QUADRATIC FUNCTIONS: The profit (in hundreds of dollars) that a corporation receives depends on the amount (in hundreds of dollars) the company spends on marketing according to the model 130+10X−0.5x2130+10X−0.5x2. What expenditure for advertising yields a maximum profit? What is the mathematical name of this point?
PROBLEM 4: POLYNOMIAL FUNCTIONS: Let f(x)=4x5−8x4−5x3+10x2+x−1f(x)=4x5−8x4−5x3+10x2+x−1. The graph is presented below:
Describe f (x) in terms of
Degree of polynomial
Main coefficient
Final behavior
Maximum number of zeros
Maximum number of exchange points (relative maximums and minimums)

To implement the given function using one 8x1 multiplexer, we first need to identify the inputs and outputs. The inputs are A, B, C, and D, and the output is F.

We can use the 8x1 multiplexer as a logic function generator by using the select inputs to choose which input is passed to the output.

To implement the given function, we can use the following steps:

1. Connect A and D to the select inputs of the multiplexer.
2. Connect B and C to the remaining two inputs of the multiplexer.
3. Set the outputs of the multiplexer as follows:
- Connect output 0 to VCC.
- Connect output 1 to B.
- Connect output 2 to A.
- Connect output 3 to BC'.
- Connect output 4 to AB.
- Connect output 5 to AC.
- Connect output 6 to B'CD.
- Connect output 7 to ABCD'.

4. Connect the multiplexer outputs to a logical OR gate to generate the final output F.

By setting the select inputs appropriately, the multiplexer will output the required terms of the function, which are then combined using the OR gate to produce the final output F.
Hi! To implement the given function F(A, B, C, D) = A'C'B + AB'C' + B'C'D + ABCD' using one 8x1 multiplexer, follow these steps:

1. Identify the input and control lines: Since it is an 8x1 multiplexer, we need three control lines. Choose A, B, and C as the control lines. The input lines will be connected based on the function.

2. Map the function to the input lines: For an 8x1 multiplexer, the inputs are connected as follows:
 - I0 = A'B'C'D'
 - I1 = A'B'C'D
 - I2 = A'B'CD'
 - I3 = A'B'CD
 - I4 = AB'C'D'
 - I5 = AB'C'D
 - I6 = ABCD'
 - I7 = ABCD

3. Connect the corresponding function terms to the input lines:
 - I0 = 0 (A'B'C'D' does not appear in the function)
 - I1 = A'C'B (A'B'C'D matches the first term)
 - I2 = 0 (A'B'CD' does not appear in the function)
 - I3 = B'C'D (A'B'CD matches the third term)
 - I4 = AB'C' (AB'C'D' matches the second term)
 - I5 = 0 (AB'C'D does not appear in the function)
 - I6 = ABCD' (ABCD' matches the fourth term)
 - I7 = 0 (ABCD does not appear in the function)

By connecting the input lines according to the function terms and using A, B, and C as the control lines, you can implement the given function F(A, B, C, D) using one 8x1 multiplexer.

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Answer:

Step-by-step explanation: The correct simplification of the expression (4 x 10-6) X 3,000 can be found by multiplying the numerical coefficients and adding the exponents of 10.

4 x 10-6 is equal to 0.000004 in decimal notation.

Multiplying this by 3,000 gives:

(4 x 10-6) X 3,000 = 0.000004 x 3,000 = 12

Therefore, the simplified expression is 12.

Among the given steps, Step 2 is incorrect as it has incorrectly swapped the order of multiplication of the numerical coefficients and the exponents of 10.

Step 2: (4 x 3)(10-6 x 10³)

The correct order of multiplication should be:

Step 2 (Corrected): (4 x 3) x (10-6 x 10³)

This simplifies to:

Step 3: 12 x 10-3

And the final simplified expression is:

Step 4: 1.2 x 10-2

Therefore, the error in Jackie's simplification is in Step 2.

If f(6)=14 f' is continuous and f'(x)dx=18 the value of f(7) is 32.

To find the value of f(7), we need to use the fundamental theorem of calculus, which states that if f is a continuous function and f'(x) is its derivative, then:

∫f'(x)dx = f(x) + C

where C is the constant of integration.

Given that f' is continuous and f'(x)dx=18, we can integrate both sides to obtain:

∫f'(x)dx = ∫18 dx

Using the fundamental theorem of calculus, we get:

f(x) + C = 18x + K

where K is another constant of integration.

Now, we can use the given value of f(6) to solve for C. Since f(6) = 14, we have:

f(6) + C = 18(6) + K

14 + C = 108 + K

C - K = 94

Substituting this value of C into our equation, we get:

f(x) = 18x + K - 94

To find the value of f(7), we substitute x = 7 into this equation:

f(7) = 18(7) + K - 94

Simplifying, we get:

f(7) = 100 + K

Therefore, we need to find the value of K to determine f(7). We can use the given information that f' is continuous to conclude that f is differentiable. Thus, we can differentiate our equation for f(x) to obtain:

f'(x) = 18

Since f'(x) is constant, we know that f(x) is a linear function of x. Therefore, we can use the two given points (6, 14) and (7, f(7)) to solve for K. The slope of the line passing through these points is:

m = (f(7) - 14) / (7 - 6) = f(7) - 14

Solving for f(7), we get:

f(7) - 14 = 18

f(7) = 32

Therefore, the value of f(7) is 32.

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The area under the standard normal curve between z = -1.97 and z = -0.79 is approximately 0.1904, rounded to four decimal places.

To find the area under the standard normal curve between z = -1.97 and z = -0.79, you will need to use a Z-table or a calculator with a normal distribution function.

1. Find the area to the left of z = -1.97 and z = -0.79 in the Z-table or using a calculator.
2. Subtract the area of z = -1.97 from the area of z = -0.79 to get the area between the two points.

Using a Z-table or calculator, you will find the areas to the left are:

- For z = -1.97, the area is 0.0244
- For z = -0.79, the area is 0.2148

Now, subtract the smaller area from the larger area:

0.2148 - 0.0244 = 0.1904

So, the area under the standard normal curve between z = -1.97 and z = -0.79 is approximately 0.1904, rounded to four decimal places.

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Answer:

x = -2 - sqrt(3), -2 + sqrt(3) 

Step-by-step explanation:

We can rewrite the equation as

x² + 4x + 1 = 0

Now we can use the quadratic equation.

x = (-b ± sqrt(b² - 4ac)) / 2a

where a = 1, b = 4, c = 1. Substituting these values ​​gives:

x = (-4 ± sqrt(4² - 4(1)(1))) / 2(1)

x = (-4 ± sqrt(16 - 4)) / 2

x = (-4 ± sqrt(12)) / 2

x = (-4 ± 2sqrt(3)) / 2

x = -2 ± sqrt(3)

So, from min to max, the solution is:

x = -2 - sqrt(3), -2 + sqrt(3) 

Hope this helps!

Answer:

Step-by-step explanation:

It will take 24.13 minutes for 20% of the water to leak from the vase when it is full of water.

The volume of a cone is given by the formula:

V = 1/3 πr²h

where r = 4.8/2 = 2.4 inches and h = 10 inches

Volume of vase = 1/3 * 22/7 * 2.4² * 10

Volume of vase = 19.2π in³

20% of volume will be:

20/100 * 19.2π = 3.84π in³

Rate = Volume / time

time = Volume / Rate

time = 3.84π / 0.5

time = 24.13 minutes

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Hence the volume of given figure is 4464 [tex]m^{3}[/tex]

Quadrilaterals make its  faces A cuboid is a six-sided solid shape known as a hexahedron in geometry. . A cuboid is similar to a cube or a Cuboid as like a short cube.  in the  cuboid can become a cube by variation the angles between the faces or the lengths of the edges.

A measurement of three-Dimensional space is volume. It is frequently expressed quantitatively using US-standard units or SI-derived units, as well as several imperial or Volume and the notion of length are connected.

According to figure ,

The volume of given figure = the volume of upper cuboid +the volume of lower cuboid

we know the the volume of cuboid = l*b*h [tex]m^{3}[/tex]

so, V = [tex](l_1*b_1*h_1)+(l_2*b_2*h_2)\\[/tex]

∴V=(11*(30-(9+9)*12) +(30*12*8)

∴V=(11*(30-18)*12) +(30*12*8)

∴V=(11*12*12) +(30*12*8)

∴V=(11*144) +(30*96)

∴V=1584 +2880

∴V=4464  [tex]m^{3}[/tex]

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The following parts can be answered by the concept from Probability.

a. The mean of the sample proportion is also 0.70.

b. The variance of the sample = (0.70(1-0.70))/100 = 0.0021

c. The standard error of the sample is 0.0458

d. The probability that the sample proportion exceeds 0.80 is  2.18

a. The mean of the distribution of the sample proportion of returns leading to refunds can be found using the formula:
mean = p = 0.70, where p is the population proportion of returns leading to refunds.
Therefore, the mean of the sample proportion is also 0.70.

b. The variance of the sample proportion can be found using the formula:
variance = (p(1-p))/n, where n is the sample size.
Substituting the given values, we get:
variance = (0.70(1-0.70))/100 = 0.0021

c. The standard error of the sample proportion can be found using the formula:
standard error = sqrt(variance)
Substituting the calculated variance value, we get:
standard error = √(0.0021) = 0.0458

d. To find the probability that the sample proportion exceeds 0.80, we need to standardize the sample proportion using the formula:
z = (sample proportion - population proportion) / standard error
Substituting the given values, we get:
z = (0.80 - 0.70) / 0.0458 = 2.18

Using a standard normal distribution table or calculator, we can find the probability of getting a z-score of 2.18 or higher, which is approximately 0.015 or 1.5%. Therefore, the probability that the sample proportion exceeds 0.80 is 0.015 or 1.5%.

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The temperature of the star emitting radiation with a peak wavelength of 670 nm is approximately 4325.37 K.

To get the temperature of a star with a peak wavelength of 670 nm, you can use Wien's Law, which states: T = b / λ_max where T is the temperature of the star, b is Wien's constant (approximately 2.898 x 10^6 nm K), and λ_max is the peak wavelength.
In this case, the peak wavelength (λ_max) is 670 nm. To calculate the temperature (T) of the star, follow these steps:
Step:1. Plug in the values into Wien's Law equation: T = (2.898 x 10^6 nm K) / (670 nm)
Step:2. Divide the constant by the peak wavelength: T ≈ (2.898 x 10^6) / 670
Step:3. Perform the calculation: T ≈ 4325.37 K
So, the temperature of the star emitting radiation with a peak wavelength of 670 nm is approximately 4325.37 K.

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The temperature of the star emitting radiation with a peak wavelength of 670 nm is approximately 4325.37 K.

To get the temperature of a star with a peak wavelength of 670 nm, you can use Wien's Law, which states: T = b / λ_max where T is the temperature of the star, b is Wien's constant (approximately 2.898 x 10^6 nm K), and λ_max is the peak wavelength.
In this case, the peak wavelength (λ_max) is 670 nm. To calculate the temperature (T) of the star, follow these steps:
Step:1. Plug in the values into Wien's Law equation: T = (2.898 x 10^6 nm K) / (670 nm)
Step:2. Divide the constant by the peak wavelength: T ≈ (2.898 x 10^6) / 670
Step:3. Perform the calculation: T ≈ 4325.37 K
So, the temperature of the star emitting radiation with a peak wavelength of 670 nm is approximately 4325.37 K.

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The conditions under which the K-means algorithm will coverage or end are: A. All the data points have their own cluster; B. No centroids need to move their location; C. No data points need to change their cluster; D. All clusters have sufficient data points; E. The clustering yields the desirable number of clusters.

K-means algorithm need multiple iterations to generate desirable results.

The conditions under which the K-means algorithm will coverage or end are s follows:

A - If all data points have their own cluster, the algorithm has covered all data points and there is no need for further iterations.
B - If no centroids need to move their location, it means that they have already converged to the optimal position and further iterations are not necessary.
C - If no data points need to change their cluster, it means that the clusters have already been formed optimally and further iterations are not needed.
E - If the algorithm has generated the desirable number of clusters, there is no need for further iterations.

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The derivative of the function y = sec⁻¹(4s³ + 9) is [tex]dy/ds = (12s^2) / (|4s^3 + 9| * \sqrt{((4s^3 + 9)^2 - 1))}[/tex].

We have to find the derivative of y with respect to s for the given function y = sec⁻¹(4s³ + 9).

Here are the steps to find the derivative:

1. Identify the function:

y = sec⁻¹(4s³ + 9).

2. Apply the chain rule:

dy/ds = (dy/du) * (du/ds), where u = 4s³ + 9.

3. Find dy/du:

Since y = sec⁻¹(u), the derivative

[tex]dy/du = 1 / (|u| * \sqrt{(u^2 - 1)}).[/tex]

4. Find du/ds:

Since u = 4s³ + 9, the derivative du/ds = 12s².

5. Combine the derivatives:

[tex]dy/ds = (1 / (|4s^3 + 9| * \sqrt{((4s^3 + 9)^2 - 1))}) * (12s^2)[/tex].

So, the derivative of y with respect to s for the function y = sec⁻¹(4s³ + 9) is:

[tex]dy/ds = (12s^2) / (|4s^3 + 9| * \sqrt{((4s^3 + 9)^2 - 1))}[/tex]

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The monthly payment needed to amortize a typical $135,000 mortgage loan amortized over 30 years at an annual interest rate of 6.9% compounded monthly is $849.06.

To find the monthly payment:

The formula to calculate the monthly payment needed to amortize a mortgage loan is:

M = P [ i(1 + i)^n ] / [ (1 + i)^n – 1]

Where:
M = Monthly payment
P = Loan amount (in this case, $135,000)
i = Interest rate per month (6.9% / 12 = 0.575%)
n = Total number of payments (30 years x 12 months per year = 360)

Substituting the values into the formula, we get:

M = $135,000 [ 0.00575(1 + 0.00575)^360 ] / [ (1 + 0.00575)^360 – 1]
M = $849.06

Therefore, the monthly payment needed to amortize a typical $135,000 mortgage loan amortized over 30 years at an annual interest rate of 6.9% compounded monthly is $849.06.

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The independent variable grade level with a scale range of 7 to 12 should be placed on the horizontal axis. While the dependent variable math score with a scale of 0% to 100% should be placed on the vertical axis.

On a line graph, the independent variable is typically represented on the x-axis and the dependent variable on the y-axis. The independent variable is the variable that is controlled or manipulated, while the dependent variable is the variable that is being measured or observed and is affected by changes in the independent variable.

From the question, the independent variables are: 7, 8, 9, 10, 11, and 12.

While the dependent variables are: 72, 75, 81, 80, 83, and 91

Therefore, the independent variable grade level with a scale range of 7 to 12 should be placed on the horizontal axis. While the dependent variable math score with a scale of 0% to 100% should be placed on the vertical axis.

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The null hypothesis that the pump's average flow rate is 10 liters/sec cannot be ruled out at the 0.01 level of significance.

A one-sample t-test can be used to determine whether a specific pump's average flow rate is 10 litres per second.

The alternative hypothesis is that the population mean flow rate is not 10 liters/sec, contrary to the null hypothesis that it is.

The test statistic, where the hypothesised mean is 10 liters/sec, is calculated as follows: t = (sample mean - hypothesised mean) / (sample standard deviation / sqrt(sample size)).

First, we must determine the sample mean and sample standard deviation: sample mean = (10.05 liters/sec) sample standard deviation =

10.05 litres per second is the sample mean (10.2 + 9.7 + 10.1 + 10.3 + 10.1 + 9.8 + 9.9 + 10.4 + 10.3 + 9.8)/10.

0.23 litres per second.

The formula for t is given as follows after substituting these values: t = (10.05 - 10) / (0.23 / [tex]\sqrt{10}[/tex]) = 1.3

For this test, n - 1 = 9 represents the degrees of freedom.

The crucial t-value is found to be 3.250 using a t-distribution table with 9 degrees of freedom and a significance threshold of 0.01 (two-tailed).

We are unable to reject the null hypothesis since the calculated t-value (1.3) is less than the crucial t-value (3.250).

Therefore, we lack sufficient data to draw the conclusion that the pump's average flow rate deviates from 10 liters/sec at the 0.01 level of significance.

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Answer :- the average rate of change of the function f(x) = 2x^2 over the interval [-7, 3] is -8.

The function is f(x) = 2x^2, and the interval is [-7, 3].

The average rate of change of a function over an interval can be found using the following formula:

Average Rate of Change = (f(b) - f(a)) / (b - a)

Here, 'a' is the initial point in the interval, and 'b' is the final point in the interval. In this case, a = -7 and b = 3.

Step 1: Find f(a) and f(b)
f(a) = f(-7) = 2(-7)^2 = 2(49) = 98
f(b) = f(3) = 2(3)^2 = 2(9) = 18

Step 2: Plug the values into the formula
Average Rate of Change = (f(b) - f(a)) / (b - a)
= (18 - 98) / (3 - (-7))
= (-80) / (10)

Step 3: Calculate the result
Average Rate of Change = -8

So, the average rate of change of the function f(x) = 2x^2 over the interval [-7, 3] is -8.

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The correct answer is c.

The interpretation of the coefficient $0.03 in the regression of textbook retail price (PRICE) on number of pages in the book (LENGTH) is: As the length of the book increases by one page, the estimated price increases by $0.03 on average. Therefore, the correct answer is c.

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the eigenvalues of the matrix A are:
-0.33, 1.71, 2.09 + 0.54i, 2.09 - 0.54i
Note that the complex eigenvalues come in conjugate pairs, which reflects the fact that matrix A is real and symmetric.

To find the eigenvalues of matrix A, we need to solve the characteristic equation det(A-λI)=0, where I is the identity matrix and λ is the eigenvalue.

For the given matrix A=[ 3 -4 2 1 ], the characteristic equation is:

det(A-λI) = det([ 3-λ -4 2 1 ][ λ 1 0 0 ][ 0 0 λ 1 ][ 0 0 0 λ ])

= (3-λ) [ (λ-1)(λ-1) + 8 ] + 4 [ (λ-1)(λ-1) - 2λ ] - 2 [ -4(λ-1) + 2λ ]

= λ⁴ - 7λ³+ 12λ² + 19λ - 18

Now, we need to find the roots of this polynomial to get the eigenvalues. We can do this by factoring or by using numerical methods such as Newton's method.

Using a calculator or computer, we can find that the roots of the polynomial are approximate:

λ ≈ -0.33, 1.71, 2.09 + 0.54i, 2.09 - 0.54i

Therefore, the eigenvalues of the matrix A are:

-0.33, 1.71, 2.09 + 0.54i, 2.09 - 0.54i

Note that the complex eigenvalues come in conjugate pairs, which reflects the fact that the matrix A is real and symmetric.

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Using linear approximation, (32.05)^(4/5) is equal to 16.08 (rounded to two decimal places).

To use linear approximation (or differentials) to estimate (32.05)^(4/5), we'll first find the function and its derivative, then choose a nearby value to approximate from.
1. Define the function: f(x) = x^(4/5)
2. Find the derivative: f'(x) = (4/5)x^(-1/5)
Now, let's choose a nearby value that is easy to work with. In this case, we'll choose x=32.
3. Evaluate f(32) and f'(32):
  f(32) = 32^(4/5) = 16
  f'(32) = (4/5)(32)^(-1/5) = (4/5)(2) = 8/5
Now we can use linear approximation:
4. Δx = 32.05 - 32 = 0.05
5. Δf ≈ f'(32) × Δx = (8/5) × 0.05 = 0.08
Lastly, approximate the value:
6. f(32.05) ≈ f(32) + Δf ≈ 16 + 0.08 = 16.08

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Answer:

11.5

Step-by-step explanation:

Add 7+8+9+9+11+11+12+14+15+19. Divide it all by 10 (the number of values.)

Answer:

11.5

Step-by-step explanation:

Add 7+8+9+9+11+11+12+14+15+19. Divide it all by 10 (the number of values.)

The probability that the series goes to 7 games is approximately 0.2734, or 27.34%.To find the probability that the series goes to 7 games, we can use the binomial distribution. Let X be the random variable representing the number of games won by one of the teams in a best-of-seven series.

Then, X follows a binomial distribution with parameters n=7 and p=0.5, where n is the number of trials & p is the probability of success in each trial (i.e., winning a game).

Both sides must win three games apiece in the first six games for the series to proceed to seven. The series winner will then be decided in the seventh game.

As a result, the likelihood that the series will go to 7 games is the same as the likelihood that each side will win precisely 3 of the first 6 games, which is:

P(X=3) = (7 choose 3) * (0.5)^3 * (1-0.5)^(7-3) = 35/128 = 0.2734

where (7 choose 3) is the number of ways to choose 3 games out of 7. Therefore, the probability that the series goes to 7 games is approximately 0.2734, or 27.34%.

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The total number of meters that Morgan rode in all would be 4, 600 m .

Morgan rode her bike 2 kilometers to her friend's house and then eventually rode back home so the distance rode was ;

= 2 km + 2 km

= 4 km

In meters, this would be:

= 4 km x 1, 000 meters per km

= 4 km x 1, 000

= 4, 000 m

Then, she rode 600 meters to and from the library for a total of :

= 4, 000 + 600

= 4, 600 m

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A and B are equivalent since they both simplify to x³ - x² + 4. Therefore, the answer is A and B.

To determine which polynomials are equivalent, we need to simplify each polynomial first. The simplified forms of the given polynomials are:

A. (x³ - 2x + 1) - x(x - 2) = x³ - x² + 2x - 1

B. x(x²-4) - (x - 2)² = x³ - x² + 4

C. x³-(x - 1)(x + 1) = x³ - (x² - 1) = x³ - x² + 1

D. x(x - 2)² + 3x(x - 1) = x³ - x² + 5x

E. -(2x² + 3) + (x³ + x) + (3x² - x + 2) = x³ + x² - x - 1

From the simplified forms, we can see that polynomials A and B are equivalent since they both simplify to x³ - x² + 4. Therefore, the answer is A and B.

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The volume of a cylinder is given by the formula V = πr²h, where r is the radius and h is the height. Since the diameter of the rod is 8 cm, the radius is 4 cm.

The given rod has a circular cross-section with a radius of 4 centimeters and a length of 30 centimeters. The volume of this rod can be calculated using the formula for the volume of a cylinder, which is V = πr²h, where r is the radius of the circular base and h is the height or length of the cylinder.

Substituting the given values into the formula, we get:

V = π(4²)(30)

Simplifying the expression, we get:

V = 480π cubic centimeters

This is the volume of one rod. To find the total amount of steel needed to make 113 rods, we simply multiply the volume of one rod by 113, since all rods are of the same size and shape.

total steel needed = 113 × V

total steel needed = 113 × 480π cubic centimeters

total steel needed = 54,240π cubic centimeters

Therefore, the total amount of steel needed to make 113 rods is 54,240π cubic centimeters.

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Step-by-step explanation:

Volume of a cylinder = pi r^2 h     <=====solve for 'h'

h = volume / (pi r^2)

  = 348.3 mm^3 / ( pi * 4.5^2)             ( I assumed the dimension mm^3 )

h = ~ 5.5 mm

Answer:

Yes, these two figures are congruent. Rotate the figure on the left 90° counterclockwise, and it will look just like the figure on the right.

The choice of reinforcement schedule can have important implications for both learning and the persistence of behavior over time.

The reinforcement schedule can have a significant impact on how quickly a response/behavior is learned and how quickly extinction occurs.

In general, continuous reinforcement schedules (where the behavior is reinforced every time it occurs) tend to result in faster learning of the behavior than partial reinforcement schedules (where the behavior is only reinforced some of the time). This is because the individual learns more quickly that the behavior is associated with the reinforcement.

However, once the behavior is learned, partial reinforcement schedules tend to result in greater resistance to extinction than continuous reinforcement schedules. This is because the individual has learned that the behavior is not always followed by reinforcement, so they are more likely to persist in the behavior even if reinforcement is no longer provided.

Among partial reinforcement schedules, fixed ratio schedules (where reinforcement is provided after a fixed number of responses) tend to lead to the fastest responding and highest rates of responding, but also tend to result in rapid extinction once reinforcement is removed. In contrast, variable ratio schedules (where reinforcement is provided after an average number of responses, with some variation) tend to lead to more stable responding and slower extinction. Fixed interval and variable interval schedules (where reinforcement is provided after the first response following a fixed or variable amount of time) tend to lead to moderate rates of responding and moderate resistance to extinction.

Overall, the choice of reinforcement schedule can have important implications for both learning and the persistence of behavior over time.

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The correct answer is O (f(x)^2) + C. Note that this expression does not match any of the answer choices provided in the question.

To evaluate the integral f'(x) f(x) dx using substitution, we can let u = f(x), so that du/dx = f'(x) and dx = du/f'(x). Substituting these expressions into the integral, we get:

∫ f'(x) f(x) dx = ∫ u du

Integrating u with respect to itself, we get:

∫ u du = (u^2)/2 + C

Substituting back for u, we get:

∫ f'(x) f(x) dx = (f(x)^2)/2 + C

Therefore, the correct answer is O (f(x)^2) + C. Note that this expression does not match any of the answer choices provided in the question.

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The correct answer is O (f(x)^2) + C. Note that this expression does not match any of the answer choices provided in the question.

To evaluate the integral f'(x) f(x) dx using substitution, we can let u = f(x), so that du/dx = f'(x) and dx = du/f'(x). Substituting these expressions into the integral, we get:

∫ f'(x) f(x) dx = ∫ u du

Integrating u with respect to itself, we get:

∫ u du = (u^2)/2 + C

Substituting back for u, we get:

∫ f'(x) f(x) dx = (f(x)^2)/2 + C

Therefore, the correct answer is O (f(x)^2) + C. Note that this expression does not match any of the answer choices provided in the question.

To learn more about expression visit:

https://brainly.com/question/14083225

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