Which of the following products is defined?

GH
FE
EH
FG
All of the above are undefined

Answer:

x = 4

Step-by-step explanation:

using the rule of exponents

• [tex]a^{-m}[/tex] = [tex]\frac{1}{a^{m} }[/tex] , then

[tex]4^{-x}[/tex] = [tex]\frac{1}{4^{x} }[/tex]

and 256 = [tex]4^{4}[/tex]

then

[tex]\frac{1}{4^{x} }[/tex] = [tex]\frac{1}{4^{4} }[/tex]

so

[tex]4^{x}[/tex] = [tex]4^{4}[/tex]

since bases on both sides are equal, bot 4 then equate exponents

x = 4

On the 7th day after the videο was pοsted, 640 peοple will see the videο.

Statistics is the discipline that cοncerns the cοllectiοn, οrganizatiοn, analysis, interpretatiοn, and presentatiοn οf data.

1 Let's start by finding the pattern in the number οf viewers. We knοw that οn the first day, 5 peοple watched the videο. On the next day, the number οf viewers dοubled tο 5 x 2 = 10. On the third day, the number οf viewers dοubled again tο 10 x 2 = 20. We can see that the number οf viewers is dοubling each day, which means we can write the number οf viewers as:

[tex]V = 5 x 2^n[/tex]

where n is the number οf days after the videο was pοsted.

Nοw we want tο find οn which day the number οf viewers will be 640. Sο we can set V equal tο 640 and sοlve fοr n:

[tex]640 = 5 x 2^n[/tex]

[tex]2^n = 128[/tex]

n = lοg2(128) = 7

2.   Tο find the first day when mοre than 20,000 peοple will see the videο, we can set V equal tο 20,000 and sοlve fοr n:

[tex]20,000 = 5 x 2^n[/tex]

2^n = 4,000

n = lοg2(4,000) ≈ 11.29

Since n represents the number οf days after the videο was pοsted, we can rοund up tο the next whοle number tο find the first day when mοre than 20,000 peοple will see the videο. Therefοre, οn the 12th day after the videο was pοsted, mοre than 20,000 peοple will see the videο if the trend cοntinues

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Answer:

61°

Step-by-step explanation:

∠4 is the vertical angle to the 61° angle. This means they will have the same measure, so ∠4 is 61°.

The probability of picking a vowel, replacing it, and then picking a consonant from the word "SLEEP" is approximately 0.096 or 9.6%.

Probability is usually expressed as a number between 0 and 1, with 0 meaning that the event is impossible and 1 meaning that the event is certain.

There are two vowels (E) and three consonants (S, L, P) in the word "SLEEP".

The probability of picking a vowel on the first draw is 2/5, because there are two vowels out of five letters total.

Since we replace the vowel we picked, the probability of picking another vowel on the second draw is also 2/5.

The probability of picking a consonant on the third draw is 3/5, because there are three consonants left out of five letters total.

Therefore, the probability of picking a vowel, replacing it, and then picking a consonant from the word "SLEEP" is:

(2/5) x (2/5) x (3/5) = 12/125 or approximately 0.096 or 9.6%.

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The probability of obtaining a reading between 0.59°C and 0.88°C is 0.7224 and 0.8106.

The sum of all possible values, weighted by the chance of each value, is equal to the mean of a discrete probability distribution of the random variable X. Each possible number of X must be multiplied by its probability P(x) before being added as a whole to determine the mean. In statistics, the mean is one measure of central trend in addition to the mode and median. The mean is simply the average of the numbers in the specified collection. It suggests that values in a specific data gathering are evenly distributed. In order to find the mean, the total values given in a datasheet must be added, and the result must be divided by the total number of values.

In this question, using the formula,

z-score = (x – μ) / σ

where:

x: individual data value

μ: population mean

σ: population standard deviation

for x=0.59

μ= 0

σ= 1

z-score= 0.59

Probability=0.7224

for x=0.88

z-score= 0.88

Probability=0.8106

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The equation can be rearranged to the form y = -qx + r. This is the equation of a straight line, which can be graphed. The point of intersection of the two lines, ak + bl = 0 and y = -qx + r, is the solution for the two variables (k and l).

The equation ak + bl = 0 is a linear equation in two variables and is solved using the method of elimination. The equation can be written in the form ax + by = c, where a, b, c are constants. To solve this equation, both sides of the equation should be divided by the coefficient of one of the variables (a or b). This will result in a equation of the form x + qy = r, where q and r are constants. Then, the equation can be rearranged to the form y = -qx + r. This is the equation of a straight line, which can be graphed. The point of intersection of the two lines, ak + bl = 0 and y = -qx + r, is the solution for the two variables (k and l). The two variables can then be calculated using the point of intersection by substituting the x and y values into the two equations. In this way, the two variables k and l can be found such that ak + bl = 0.

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What are the integers k and l such that ak + bl = 0?

Answer: 39

39 is the only answer option greater than 17

Answer:

3/4 of a bag of dog food will last 3 days.

A) Relative frequency is number of times an event happened over total number of events:

Answer is 21/50 = 0.42

B) On a 6 sides die, there are 3 even numbers and 3 odd numbers, so the theoretical probability of landing on odd would be 3/6 = 0.50

C) Because the die has an equal amount of chance landing on even or odd, both are 3/6, then the dice is not biased.

Since Kingsley wanted an equation to convert from inches to centimeters, the correct answer is B) c = 2.54i.

What is equation ?

An equation is a statement that asserts the equality of two expressions, usually separated by an equals sign (=). The expressions on either side of the equals sign may contain one or more variables, which are unknown values that can be determined by solving the equation.

Kingsley wants to convert a given length in inches to centimeters. He knows that 1 inch is about 2.45 centimeters.

Let's call the length in inches "i" and the length in centimeters "c".

We want to find an equation that relates i and c. We know that 1 inch is about 2.45 centimeters, so we can write:

1 inch = 2.45 centimeters

To convert from inches to centimeters, we can multiply the length in inches by 2.45. So:

c = 2.45i

This is the equation Kingsley can use to convert any given length in inches to centimeters.

Alternatively, we can rearrange this equation to solve for i:

c = 2.45i

Divide both sides by 2.45:

c/2.45 = i

So the equation for converting from centimeters to inches is:

i = c/2.45

Therefore,  since Kingsley wanted an equation to convert from inches to centimeters, the correct answer is B) c = 2.54i.

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Answer:

the answer will be 117

Step-by-step explanation:

you need to multiply

The employee will contribute 4% of the first $1000, which is $40. Then, the employee will contribute 3% of the next $1000, which is $30. Following that, the employee will contribute 2% of the next $1000, which is $20. Finally, the employee will contribute 1% of the remaining $17,000, which is $170. Therefore, the employee will contribute a total of $260 to the fund.

An employee who earned $20,000 will contribute $260 to the welfare fund.

To calculate the contribution to the welfare fund for an employee who earned $20,000, we can break down the earnings into different tiers based on the given rates.

The first $1000 will have a contribution rate of 4%.

Contribution for the first $1000 = 4% of $1000 = $40.

The next $1000 will have a contribution rate of 3%.

Contribution for the next $1000 = 3% of $1000 = $30.

The next $1000 will have a contribution rate of 2%.

Contribution for the next $1000 = 2% of $1000 = $20.

The remaining amount above $3000 ($20,000 - $3000 = $17,000) will have a contribution rate of 1%.

Contribution for the remaining amount = 1% of $17,000 = $170.

Now, let's sum up the contributions for each tier:

$40 + $30 + $20 + $170 = $260.

Therefore, an employee who earned $20,000 will contribute $260 to the welfare fund.

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When a shape is dilated, the size of the shape changes. The true statement is (d) The scale factor is 2.5.

Dilation:

Dilation is the process of changing the size of an object or shape by reducing or increasing its size by a specific scale factor. For example, a circle with a radius of 10 units shrinks to a circle with a radius of 5 units. Applications of this method are in photography, arts and crafts, sign making and more.

According to the Question:

How to determine the scale factor

In figure A, we have:

Length = 0.6

In figure B, we have:

Length =1.5

The scale factor is then calculated as:

K = 1.5/0.6

Dividing the equation:

k = 2.5

Hence, the true statement is (d) The scale factor is 2.5.

Complete Question:

The first figure is dilated to form the second figure. Which statement is true?

The scale factor is 0.4.

The scale factor is 0.9.

The scale factor is 2.1.

The scale factor is 2.5.

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The Davenports' medical expenses contribute to their total itemized deductions are $375 (7.5% for 2019).

The costs you incurred for state and local income or sales taxes, real estate taxes, personal property taxes, mortgage interest, and disaster losses are all included in itemised deductions. You can also count charitable donations and a portion of your out-of-pocket medical and dental costs.

Currently for the 2019 (due 2020), you can deduct medical expenses that exceed 7.5% of your AGI, but back then in 2019, the threshold was 7.5%, not 10%.

So the Davenports can only deduct

$5,250 - ($65,000 x 7.5%) = $375

if they decided to itemize their deductions.

The threshold will increase back to 10% starting 2020 (due 2021) tax returns.

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The value of x is 5

Triangles with the same shape but different sizes are said to be similar triangles. To be more specific, two triangles are comparable if their respective sides are proportionate and their corresponding angles are congruent.

Two triangles are similar if corresponding angles are congruent and corresponding sides are proportional.

from the below figure, both the triangles are similar, ∆ABC ≈ ∆EFB

By using Thales's theorem, the ratio of the sides of triangles are;

BE/EA = BF/FC

15/30 = x/10    

x = 5

Therefore the value of x is 5

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The ratio of triangle sides can be calculated using Thales' theory, and it is the value of x is 5

Similar triangles are those with the same shape but varying sizes. To be more precise, two triangles are comparable if their matching angles and respective sides are congruent.

If matching sides are proportional and corresponding angles are congruent, two triangles are similar.

Both triangles in the following figure are comparable ∆ABC ≈ ∆EFB

The ratio of triangle sides can be calculated using Thales' theory, and it is;

BE/EA = BF/FC

15/30 = x/10    

x = 5

Therefore the value of x is 5

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a) The domain of the function is  [0, 7.1875], while the range is [0,23]. Considering the domain and the range, the graph of the function is given by the image presented at the end of the answer.

b) The trip had a duration of 5.1875 hours.

The function for this problem is defined as follows:

g(t) = 23 - 3.2t.

The domain is the set of input values that can be assumed by the function. The time cannot have negative measures, hence the lower bound of the domain is of zero, while the gas cannot be negative, hence the upper bound of the domain is given as follows:

23 - 3.2t = 0

3.2t = 23

t = 23/3.2

t = 7.1875 hours.

The range is given by the set of all output values assumed the function, which are the values of the gas, hence it is [0,23].

The graph is a linear function between points (0, 23) and (7.1875, 0).

At the end of the trip there were 6.4 gallons left, hence the length of the trip is obtained as follows:

23 - 3.2t = 6.4

t = (23 - 6.4)/3.2

t = 5.1875 hours.

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The joint probability density function for the random variables Y1 and Y2 E(Y1+Y2) and V(Y1+Y2) is 41/144.

To find E(Y1+Y2), we need to integrate the sum of Y1 and Y2 over their joint probability density function:

E(Y1+Y2) = ∫∫ (y1 + y2) f(y1,y2) dy1 dy2

= ∫∫ (y1 + y2) (2) dy1 dy2, where the limits of integration are 0 to 1 for both y1 and y2 and y1+y2 <=1

= ∫[tex]0^1[/tex] ∫[tex]0^{(1-y1)}[/tex](y1 + y2) (2) dy2 dy1

= ∫[tex]0^1[/tex] (2y1 + 1) (1-y1)² dy1

= 5/12

To find V(Y1+Y2), we can use the formula V(Y1+Y2) = E[(Y1+Y2)²] - [E(Y1+Y2)]².

First, we need to find E[(Y1+Y2)^2]:

E[(Y1+Y2)²] = ∫∫ (y1+y2)² f(y1,y2) dy1 dy2

= ∫∫ (y1² + y2² + 2y1y2) (2) dy1 dy2, where the limits of integration are 0 to 1 for both y1 and y2 and y1+y2 = 1

= ∫[tex]0^1[/tex] ∫[tex]0^{(1-y1)}[/tex] (y1² + y2² + 2y1y2) (2) dy2 dy1

= ∫[tex]0^1[/tex] (1/3)y1³ + (1/2)y1² + (1/2)y1

(1/3)y1 + (1/4) dy1

= 7/12

Next, we need to find [E(Y1+Y2)]²:

[E(Y1+Y2)]² = (5/12)² = 25/144

Therefore, V(Y1+Y2) = E[(Y1+Y2)²] - [E(Y1+Y2)]² = (7/12) - (25/144) = 41/144.

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Answer: numbers 1,3 and 4

Step-by-step explanation:

Samir's new balance is $5,044.17.

To calculate Samir's new balance, add the previous balance, subtract the payment, add the new transaction, and multiply by the interest rate for one period. The following formula can be used to calculate the interest for a single period:

balance * APR / 12 = interest

where APR stands for annual percentage rate and 12 represents the number of months in a year.

When we apply this formula to Samir's balance and APR, we get:

5336.22 * 0.29 / 12 = 128.95 in interest

As a result, the total new balance is:

5336.22 - 607 + 186 + 128.95 = 5044.17

We get the following when we round to the nearest hundredth:

$5,044.17

As a result, Samir now has a balance of $5,044.17.

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A quadratic function in standard form that passes through the points [tex](-8,0), (-5,-3), and (-2,0)[/tex] is equals to the [tex]f(x) = (1/3)( x^{2} + 10x + 16)[/tex].

f(x) = ax2 + bx + c, in which a, b, and c are integers and an is not equal to zero, is a quadratic function. A parabola is the shape of a quadratic function's graph.

In other terms, you have a quadratic equation if a times the squares of the expression after b plus b twice that same equation not square plus c equals 0.

[tex]f(x) = ax^{2} + bx + c ----(1)[/tex]

is determined by three points and must be [tex]a[/tex] not equal [tex]0[/tex]. That is for determining the f(x) we have to determine value of three values a, b, and c. Now, we have three ordered pairs [tex](-8,0), (-5,-3)[/tex], and [tex](-2,0)[/tex] and we have to determine quadratic function passing through these points. So, firstly, plug the coordinates of point[tex]( -8,0), x = -8, y = f(x) = 0[/tex] in equation [tex](1)[/tex],

[tex]= > 0 = a(-8)^{2} + b(-8) + c[/tex]

[tex]= > 64a - 8b + c = 0 -------(2)[/tex]

Similarly, for second point [tex]( -5,-3) , f(x) = -3, x = -5[/tex]

[tex]= > - 3 = a(-5)^{2} + (-5)b + c[/tex]

[tex]= > 25a - 5b + c = -3 --(3)[/tex]

Continue for third point [tex](-2,0)[/tex]

[tex]= > 0 = a(-2)^{2} + b(-2) + c[/tex]

[tex]= > 4a -2b + c = 0 --(4)[/tex]

So, we have three equations and three values to determine.

Subtract equation [tex](4)[/tex] from [tex](2)[/tex]

[tex]= > 64 a - 8b + c - 4a + 2b -c = 0[/tex]

[tex]= > 60a - 6b = 0[/tex]

[tex]= > 10a - b = 0 --(5)[/tex]

subtract equation [tex](4)[/tex] from [tex](3)[/tex]

[tex]= > 21a - 3b = -3 --(6)[/tex]

from equation (4) and (5),

[tex]= > 3( 10a - b) - 21a + 3b = -(- 3)[/tex]

[tex]= > 30a - 3b - 21a + 3b = 3[/tex]

[tex]= > 9a = 3[/tex]

[tex]= > a = 1/3[/tex]

from [tex](5)[/tex] , [tex]b = 10a = 10/3[/tex]

from [tex](4)[/tex], [tex]c = 2b - 4a = 20/3 - 4/3 = 16/3[/tex]

So,[tex]f(x)= (1/3)( x^{2} + 10x + 16)[/tex]

Hence, required values are [tex]1/3, 10/3,[/tex] and [tex]16/3[/tex].

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Answer:

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

Given 3 points of a quadratic function and two of them lie on the x-axis:

These two points are representing the roots of the function. With known roots we can show the function in the factor form:

Substitute the roots into the equation and use the third point with coordinates x = - 5, f(x) = - 3, find the value of a:

This gives us the function in the factor form:

Convert this into standard form of f(x) = ax² + bx + c:

The experimental probability that the card selected was a K or 6 is 17/100 or 0.17.

What is probability?

Probability is a measure of the likelihood or chance of an event occurring. It is expressed as a number between 0 and 1, where 0 indicates that an event is impossible and 1 indicates that it is certain. The probability of an event can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes. In other words, it is the ratio of the number of desired outcomes to the total number of outcomes.

The frequency of card 6 is 7 and the frequency of card K is 12. However, the card K is also counted in the total count for JQK, so we need to subtract 2 from the frequency of K to get the actual count of K.

Actual count of K = 12 - 2 = 10

Total count of 6 and K = 7 + 10 = 17

The experimental probability of drawing a K or 6 is the frequency of drawing K or 6 divided by the total number of draws:

Experimental probability = (frequency of K or 6) / (total number of draws)

Experimental probability = 17 / 100

Therefore, the experimental probability that the card selected was a K or 6 is 17/100 or 0.17.

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The function g(t) is defined as:

g(t) = 1/√(16-t^2)

The function is continuous for all values of t that satisfy the following conditions:

The denominator is non-zero:

The denominator of the function is √(16-t^2). Therefore, the function is undefined when 16-t^2 < 0, or when t is outside the interval [-4,4].

There are no vertical asymptotes:

The function does not have any vertical asymptotes, because the denominator is always positive.

Thus, the function g(t) is continuous on the interval [-4,4].

The values of x for which the series converges is x ∈ (-1/5, 1/5). The sum of the series for those values of x is (-5x)/(1 + 5x).

The series is [tex]\Sigma^{\infty}_{n=1}(-5)^nx^n[/tex].

We can write this series as [tex]\Sigma^{\infty}_{n=1}(-5x)^n[/tex].

This is a infinite geometric series with first term a = -5x and common ration r = -5x.

It is convergent when

|r| < 1

|-5x| < 1

|-5| |x| < 1

5|x| < 1

Divide by 5 on both side, we get

|x| < 1/5

The series is convergent when x ∈ (-1/5, 1/5).

Sum of the series is

Sₙ = a/1 - n

Sₙ = (-5x)/{1 - (-5x)}

Sₙ = (-5x)/(1 + 5x)

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The complete question is:

Find the values of x for which the series converges. Find the sum of the series for those values of x.

[tex]\Sigma^{\infty}_{n=1}(-5)^nx^n[/tex]

The company can produce approximately 425 widgets for $2127.

What is cost function ?

The key concept used here is the concept of cost functions, which is an important concept in economics and business. A cost function is a mathematical function that expresses the total cost of production as a function of the level of output produced. In this case, the cost function is a linear function of the form C = a + bx, where C is the total cost, a is the fixed cost, b is the variable cost per unit, and x is the level of output.

Finding the number of widgets the company can produce given a fixed cost and a variable cost per widget :

To solve this problem, we can set up an equation that relates the total cost to the number of widgets produced.

Let x be the number of widgets produced.

The total cost C is given by:

C = fixed cost + variable cost

C = 1277 + 1.93x

We want to find the number of widgets produced for a total cost of $2127. So we can set up an equation:

2127 = 1277 + 1.93x

Subtracting 1277 from both sides gives:

850 = 1.93x

Dividing both sides by 1.93 gives:

x ≈ 439.9

Since we can't produce a fractional number of widgets, we need to round down to the nearest integer. Therefore, the company can produce approximately 425 widgets for $2127.

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a) The first four nonzero terms of the power series for f(x) about x=0 are

e^6 - 2x^2 + (2x^4)/2! - (2x^6)/3!

The general term of the power series is (-2)^n (2x)^(2n) / (2n)!

b) The interval of convergence of the power series is (-∞, ∞).

c) To estimate the error between f(x) and its partial sum g(x) given by the sum of the first four nonzero terms of the power series, we can use the Lagrange form of the remainder

|R4(x)| = |f(x) - g(x)| ≤ M |x|^5 / 5!

a) To find the power series for f(x) about x = 0, we can use the Maclaurin series formula

f(x) = Σ[n=0 to ∞] (fⁿ(0)/n!) xⁿ

where fⁿ(0) denotes the nth derivative of f evaluated at x=0.

In this case, we have

f(x) = e^6(-2x^2)

fⁿ(x) = dⁿ/dxⁿ(e^6(-2x^2)) = (-2)^n(2x)^ne^6(-2x^2)

So, we can write the power series as

f(x) = Σ[n=0 to ∞] ((-2)^n(2x)^n e^6(0))/n!)

= Σ[n=0 to ∞] ((-2)^n (2x)^n /n!)

To find the first four nonzero terms, we substitute n = 0, 1, 2, and 3 into the above formula

f(0) = e^6

f'(0) = 0

f''(0) = 24

f'''(0) = 0

So, the first four nonzero terms of the power series are:

e^6 - 2x^2 + (2x^4)/2! - (2x^6)/3!

The general term of the power series is

(-2)^n (2x)^(2n) / (2n)!

b) To find the interval of convergence of the power series, we can use the ratio test

lim [n→∞] |((-2)^(n+1) (2x)^(2n+2) / (2n+2)! ) / ((-2)^n (2x)^(2n) / (2n)!)|

= lim [n→∞] |-4x^2/(2n+1)(2n+2)|

= lim [n→∞] 4x^2/(2n+1)(2n+2)

Since this limit depends on the value of x, we need to consider two cases

i) If x = 0, then the power series reduces to the constant term e^6, and the interval of convergence is just x=0.

ii) If x ≠ 0, then the series converges absolutely if and only if the limit is less than 1 in absolute value

|4x^2/(2n+1)(2n+2)| < 1

This is true for all values of x as long as n is sufficiently large. So, the interval of convergence is the entire real line (-∞, ∞).

c) We can use the Lagrange form of the remainder to estimate the error between f(x) and its partial sum g(x) given by the sum of the first four nonzero terms of the power series

|R4(x)| = |f(x) - g(x)| ≤ M |x|^5 / 5!

where M is an upper bound for the fifth derivative of f(x) on the interval [-0.6, 0.6].

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Step-by-step explanation:

"increase" means to take the original 100% and put an additional 3.5% of these 100% on top of it.

so, we have to calculate

100% + 3.5% of £16470.45

100% of £16470.45 = £16470.45 × 100/100

3.5% of £16470.45 = £16470.45 × 3.5/100

the sum is therefore

£16470.45 × (100/100 + 3.5/100) =

= £16470.45 × (1 + 0.035) = £16470.45 × 1.035 =

= £17,046.91575 ≈ £17,046.92

Given that (1, 2, 3] System{1, 4, 7,6] for a system known to be LTI, the impulse response of the system: h[n] = (1/2)*δ[n] + δ[n-1] + (3/2)*δ[n-2]

To compute the impulse response h[n] of a linear time-invariant (LTI) system given its input-output relationship, we can use the convolution sum:

y[n] = x[n] * h[n]

y[n] = (1/2)*(x[n] + 2x[n-1] + 3x[n-2])

y[n] = (1/2)*(δ[n] + 2δ[n-1] + 3δ[n-2])

y[n] = (1/2)*δ[n] + δ[n-1] + (3/2)*δ[n-2]

Thus, the impulse response of the system is:

h[n] = (1/2)*δ[n] + δ[n-1] + (3/2)*δ[n-2],where δ[n] is the impulse signal.

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Therefore, the volume of the sphere to the nearest hundredth is 724,775.70 cubic millimeters.

Volume is a measurement of the amount of space occupied by a three-dimensional object. It is often expressed in units such as cubic meters (m³), cubic centimeters (cm³), cubic feet (ft³), or gallons (gal), depending on the context. The volume of a solid object can be calculated by multiplying its length, width, and height or using a specific formula depending on the shape of the object. For example, the volume of a rectangular box can be calculated as length x width x height, while the volume of a cylinder can be calculated as π x radius² x height. In general, volume is an important concept in many fields, including physics, chemistry, engineering, and architecture. It is often used to describe the capacity of containers, the displacement of fluids, and the amount of material used in construction or manufacturing.

Here,

The formula for the volume of a sphere is given as V = (4/3)πr³, where r is the radius of the sphere and π is approximately 3.14.

Substituting the given value of the radius, we get:

V = (4/3) x 3.14 x 48³

V ≈ 724,775.68 cubic millimeters

Rounding this value to the nearest hundredth, we get:

V ≈ 724,775.68 ≈ 724,775.70 cubic millimeters (rounded to two decimal places)

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The answer of the given question is (a) TR = p(40 - (4/3)p), or TR = 40p - (4/3)p² , (b)  TC = 9q + (3/10)q² + 30 , (c) π = 40p - (4/3)p² - 9q - (3/10)q² - 30 , (d) TR ≈ 342.67 , (e) the profit when 20 units are produced is approximately RM188.27 , (f)  the profit when 7 units are produced is approximately -RM24.44, indicating a loss , (g) the output required to obtain a profit of RM100 is approximately 8.78 units.

An equation is  mathematical statement that asserts yhe equality of two expressions. It typically consists of variables, constants, and mathematical operations like addition, subtraction, multiplication, and division, among others. Equations are often used to solve problems, to model real-world phenomena, and to describe mathematical relationships.

(a) The total revenue is given by TR = p x q. Substituting the demand equation q = 40 - (4/3)p, we get TR = p(40 - (4/3)p), or TR = 40p - (4/3)p².

(b) The total cost is given by TC = q x AC. Substituting the given average cost function, we get TC = 9q + (3/10)q² + 30.

(c) The profit is given by π = TR - TC. Substituting the expressions we found in parts (a) and (b), we get π = 40p - (4/3)p² - 9q - (3/10)q² - 30.

(d) At the break even point, the firm earns zero profit, so we set π = 0 and solve for q. Substituting the expression we found in part (a) for p, we get:

0 = 40p - (4/3)p² - 9q - (3/10)q² - 30

0 = 40(40/3 - (3/4)q) - (4/3)(40/3 - (3/4)q)² - 9q - (3/10)q² - 30

0 = 533.33 - 51.25q - 0.22q^2

Solving for q using the quadratic formula, we get:

q = (51.25 ± sqrt(51.25² - 4(-0.22)(533.33))) / 2(-0.22)

q ≈ 22.75 or q ≈ 206.58

We reject the solution q ≈ 206.58 because it is outside the relevant range of output, which is between 0 and 40. Therefore, the total output at the break even point is approximately 22.75 units. To find the total revenue at the break even point, we substitute q = 22.75 into the demand equation from part (a) and get:

p = (40/3) - (3/4)q

p ≈ 15.08

TR = p x q

TR ≈ 342.67

(e) To find the profit when 20 units are produced, we substitute q = 20 into the expression for profit we found in part (c) and get:

π = 40p - (4/3)p² - 9q - (3/10)q² - 30

π ≈ 188.27

Therefore, the profit when 20 units are produced is approximately RM188.27.

(f) To find the profit when 7 units are produced, we substitute q = 7 into the expression for profit we found in part (c) and get:

π = 40p - (4/3)p² - 9q - (3/10)q² - 30

π ≈ -24.44

Therefore, the profit when 7 units are produced is approximately -RM24.44, indicating a loss.

(g) To find the output required to obtain a profit of RM100, we set the profit equation equal to 100 and solve for q:

Profit = TR - TC

100 = pq - ACq

100 = (40-(4/3)p)*q - (9+(3/10)q+30/q)*q

100 = (40-(4/3)p - 9q - 3q²/10)

Multiplying by 10 and rearranging terms, we get a quadratic equation in q:

3q² + 91q - 310 = 0

Solving for q using the quadratic formula, we get:

q = (-91 ± sqrt(91² - 43(-310)))/(2*3)

q ≈ 8.78 or q ≈ -29.44

Since the quantity produced cannot be negative, the output required to obtain a profit of RM100 is approximately 8.78 units.

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Step-by-step explanation:

There are many possible solutions to this problem, but one possible set of values for P, Q, R, S, T, and U is:

P = 2

Q = 5

R = 1

S = 8

T = 9

U = 9

With these values, we have:

PQR = 251

STU = 156

And the sum of PQR and STU is indeed 407.