Find the derivative for f(x)=3x^2 −1 using first principle.

To rewrite the function v(t) according to the given student number, we replace a, b, and c with the respective values obtained from the last three digits. Then, we find v(f) by substituting f into the rewritten function. Finally, we sketch the graphs of v(t) and v(f).

Let's assume the student number is 1182836. In this case, a is 6, b is 3, and c is 8. Now, we rewrite the function v(t) accordingly:

v(t) = 6sin(20πt) - 3n(t/8) + 3n(t/8)cos(10πt) + 6sin(t/3) + 6∧(t/4)cos(4πt)

To find v(f), we substitute f into the rewritten function:

v(f) = 6sin(20πf) - 3n(f/8) + 3n(f/8)cos(10πf) + 6sin(f/3) + 6∧(f/4)cos(4πf)

To sketch the graphs of v(t) and v(f), we need to plot the function values against the corresponding values of t or f. The graph of v(t) will have the horizontal axis representing time (t) and the vertical axis representing the function values. The graph of v(f) will have the horizontal axis representing frequency (f) and the vertical axis representing the function values.

The specific shape of the graphs will depend on the values of t or f, as well as the constants and trigonometric functions involved in the function v(t) or v(f). It would be helpful to use graphing software or a graphing calculator to accurately sketch the graphs.

In summary, we rewrite the function v(t) according to the student number, substitute f to find v(f), and then sketch the graphs of v(t) and v(f) using the corresponding values of t or f.

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Answer:

B) x=8

Step-by-step explanation:

The two marked angles are alternate exterior angles since they are outside the parallel lines and opposites sides of the transversal. Thus, they will contain the same measure, so we can set them equal to each other:

[tex]11+7x=67\\7x=56\\x=8[/tex]

Therefore, B) x=8 is correct.

For the given equations, the center and radius of the circles are as follows:

a) Center: (3, 5), Radius: 4

b) Center: (-4, 1), Radius: 2

a) The equation (x-3)² + (y-5)²=16 is in the standard form of a circle equation, (x-h)² + (y-k)² = r², where (h, k) represents the center of the circle and r represents the radius.

Comparing the given equation with the standard form, we can identify that the center is at (3, 5) and the radius is [tex]\sqrt{16}[/tex]=4.

b) Similarly, for the equation (x+4)² + (y-1)² =4 we can identify the center as (-4, 1) and the radius as [tex]\sqrt{4}[/tex] =2.

To sketch the graphs, start by marking the center point on the coordinate plane according to the determined coordinates.

Then, plot points on the graph that are at a distance equal to the radius from the center in all directions. Connect these points to form a circle shape.

For equation (a), the circle will have a center at (3, 5) and a radius of 4. For equation (b), the circle will have a center at (-4, 1) and a radius of 2.

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A. As x approaches negative infinity, f(x) approaches negative infinity.

B. As x approaches positive infinity, f(x) approaches positive infinity.

C. The zeros (x-intercepts) of the function are x = -1 and x = 2.

D. The y-intercept of the function is -8.

a. To describe the end behavior of the polynomial function f(x) = (x + 1)² (x - 2), we look at the highest degree term, which is (x + 1)² (x - 2). Since the degree of the polynomial is odd (degree 3), the end behavior will be as follows:

As x approaches negative infinity, f(x) approaches negative infinity.

As x approaches positive infinity, f(x) approaches positive infinity.

b. To find the number of turning points, we can look at the degree of the polynomial. Since the degree is 3, there can be at most 2 turning points.

c. To find the zeros (x-intercepts) of the function, we set f(x) equal to zero and solve for x:

(x + 1)² (x - 2) = 0

Setting each factor equal to zero, we have:

x + 1 = 0 or x - 2 = 0

Solving these equations, we find:

x = -1 or x = 2

Therefore, the zeros (x-intercepts) of the function are x = -1 and x = 2.

d. To find the y-intercept of the function, we substitute x = 0 into the function:

f(0) = (0 + 1)² (0 - 2)

f(0) = (1)² (-2)

f(0) = 4(-2)

f(0) = -8

Therefore, the y-intercept of the function is -8.

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2a) Monthly payment = $422.12 2b)Total amount paid = $25,327.20 2c)  Interest paid = $2,827.20 3) $2,871.71 4a) Monthly deposit = $875.15 4b)$656,287.50 5a) $173.25  5b)Account balance = $2273.25

In these problems, we will be using financial formulas to calculate monthly payments, total payments, interest paid, and account balances. The formulas used are as follows:

PMT: Monthly payment

PV: Present value (loan amount or initial deposit)

RATE: Interest rate per period

NPER: Total number of periods

Here are the steps to solve each problem:

Problem 2a:

Formula: PMT(RATE, NPER, PV)

Inputs: RATE = 4%/12, NPER = 5*12, PV = $22,500

Calculation: PMT(4%/12, 5*12, $22,500)

Answer: Monthly payment = $422.12 (rounded to the nearest cent)

Problem 2b:

Calculation: Monthly payment * NPER

Answer: Total amount paid = $422.12 * (5*12) = $25,327.20

Problem 2c:

Calculation: Total amount paid - PV

Answer: Interest paid = $25,327.20 - $22,500 = $2,827.20

Problem 3:

Formula: PMT(RATE, NPER, PV)

Inputs: RATE = 6%/12, NPER = 25*12, PV = $400,000

Calculation: PMT(6%/12, 25*12, $400,000)

Answer: Monthly withdrawal = $2,871.71

Problem 4a:

Formula: PMT(RATE, NPER, PV)

Inputs: RATE = 9%/12, NPER = 25*12, PV = 0 (assuming starting from $0)

Calculation: PMT(9%/12, 25*12, 0)

Answer: Monthly deposit = $875.15

Problem 4b:

Calculation: Monthly deposit * NPER - PV

Answer: Interest earned = ($875.15 * (25*12)) - $0 = $656,287.50

Problem 5a:

Formula: I = PRT

Inputs: P = $2100, R = 5.5%, T = 18/12 (convert months to years)

Calculation: I = $2100 * 5.5% * (18/12)

Answer: Interest earned = $173.25

Problem 5b:

Calculation: P + I

Answer: Account balance = $2100 + $173.25 = $2273.25

By following these steps and using the appropriate formulas, you can solve each problem and obtain the requested results.

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Answer:

y=-4/9x^2-20/9x+56/9

Step-by-step explanation:

Using the recurrence relation, we can calculate f(1), f(2), f(3), and f(4).

a. f(0) = 1, f(4) = 37 b. f(0) = 9, f(4) = 45

c. f(0) = 13, f(4) = 49 d. f(0) = 159, f(4) = 195

To find the value of f(4) for each initial condition, we can use the given recurrence relation f(n+1) = f(n) + 9 iteratively.

a. If f(0) = 1, we can compute f(1) = f(0) + 9 = 1 + 9 = 10, f(2) = f(1) + 9 = 10 + 9 = 19, f(3) = f(2) + 9 = 19 + 9 = 28, and finally f(4) = f(3) + 9 = 28 + 9 = 37.

Therefore, when f(0) = 1, we have f(4) = 37.

b. If f(0) = 9, we can similarly compute f(1) = f(0) + 9 = 9 + 9 = 18, f(2) = f(1) + 9 = 18 + 9 = 27, f(3) = f(2) + 9 = 27 + 9 = 36, and finally f(4) = f(3) + 9 = 36 + 9 = 45.

Therefore, when f(0) = 9, we have f(4) = 45.

c. If f(0) = 13, we proceed as before to find f(1) = f(0) + 9 = 13 + 9 = 22, f(2) = f(1) + 9 = 22 + 9 = 31, f(3) = f(2) + 9 = 31 + 9 = 40, and finally f(4) = f(3) + 9 = 40 + 9 = 49.

Therefore, when f(0) = 13, we have f(4) = 49.

d. If f(0) = 159, we can compute f(1) = f(0) + 9 = 159 + 9 = 168, f(2) = f(1) + 9 = 168 + 9 = 177, f(3) = f(2) + 9 = 177 + 9 = 186, and finally f(4) = f(3) + 9 = 186 + 9 = 195.

Therefore, when f(0) = 159, we have f(4) = 195.

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Therefore, the value of the 100th term is 405 (option a).

To find the formula for an arithmetic sequence, we can use the formula:

[tex]a_n = a_1 + (n - 1)d,[/tex]

where:

an represents the nth term of the sequence,

a1 represents the first term of the sequence,

n represents the position of the term in the sequence,

d represents the common difference between consecutive terms.

Given that the 4th term is 21 and the 9th term is 41, we can set up the following equations:

[tex]a_4 = a_1 + (4 - 1)d[/tex]

= 21,

[tex]a_9 = a_1 + (9 - 1)d[/tex]

= 41.

Simplifying the equations, we have:

[tex]a_1 + 3d = 21[/tex], (equation 1)

[tex]a_1 + 8d = 41.[/tex] (equation 2)

Subtracting equation 1 from equation 2, we get:

[tex]a_1 + 8d - (a)1 + 3d) = 41 - 21,[/tex]

5d = 20,

d = 4.

Substituting the value of d back into equation 1, we can solve for a1:

[tex]a_1 + 3(4) = 21,\\a_1 + 12 = 21,\\a_1 = 21 - 12,\\a_1 = 9.\\[/tex]

Therefore, the formula for the arithmetic sequence is:

[tex]a_n = 9 + 4(n - 1).[/tex]

To determine the value of the 100th term (a100), we substitute n = 100 into the formula:

[tex]a_{100} = 9 + 4(100 - 1),\\a_{100} = 9 + 4(99),\\a_{100 }= 9 + 396,\\a_{100} = 405.[/tex]

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The resulting equation written in standard form is 16x² - 22x + 6 = 0.

Given that, y = 12 + 2x is the linear equation and y = -16x² + 24x + 6 is a quadratic equation.

The standard form of the quadratic equation is ax² + bx + c = 0, where 'a' is the leading coefficient and it is a non-zero real number.

Now,

[tex]\sf y=-16x^2+24x+6[/tex]

Substitute,

[tex]\sf y=12+2x[/tex] in [tex]\sf y=-16x^2+24x+6[/tex].

[tex]\sf 12+2x=-16x^2+24x+6[/tex]

[tex]\rightarrow \sf 16x^2-24x-6+12+2x=0[/tex]

[tex]\rightarrow\bold{16x^2 - 22x + 6 = 0}[/tex]

Therefore, the resulting equation written in standard form is 16x² - 22x + 6 = 0.

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Kevin and Randy Muise have 53 quarters and 23 nickels.

We can solve this problem by using the following steps:

Let x be the number of quarters and y be the number of nickels.

We know that x + y = 76 and .25x + .05y = 13.40

Solve for x and y using simultaneous equations.

x = 53 and y = 23

Here is a more detailed explanation of each step:

Let x be the number of quarters and y be the number of nickels. This is a good first step because it allows us to represent the unknown values with variables.

We know that x + y = 76 and .25x + .05y = 13.40. This is because we are given that there are a total of 76 coins in the jar, and that each quarter is worth 25 cents and each nickel is worth 5 cents.

We can solve for x and y using simultaneous equations. To do this, we can either use the elimination method or the substitution method. In this case, we will use the elimination method.

To use the elimination method, we need to get one of the variables to cancel out. We can do this by multiplying the first equation by -.25 and the second equation by 20. This gives us:

-.25x - .25y = -19

4x + 10y = 268

Adding these two equations together, we get:

3.75y = 249

y = 66

Now that we know y, we can substitute it into the first equation to solve for x. This gives us:

x + 66 = 76

x = 10

Therefore, Kevin and Randy Muise have 53 quarters and 23 nickels.

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For the function, y = -1 + cos(t) with 0 ≤ t ≤ 27 and 2 ≤ b ≤ 6, we can calculate its graph as follows:We have the following restrictions to apply to t and b:0 ≤ t ≤ 27 (restrictions on t)2 ≤ b ≤ 6 (restrictions on .

b)Now, let us calculate the derivative of the function, with respect to t:dy/dt = -sin(t)Let us set the derivative equal to zero, to find the stationary points of the function:dy/dt

= 0

=> -sin(t)

= 0

=> t

= 0, πNow, we calculate the second derivative: d²y/dt²

= -cos(t)At t

= 0, we have d²y/dt²

= -cos(0)

= -1 < 0, so the function has a local maximum at t =

.At t

= π, we have d²y/dt²

= -cos(π)

= 1 > 0, so the function has a local minimum at t

= π.

Therefore, the lowest point of the graph occurs when t = π.The value of b determines the amplitude of the function. As b increases, the amplitude increases. This means that the peaks and valleys of the graph become more extreme, while the midline (y

= -1) remains the same.Here is the graph of the function for b

= 2 (red), b

= 4 (blue), and b

= 6 (green):As you can see, as b increases, the amplitude of the graph increases, making the peaks and valleys more extreme.

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Given, Jerome wants to invest $20,000 as part of his retirement plan.

He can invest the money at 5.1% simple interest for 32 yr, or he can invest at 3.7% interest compounded continuously for 32yr. We have to determine which investment plan results in more total interest.

Let us solve the problem.

To determine which investment plan will result in more total interest, we can use the following formulas for simple interest and continuously compounded interest.

Simple Interest formula:

I = P * r * t

Continuous Compound Interest formula:

I = Pe^(rt) - P,

where e = 2.71828

Given,P = $20,000t = 32 yr

For the first investment plan, r = 5.1%

Simple Interest formula:

I = P * r * tI = $20,000 * 0.051 * 32I = $32,640

Total interest for the first investment plan is $32,640.

For the second investment plan, r = 3.7%

Continuous Compound Interest formula:

I = Pe^(rt) - PI = $20,000(e^(0.037*32)) - $20,000I = $20,000(2.71828)^(1.184) - $20,000I = $48,124.81 - $20,000I = $28,124.81

Total interest for the second investment plan is $28,124.81.

Therefore, 5.1% simple interest investment plan results in more total interest.

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the compound comparison "2 > 3 or 3 < 4" is True.

The compound comparison "2 > 3 or 3 < 4" evaluates to True. The comparison "2 > 3" is False because 2 is not greater than 3. However, the comparison "3 < 4" is True because 3 is less than 4. In a compound comparison with the "or" operator, if at least one of the individual comparisons is True, then the whole compound comparison is considered True.

In this case, the second comparison "3 < 4" is True, which means that the compound comparison "2 > 3 or 3 < 4" is also True.

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a) The account will contain approximately $7345.56 after 7 years.

b) There will be approximately 76,800 bacteria in the colony after 7 hours.

a) i) The initial amount in the account is $5000, so a_0 = $5000. The amount in the account at the end of n years can be expressed as a_n = (1.07)a_{n-1}, since the interest compounds annually and increases the account balance by 7% each year.

ii) To find the amount in the account after 7 years, we substitute n = 7 into the recurrence relation:

a_7 = (1.07)a_{6}

   = (1.07)((1.07)a_{5})

   = (1.07)((1.07)((1.07)a_{4}))

   = [tex](1.07)^7a_{0}[/tex]

   = [tex](1.07)^7($5000)[/tex]

   ≈ $7345.56

Therefore, the account will contain approximately $7345.56 after 7 years.

b) i) The recurrence relation for the number of bacteria after n hours is a_n = 2a_{n-1}, as the colony doubles in size every hour.

ii) If 150 bacteria are used to begin a new colony, we substitute n = 7 into the recurrence relation:

a_7 = 2a_{6}

   = 2(2a_{5})

   = 2(2(2a_{4}))

   = [tex]2^7a_{0}[/tex]

   = 2[tex]^7(150)[/tex]

   = [tex]2^8(75)[/tex]

   =[tex]2^9(37.5)[/tex]

   ≈ 76,800

Therefore, there will be approximately 76,800 bacteria in the colony after 7 hours.

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The claimed inductive hypothesis is: For every k ≥ 4, if 2^k ≥ k², then 2^(k+1) ≥ (k+1)².

Let's discuss the given proof and find out what should be claimed in the inductive hypothesis:We are given that For all n ≥ 4, 2^n ≥ n². We need to show that 2^(k+1) ≥ (k+1)² if 2^k ≥ k² holds for k ≥ 4. It is assumed that 2^k ≥ k² is true for k = n.Now, we need to show that 2^(k+1) ≥ (k+1)² is also true. We will use the given hypothesis to prove it as follows:2^(k+1) = 2^k * 2 ≥ k² * 2 (since 2^k ≥ k² by hypothesis)Now, we need to show that k² * 2 ≥ (k+1)² i.e. k² * 2 ≥ k² + 2k + 1 (expand the right-hand side)This simplifies to 2k ≥ 1 or k ≥ 1/2. We know that k ≥ 4 by hypothesis, so this is certainly true. Hence, 2^(k+1) ≥ (k+1)² holds for k ≥ 4. Thus, the claimed inductive hypothesis is: For every k ≥ 4, if 2^k ≥ k², then 2^(k+1) ≥ (k+1)².

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(i) The number of pupils who play Football only is 270.

(ii) The total number of pupils who play either Football or Hockey is 420

To solve this problem, we can use the principle of inclusion-exclusion.

Let's define the following:

F = Number of pupils who play Football

H = Number of pupils who play Hockey

Given information:

F = 400 (Number of pupils who play Football)

H = 150 (Number of pupils who play Hockey)

Number of pupils who play both Football and Hockey = 130

(i) Number of pupils who play Football only:

This can be calculated by subtracting the number of pupils who play both Football and Hockey from the total number of pupils who play Football:

Number of pupils who play Football only = F - (Number of pupils who play both Football and Hockey) = 400 - 130 = 270.

(ii) Total number of pupils who play either Football or Hockey:

To find this, we need to add the number of pupils who play Football and the number of pupils who play Hockey and then subtract the number of pupils who play both Football and Hockey to avoid double counting:

Total number of pupils who play either Football or Hockey = F + H - (Number of pupils who play both Football and Hockey) = 400 + 150 - 130 = 420.

So, the answers to the questions are:

(i) The number of pupils who play Football only is 270.

(ii) The total number of pupils who play either Football or Hockey is 420.

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The probabilities are: 3. P(z > 1.34) = 0.0901 (option A), P(z > 1.79) = 0.0367 (option A). Let's determine:

To determine the probabilities P(z > 1.34) and P(z > 1.79), where z is a standard normal random variable, we can follow these steps:

P(z > 1.34) refers to the probability of obtaining a z-value greater than 1.34 under the standard normal distribution.

Look up the z-table or use a statistical software to find the corresponding area under the standard normal curve for the given z-values.

In the z-table, find the row corresponding to the first decimal place of the z-value. In this case, it is 1.3 for 1.34 and 1.7 for 1.79.

Locate the column corresponding to the second decimal place of the z-value. In this case, it is 0.04 for 1.34 and 0.09 for 1.79.

The intersection of the row and column in the z-table gives the area to the left of the z-value. Subtracting this value from 1 will give the area to the right, which is the desired probability.

For P(z > 1.34), we find the value 0.0901, corresponding to option A.

For P(z > 1.79), we find the value 0.0367, corresponding to option A.

Therefore, the probabilities are:

3. P(z > 1.34) = 0.0901 (option A)

P(z > 1.79) = 0.0367 (option A)

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The equation of the line passing through P(−6,−5) is 7y + x + 42 = 0 in standard form. The equation of the line passing through P(4, 2) is -y + 2 = 0 in standard form. The equation of the line passing through P(−8,−5) is x + 8 = 0 in standard form.

1. To write the equation of a line in standard form (Ax + By = C), we need to determine the values of A, B, and C. We are given the slope (m = -1/7) and a point on the line (P(-6, -5)).

Using the point-slope form of a linear equation, we have y - y1 = m(x - x1), where (x1, y1) is the given point. Plugging in the values, we get y - (-5) = (-1/7)(x - (-6)), which simplifies to y + 5 = (-1/7)(x + 6).

To convert this equation to standard form, we multiply both sides by 7 to eliminate the fraction and rearrange the terms to get 7y + x + 42 = 0. Thus, the equation of the line is 7y + x + 42 = 0 in standard form.

2. Since the slope (m) is given as 0, the line is horizontal. A horizontal line has the same y-coordinate for every point on the line. Since the line passes through P(4, 2), the equation of the line will be y = 2.

To convert this equation to standard form, we rearrange the terms to get -y + 2 = 0. Multiplying through by -1, we have y - 2 = 0. Therefore, the equation of the line is -y + 2 = 0 in standard form.

3. When the slope (m) is undefined, it means the line is vertical. A vertical line has the same x-coordinate for every point on the line. Since the line passes through P(-8, -5), the equation of the line will be x = -8.

In standard form, the equation becomes x + 8 = 0. Therefore, the equation of the line is x + 8 = 0 in standard form.

In conclusion, we have determined the equations of lines with different slopes and passing through given points. By understanding the slope and the given point, we can use the appropriate forms of equations to represent lines accurately in standard form.

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(a) The values of x that satisfy y = -√2/2 in the interval [0, 4π] are: x = π/4, 3π/4, 5π/4, 7π/4, 9π/4, 11π/4, 13π/4, 15π/4.

(b) All x-values except those listed in part (a) satisfy y > -√2/2 in the interval [0, 4π].

(c) All x-values except those listed in part (a) satisfy y < -√2/2 in the interval [0, 4π].

To find the values of x that satisfy the given conditions, we need to examine the graph of the equation y = sin(x) on the interval [0, 4π].

(a) For y = -√2/2:

Looking at the unit circle or the graph of the sine function, we can see that y = -√2/2 corresponds to two points in each period: -π/4 and -3π/4.

In the interval [0, 4π], we have four periods of the sine function, so we need to consider the following values of x:

x₁ = π/4, x₂ = 3π/4, x₃ = 5π/4, x₄ = 7π/4, x₅ = 9π/4, x₆ = 11π/4, x₇ = 13π/4, x₈ = 15π/4.

Therefore, the values of x that satisfy y = -√2/2 in the interval [0, 4π] are:

x = π/4, 3π/4, 5π/4, 7π/4, 9π/4, 11π/4, 13π/4, 15π/4.

(b) For y > -√2/2:

Since -√2/2 is the minimum value of the sine function, any value of x that produces a y-value greater than -√2/2 will satisfy the condition.

In the interval [0, 4π], all x-values except those listed in part (a) will satisfy y > -√2/2.

(c) For y < -√2:

Again, since -√2/2 is the minimum value of the sine function, any value of x that produces a y-value less than -√2/2 will satisfy the condition.

In the interval [0, 4π], all x-values except those listed in part (a) will satisfy y < -√2/2.

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The probability that the sample mean is between 17.5 and 18.5 is approximately 0.369, rounded to three decimal places.

To find the probability that the sample mean is between 17.5 and 18.5 for a random sample of 16 smartphone owners, we need to use the properties of the normal distribution.

The population mean is 18 and the population standard deviation is 6, we can calculate the standard error of the mean using the formula: standard deviation / sqrt(sample size). In this case, the standard error is 6 / sqrt(16) = 6 / 4 = 1.5.

Next, we need to calculate the z-scores for the lower and upper limits of the sample mean range. The z-score formula is given by: (sample mean - population mean) / standard error.

For the lower limit (17.5), the z-score is (17.5 - 18) / 1.5 = -0.333.

For the upper limit (18.5), the z-score is (18.5 - 18) / 1.5 = 0.333.

Using a standard normal distribution table or a calculator, we can find the probability corresponding to these z-scores. The probability that the sample mean is between 17.5 and 18.5 is approximately 0.369, rounded to three decimal places.

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According to a report from a business intelligence company, smartphone owners are using an average of 18 apps per month. Assume that number of apps used per month by smartphone owners is normally distributed and that the standard deviation is 6. Complete parts (a) through (d) below. a. If you select a random sample of 16 smartphone owners, what is the probability that the sample mean is between 17.5 and 18.5? (Round to three decimal places as needed.)

The remainder when dividing [tex]\(x^9 - 2\)[/tex] by [tex](x - \sqrt[3]{3})[/tex] is 25. (Option d)

To find the remainder when dividing [tex]\(x^9 - 2\)[/tex] by [tex](x - \sqrt[3]{3})[/tex], we can use the Remainder Theorem. According to the theorem, if we substitute [tex]\(\sqrt[3]{3}\)[/tex] into the polynomial, the result will be the remainder.

Let's substitute [tex]\(\sqrt[3]{3}\)[/tex] into [tex]\(x^9 - 2\)[/tex]:

[tex]\(\left(\sqrt[3]{3}\right)^9 - 2\)[/tex]

Simplifying this expression, we get:

[tex]\(3^3 - 2\)\\\(27 - 2\)\\\(25\)[/tex]

Therefore, the remainder when dividing [tex]\(x^9 - 2\) by \((x - \sqrt[3]{3})\)[/tex] is 25. Hence, the correct option is (d) 25.

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Explanation:We are given that the two airplanes leave an airport at the same time, with an angle between them of 135 degrees and that one airplane travels at 421 mph and the other travels at 335 mph.

We are also asked to find how far apart the planes are after 3 hours

First, we need to find the distance each plane has traveled after 3 hours.Using the formula d = rt, we can find the distance traveled by each plane. Let's assume that the first plane (traveling at 421 mph) is represented by vector AB, and the second plane (traveling at 335 mph) is represented by vector AC. Let's call the angle between the two vectors angle BAC.So, the distance traveled by the first plane in 3 hours is dAB = 421 × 3 = 1263 milesThe distance traveled by the second plane in 3 hours is dAC = 335 × 3 = 1005 miles.

Now, to find the distance between the two planes after 3 hours, we need to use the Law of Cosines. According to the Law of Cosines, c² = a² + b² - 2ab cos(C), where a, b, and c are the lengths of the sides of a triangle, and C is the angle opposite side c. In this case, we have a triangle ABC, where AB = 1263 miles, AC = 1005 miles, and angle BAC = 135 degrees.

We want to find the length of side BC, which represents the distance between the two planes.Using the Law of Cosines, we have:BC² = AB² + AC² - 2(AB)(AC)cos(BAC)BC² = (1263)² + (1005)² - 2(1263)(1005)cos(135)BC² = 1598766BC = √(1598766)BC ≈ 1263.39Therefore, the planes are approximately 1263.39 miles apart after 3 hours. This is the final answer.

We used the Law of Cosines to find the distance between the two planes after 3 hours. We found that the planes are approximately 1263.39 miles apart after 3 hours.

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The Kretovich Company's annual sales were $7,250,000.

To find out the annual sales of the Kretovich Company, given quick ratio, current ratio, days sales outstanding, total current assets, and cash and marketable securities, the following formula is used:

Annual sales = (Total current assets - Cash and marketable securities) / (Days sales outstanding / 365)

Quick ratio = (Cash + Marketable securities + Receivables) / Current liabilities

And, Current ratio = Current assets / Current liabilities

To solve the above question, we will first find out the total current liabilities and total current assets.

Let the total current liabilities be CL

So, quick ratio = (Cash + Marketable securities + Receivables) / CL1.4 = (115,000 + R) / CL

Equation 1: R + 115,000 = 1.4CLWe also know that, Current ratio = Current assets / Current liabilities

So, 3 = Total current assets / CL

So, Total current assets = 3CL

We have been given that, Total current assets = $840,000

We can find the value of total current liabilities by using the above two equations.

3CL = 840,000CL = $280,000

Putting the value of CL in equation 1, we get,

R + 115,000 = 1.4($280,000)R = $307,000

We can now use the formula to find annual sales.

Annual sales = (Total current assets - Cash and marketable securities) / (Days sales outstanding / 365)= ($840,000 - $115,000) / (36.5/365)= $725,000 / 0.1= $7,250,000

Therefore, the Kretovich Company's annual sales were $7,250,000.

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The values of a and b that make the function f(x) continous on all real numbers are: a can be any value, b = 0, and

h = 3/2.

To make the function f(x) continuous on all real numbers, we need to ensure that the three pieces of the function connect smoothly at the transition points (-2 and 3). For continuity, the function values on either side of these transition points should be equal.

Let's start with the transition point x = -2:

1. Evaluate f(x) for x < -2:

  f(x) = x² + bx - 3

2. Evaluate f(x) for x > -2:

  f(x) = az - 2h

Since f(x) needs to be continuous at x = -2, the function values on both sides of -2 should be equal:

x² + bx - 3 = az - 2h

Next, let's consider the transition point x = 3:

1. Evaluate f(x) for x < 3:

  f(x) = x² + bx - 3

2. Evaluate f(x) for x > 3:

  f(x) = 7 - 2bx

Since f(x) needs to be continuous at x = 3, the function values on both sides of 3 should be equal:

x² + bx - 3 = 7 - 2bx

Now we have two equations:

1. x² + bx - 3 = az - 2h   ----(1)

2. x² + bx - 3 = 7 - 2bx   ----(2)

To find the values of a and b, we can solve these equations simultaneously.

From equation (1), we have:

az - 2h = x² + bx - 3   ----(3)

Rearranging equation (3) gives:

az = x² + bx - 3 + 2h   ----(4)

From equation (2), we have:

7 - 2bx = x²+ bx - 3   ----(5)

Rearranging equation (5) gives:

x² + 3bx - 10 = 0   ----(6)

Now we have two equations to solve: (4) and (6).

To ensure the continuity of the function, the discriminant of equation (6) should be non-negative:

Discriminant (D) = (3b)² - 4(1)(-10)

               = 9b² + 40 ≥ 0

Solving this inequality, we find:

9b² + 40 ≥ 0

9b² ≥ -40

b² ≥ -40/9

b² ≥ 40/9

Since b² is non-negative, we can conclude that 40/9 ≥ 0. Therefore, any value of b can satisfy the inequality.

As for a, we can substitute the value of b into equation (4) and solve for a:

az = x² + bx - 3 + 2h

az = x² + (b/2)x + (b/2)x - 3 + 2h

az = x(x + (b/2)) + (b/2)(x - 3) + 2h

For the expression to be valid for all real numbers, the coefficient of x and the constant term must be zero.

Coefficient of x: (b/2)(x - 3) = 0

Since b can be any value, (b/2) = 0

Thus, b = 0

Constant term: 2h - 3 = 0

2h = 3

h = 3/2

Now we have found the

values of a and b that make the function f(x) continuous:

a can be any value,

b = 0, and

h = 3/2.

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Over the course of 30 years, Susie will pay approximately $180,906.00 in interest on her $150,000 mortgage.

To calculate the total interest paid over the 30-year mortgage, we first need to determine the total amount paid. Susie pays $501.41 every month for 30 years, which is a total of 12 * 30 = 360 payments.

The total amount paid is then calculated by multiplying the monthly payment by the number of payments: $501.41 * 360 = $180,516.60.

To find the interest paid, we subtract the original loan amount from the total amount paid: $180,516.60 - $150,000 = $30,516.60.

Therefore, over the 30 years of payments, Susie will pay approximately $30,516.60 in interest on her $150,000 mortgage. Rounding this to the nearest cent gives us $30,516.00.

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It will cost $356.4 to carpet a room that measures 3.6 feet by 11 feet, if the carpet costs $9 per square foot.

To find out the total cost to carpet a room that measures 3.6 feet by 11 feet, if the carpet costs $9 per square foot, Calculate the area of the room

Area of the room = Length × Width

Area of the room = 3.6 ft × 11 ft

Area of the room = 39.6 square feet

Find the cost of carpeting the room

Now, we know that the cost of the carpet is $9 per square foot.

Total cost = Area of the room × Cost per square foot

Total cost = 39.6 square feet × $9Total cost = $356.4

So, it will cost $356.4 to carpet a room that measures 3.6 feet by 11 feet, if the carpet costs $9 per square foot.

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a. The coordinates of the centre of the circle are (0,0).

b. The radius is 8.

A circle is defined by the equation x² + y² = 64.

We are to find the coordinates of the centre and the radius.

Given equation of the circle is x² + y² = 64

We know that the equation of a circle is given by

(x - h)² + (y - k)² = r²,

where (h, k) are the coordinates of the centre and r is the radius of the circle.

Comparing this with x² + y² = 64,

we get:

(x - 0)² + (y - 0)² = 8²

Therefore, the centre of the circle is at the point (0, 0).

Using the formula, r² = 8² = 64,

we get the radius, r = 8.

Therefore, a. The coordinates of the centre are (0,0). b. The radius is 8.

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(a) The profit function is given by P(x) = R(x) - C(x), where R(x) is the revenue function.
(b) To maximize profit, the company should manufacture the number of infant car seats that corresponds to the maximum point of the profit function. The maximum monthly profit can be determined by evaluating the profit function at this point. The price for each infant car seat can be found by substituting the optimal production level into the price-demand equation.

(a) The profit function, P(x), is calculated by subtracting the cost function, C(x), from the revenue function, R(x). The revenue function is determined by multiplying the price, p, by the quantity sold, x. In this case, the price-demand equation x = 9000 - 30p gives us the quantity sold as a function of the price. So, the revenue function is R(x) = p * x. Substituting the given price-demand equation into the revenue function, we have R(x) = p * (9000 - 30p). Therefore, the profit function is P(x) = R(x) - C(x) = p * (9000 - 30p) - (150000 + 30x).
(b) To maximize profit, we need to find the production level that corresponds to the maximum point on the profit function. This can be done by finding the critical points of the profit function (where its derivative is zero or undefined) and evaluating them within the feasible range. Once the optimal production level is determined, we can calculate the maximum monthly profit by substituting it into the profit function. The price for each infant car seat can be obtained by substituting the optimal production level into the price-demand equation x = 9000 - 30p and solving for p.

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In summary, the Fourier coefficients for the periodic function y(t) = -3 on the interval -5 ≤ t ≤ 5 are:

c₀ = -3 (DC component)

cₙ = 0 for n ≠ 0 (other coefficients)

To find the Fourier coefficients of the periodic function y(t) = -3 on the interval -5 ≤ t ≤ 5, we can use the formula for Fourier series coefficients:

cn = (1/T) ∫[t₀-T/2, t₀+T/2] y(t) [tex]e^{(-i2\pi nt/T)}[/tex] dt

where T is the period of the function and n is an integer.

In this case, the function y(t) is constant, y(t) = -3, and the period is T = 10 (since the interval -5 ≤ t ≤ 5 spans 10 units).

To find the Fourier coefficient c₀ (corresponding to the DC component or the average value of the function), we use the formula:

c₀ = (1/T) ∫[-T/2, T/2] y(t) dt

Substituting the given values:

c₀ = (1/10) ∫[-5, 5] (-3) dt

  = (-3/10) [tex][t]_{-5}^{5}[/tex]

  = (-3/10) [5 - (-5)]

  = (-3/10) [10]

  = -3

Therefore, the DC component (c₀) of the Fourier series of y(t) is -3.

For the other coefficients (cₙ where n ≠ 0), we can calculate them using the formula:

cₙ = (1/T) ∫[-T/2, T/2] y(t)[tex]e^{(-i2\pi nt/T) }[/tex]dt

Since y(t) is constant, the integral becomes:

cₙ = (1/T) ∫[-T/2, T/2] (-3) [tex]e^{(-i2\pi nt/T)}[/tex] dt

  = (-3/T) ∫[-T/2, T/2] [tex]e^{(-i2\pi nt/T)}[/tex] dt

The integral of e^(-i2πnt/T) over the interval [-T/2, T/2] evaluates to 0 when n ≠ 0. This is because the exponential function oscillates and integrates to zero over a symmetric interval.

all the coefficients cₙ for n ≠ 0 are zero.

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The particular solution that satisfies the given initial conditions is y(x) = y(x) = y(x) = e^2x + 6e^(-3x).

To verify that y1 = e^2x and y2 = e^(-3x) are solutions to the differential equation y'' + y' - 6y = 0, we substitute them into the equation:

For y1:

y'' + y' - 6y = (e^2x)'' + (e^2x)' - 6(e^2x) = 4e^2x + 2e^2x - 6e^2x = 0

For y2:

y'' + y' - 6y = (e^(-3x))'' + (e^(-3x))' - 6(e^(-3x)) = 9e^(-3x) - 3e^(-3x) - 6e^(-3x) = 0

Both y1 and y2 satisfy the differential equation.

To find a particular solution that satisfies the initial conditions y(0) = 7 and y'(0) = -1, we express y(x) as y(x) = c1y1 + c2y2, where c1 and c2 are constants. Substituting the initial conditions into this expression, we have:

y(0) = c1e^2(0) + c2e^(-3(0)) = c1 + c2 = 7

y'(0) = c1(2e^2(0)) - 3c2(e^(-3(0))) = 2c1 - 3c2 = -1

Solving this system of equations, we find c1 = 1 and c2 = 6. Therefore, the particular solution that satisfies the given initial conditions is y(x) = y(x) = y(x) = e^2x + 6e^(-3x).

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