes radioactive decay, after 6 half-lives only 0.006336 g of the original sample will remain. This calculation is based on the formula for calculating the amount of a radioactive substance remaining after a certain number of half-lives, which takes into account the decay rate of the substance.

Hence,0.006336 g of this sample is left after 6 half-lives.

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The actual error ≈ 0.00002082 and Error Bound E ≈ 0.00002083. Comparing the actual error to the error bound E, we can see that they are very close in magnitude.

To determine a formula of the form √(1+x) = 1 + 0.5x ± E using the degree 1 Taylor polynomial of f(x) = √(1+x) with remainder, we start by finding the degree 1 Taylor polynomial:

P1(x) = f(a) + f'(a)(x - a)

where a = 0 (the point of expansion). Let's calculate the derivatives:

f(x) = √(1+x)

f'(x) = 1/(2√(1+x))

Substituting a = 0 and f(a) = f(0) = √1 = 1, we have:

P1(x) = 1 + f'(0)(x - 0)

     = 1 + (1/2)(x)

     = 1 + 0.5x

The remainder term R1(x) is given by:

R1(x) = (x - a)²/2! * f''(c)

To find the error bound E, we need to evaluate the second derivative f''(c) for some value c between 0 and x. Taking the second derivative of f(x) = √(1+x), we get:

f''(x) = -1/(4(1+x)^(3/2))

Substituting x = 0.02 (since we're approximating √1.02), we have:

f''(c) = -1/(4(1+c)^(3/2))

To find the error E, we evaluate the remainder term using the maximum value of f''(c) in the interval [0, 0.02]. To approximate this, we use a calculator:

E = |R1(0.02)| = |0.02 - 0|²/2! * |-1/(4(1+c)^(3/2))|

Calculating this expression, we find E ≈ 0.00002083.

Using a calculator, we can evaluate the actual error by subtracting the approximation 1 + 0.5(0.02) from the actual value of √1.02:

Actual error = √1.02 - (1 + 0.5(0.02))

Calculating this, we find the actual error ≈ 0.00002082.

Comparing the actual error to the error bound E, we can see that they are very close in magnitude.

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For Question 9, the equation y' = (y-1)(y-2) has two stable equilibrium solutions. For Question 10, the equation y' = cos(y) does not have infinitely many equilibrium solutions.

Question 9 asks about the equation y' = (y-1)(y-2) and the type of equilibrium solutions it possesses. An equilibrium solution occurs when y' (the derivative of y with respect to some independent variable) equals zero. By setting (y-1)(y-2) equal to zero and solving for y, we find two values: y = 1 and y = 2. To determine the stability of these equilibrium solutions, we analyze the sign of the derivative around these points. Since (y-1)(y-2) is positive for y > 2 and negative for 1 < y < 2, we can conclude that y = 1 is a stable equilibrium solution, while y = 2 is an unstable equilibrium solution.

Question 10 deals with the equation y' = cos(y) and whether it has infinitely many equilibrium solutions. To find equilibrium solutions, we set cos(y) equal to zero and solve for y. The solutions are y = (2n+1)π/2, where n is an integer. However, these equilibrium solutions do not extend to infinity. Therefore, the statement "Equation y' = cos(y) has infinitely many equilibrium solutions" is false.

Understanding the stability and existence of equilibrium solutions in differential equations is crucial for analyzing the behavior and long-term dynamics of the system described by the equation.

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The area rolled during one complete revolution of the lawn roller is approximately  38 square feet (nearest whole number).

To find the area rolled, we need to calculate the lateral surface area of the cylindrical roller. The formula for the lateral surface area of a cylinder is given by A = 2πrh, where π is the mathematical constant pi (approximately 3.14159), r is the radius, and h is the height (length) of the cylinder.

Given that the radius of the roller is 18 inches, we need to convert it to feet by dividing it by 12 since there are 12 inches in a foot. So the radius (r) becomes 18/12 = 1.5 feet.

The height (length) of the roller is given as 4 feet. Therefore, h = 4 feet.

Plugging the values into the formula, we have A = 2π(1.5)(4) = 12π square feet.

Now, to find the area rolled during one complete revolution, we multiply the lateral surface area by the number of revolutions, which is 1. So the total area rolled is 12π square feet.

Using the calculator value of π, which is approximately 3.14159, we can approximate the area rolled as 12(3.14159) = 37.69908 square feet.

Rounding to the nearest whole number, the area rolled during one complete revolution of the lawn roller is approximately 38 square feet.

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The correct option is (a) 0.3170.

What is probability?

Probability is a measure or quantification of the likelihood that a specific event will occur. It is a way of expressing uncertainty in terms of numerical values between 0 and 1, where 0 represents impossibility (an event will not occur) and 1 represents certainty (an event will definitely occur).

To find the probability P (13.7 < X < 30.7) for a normal random variable X with mean μ=35 and standard deviation σ=10.

we can use the standard normal distribution.

First, we need to standardize the values using the z-score formula:

z= x-μ / σ

For the lower value, 13.7:

z = (13.7 - 35)/ 10

 = -2.13

For the upper value, 30.7:

z₂ = (30.7 - 35) / 10

    = -0.43

Next, we look up the corresponding probabilities for these z-scores in the standard normal distribution table or use a calculator.

Using the table or calculator, we find:

P (z < -2.13) ≈ 0.0166 (rounded to four decimal places)

P (z < -0.43) ≈ 0.3336 (rounded to four decimal places)

Finally, we subtract the lower probability from the upper probability to find the desired probability:

P (13.7 < X < 30.7) = P(z₁ < z < z₂)

                               ≈0.3336−0.0166

                               ≈0.3170

Therefore, the answer is (a) 0.3170.

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the next number in the series is 17.

The given series of numbers is a sequence of prime numbers. To find the next number, we need to identify the next prime number after 13.

The next prime number after 13 is 17.

what is series?

In mathematics, a series is the sum of the terms of a sequence. It is a sequence of numbers that are added together in a specific order. Each term in the series is typically obtained by applying a rule or formula to the preceding terms.

For example, the series 1 + 2 + 3 + 4 + 5 + ... is the sum of all positive integers. In this case, the terms of the series are generated by adding the next positive integer to the sum of the previous terms.

Series can be finite, meaning they have a specific number of terms, or they can be infinite, meaning they continue indefinitely.

Series are an important concept in mathematics and have applications in various fields such as calculus, number theory, and probability.

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The volumes of the solids are 1) 4986 m³, 2) 134 km³ and 3) 4179 in³.

Given are the solids in shapes of spheres, cylinders and cone we need to find the volumes,

So,

Volume of a Sphere:

V = (4/3) × π × r³

Where V is the volume and r is the radius of the sphere.

Volume of a Cylinder:

V = π × r² × h

Where V is the volume, r is the radius of the base, and h is the height of the cylinder.

Volume of a Cone:

V = (1/3) × π × r² × h

Where V is the volume, r is the radius of the base, and h is the height of the cone.

1) Sphere with diameter 21.2 m,

Volume = V = (4/3) × π × (21.2/2)³ = 4986 m³

2) Cone with base diameter and height of 8 km,

Volume = (1/3) × π × (8/2)² × 8 = 134 km³

3) Cylinder with base radius and height of 11 in,

Volume = π × 11² × 11 = 4179 in³

Using the similar process you can find the rest volumes.

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Answer:

------------------------

As per diagram, point E has coordinates (5, 2).

Reflection across the y-axis results in the x-coordinate flip the sign, while the y-coordinate remains unchanged.

Hence the point E' is (- 5, 2).

Answer:

(2) - [tex]\vec v= < 15.5885, -9 >[/tex]

(3) - [tex]\vec u= < -59.9371, 148.349 >[/tex]

Step-by-step explanation:

Problem #2:

Given the vector in magnitude-angle form, find it in component form.

Call the vector, vector "v."

[tex]||\vec v||= 18 \ at \ -30 \textdegree\\\\\rightarrow \boxed{\vec v= < ||\vec v||\cos\theta,||\vec v||\sin\theta > }\\\\\Longrightarrow \vec v= < (18)\cos( -30 \textdegree),(18)\sin( -30 \textdegree) > \\\\\therefore \boxed{\boxed{ \vec v= < 15.5885, -9 > }}[/tex]

Problem #3:

Given the vector in magnitude-angle form, find it in component form.

Call the vector, vector "u."

[tex]||\vec u||= 160 \ at \ 112 \textdegree\\\\\rightarrow \boxed{\vec u= < ||\vec u||\cos\theta,||\vec u||\sin\theta > }\\\\\Longrightarrow \vec u= < (160)\cos( 112 \textdegree),(160)\sin( 112 \textdegree) > \\\\\therefore \boxed{\boxed{ \vec u= < -59.9371, 148.349 > }}[/tex]

The system of equations are solved and the numbers are 15 and 7

Given data ,

The difference of the two numbers is 8, which can be expressed as:

x - y = 8

It is also given that twice the first number (2x) added to three times the second number (3y) equals 51:

2x + 3y = 51

We now have a system of two equations with two variables. We can solve this system using various methods, such as substitution or elimination.

Let's solve the system using the substitution method:

From equation (1), we can express x in terms of y:

x = y + 8

Substituting this expression for x into equation (2), we get:

2(y + 8) + 3y = 51

2y + 16 + 3y = 51

5y + 16 = 51

5y = 51 - 16

5y = 35

y = 35/5

y = 7

Substituting the value of y back into equation (1):

x - 7 = 8

x = 8 + 7

x = 15

Hence , the two numbers are x = 15 and y = 7

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The Hamming norm of a vector is defined as the count of non-zero elements in the vector.

For vector u = (1, 1, 0, 1, 0, 0, 0), we can see that there are three non-zero elements: 1, 1, and 1. Thus, the Hamming norm of u is 3.

For vector v = (1, 0, 0, 1, 1, 0, 1), we observe that there are four non-zero elements: 1, 1, 1, and 1. Hence, the Hamming norm of v is 4.

The Hamming norm is a measure of the "sparsity" or the number of active components in a vector. It is particularly useful in binary or sparse data analysis. In the given vectors, the Hamming norm indicates the number of non-zero entries, providing information about the magnitude of their deviation from the zero vector.

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The simplified expression is 2x⁵.

To simplify the expression [tex]\left(4x^{9/2}\right)\left(\frac12x^{1/2}\right)[/tex], we can multiply the coefficients and combine the variables with the same base.

Multiplying the coefficients: [tex]4 \times \frac12 = 2[/tex]

Multiplying the variables with the same base:

[tex]$x^{9/2} \times x^{1/2} = x^{\left(\frac92 + \frac12\right)} = x^{10/2} = x^5$[/tex]

Putting it all together, the simplified expression is [tex]2x^5[/tex]

Hence the simplified expression is 2x⁵.

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The correct answer is d) p = 20 and q = 18. For these values of p and q

[tex]x^{36} + px^q + 100[/tex] is a perfect square for all integer values of x

To explain why p = 20 and q = 18 are the correct values, let's analyze the expression [tex]x^{36} + px^q + 100[/tex]. For this expression to be a perfect square for all integer values of x, it must be in the form (ax^18 + b)^2, where a and b are integers.

Expanding (ax^18 + b)^2 gives us [tex]ax^{36} + 2abx^{18} + b^2[/tex]. Comparing this with the given expression [tex]x^{36} + px^q + 100[/tex], we can deduce the following:

1. The constant term in both expressions must be the same, which gives us b^2 = 100. The only possible integer value for b is 10, as it is the only square root of 100.

2. The coefficient of x^36 in both expressions must also be the same, which gives us a^2 = 1. The only possible integer value for a is 1.

3. The coefficient of x^18 in the expanded form is 2ab, which should be equal to px^q in the given expression. Therefore, we have 2ab = px^q. Since a = 1, this simplifies to 2b = px^q.

We know that b = 10, so we can substitute it into the equation: 2 * 10 = px^q. Simplifying further, we get 20 = px^q.

Now, we need to find a value for p and q that satisfies the equation for all integer values of x. If we set q = 18, then x^q = x^18, and the equation becomes 20 = px^18. This is satisfied for any value of x.

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The statement "Choosing a smoothie containing apple and choosing a smoothie containing blueberry are not mutually exclusive events." can be inferred from the calculation.

Firstly, we identify the total number of each type of smoothies available. We have 12 apple smoothies, 7 blueberry, 8 other, and 3 smoothies that are common to both apple and blueberry. This brings our total smoothies to 30.

a) To find the probability that a smoothie contains either apple or blueberry, we need to consider the apple smoothies and smoothies that are common to both apple and blueberry then blueberry smoothies and smoothies that are common to both apple and blueberry. So, we add up the numbers of these smoothies and divide by the total number of smoothies.

P(contains apple) = (12 apple + 3 common) / 30 total = 15 / 30 which equals 0.5

P(contains blueberry) = (7 blueberries + 3 common) / 30 total = 10 / 30 which equals 0.33

Then, we find P(contains apple) + P(contains blueberry) = 0.5 + 0.33 which equals 0.83.

b) To find the probability that a smoothie contains apple or blueberry, we add up the number of apple smoothies, blueberry smoothies and smoothies common to both, then divide by the total number of smoothies.

P(contains apple or blueberry) = (12 apple + 7 blueberries + 3 common) / 30 total = 22/30 which equals 0.73.

c) The events of choosing a smoothie containing apple and choosing a smoothie containing blueberry are mutually exclusive if P(contains apple) + P(contains blueberry) is equal to P(contains apple or blueberry). As 0.83 is not equal to 0.73, these are not mutually exclusive events.

Therefore, the statement "Choosing a smoothie containing apple and choosing a smoothie containing blueberry are not mutually exclusive events." can be inferred from the calculation.

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The tank height will be 2.3125 feet (2 feet for water height + 0.3125 feet for displacement height + 1 foot for splashing prevention).

To determine the height of the water in the tank, we first need to calculate the volume of the tank. A 4' by 4' square base gives us an area of 16 square feet.

Multiplying this by the height of the water will give us the volume of water in the tank. Let's assume we want the water to be 2 feet deep, so the volume of water will be 32 cubic feet.

Next, we need to calculate the displacement height. When the teacher falls in, they will displace a certain amount of water. Since the teacher is entirely submerged, their volume will be equal to the volume of water displaced.

Assuming the teacher has a volume of 5 cubic feet, this is the amount of water that will be displaced, causing the water level to rise by 5/16 or 0.3125 feet.

To prevent splashing, we need to add an additional foot to the height of the tank beyond the displacement height. This calculation should be labeled "Tank Height" and included on the design sheet.

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The coordinates of T" include the following: D. (8, 10).

In Mathematics and Geometry, the translation of a geometric figure upward means adding a digit to the value on the positive y-coordinate (y-axis) of the pre-image.

(x, y)                                                    →                  (x - 1, y + 3)

Coordinate T (5, 2)                             →                  T' (5 - 1, 2 + 3) = T' (4, 5).

Next, we would dilate the coordinates of the vertices by applying a scale factor of 2 that is centered at the origin as follows:

Coordinate T' (4, 5) → (4 × 2, 5 × 2) = Coordinate T" (8, 10).

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