use f(x, y, z) = x2 yz, f(x, y, z) = xy, yz, xz , and g(x, y, z) = −sin(z), exz, y . compute (f ✕ g)(5, −1, 8). (your instructors prefer angle bracket notation < > for vectors.)

Answer:

0

Step-by-step explanation:

ny^2 + 2y - 4 = 0

ny^2 + 2y = 4

y(ny + 2) = 4

y = 4

ny + 2 = 4

ny = 2, 0 = 2

The only possible solution to make this expression incorrect is if 0 = 2, so n is equal to 0.

To compute the average velocity over each time interval, we use the formula: average velocity = (h(t2) - h(t1))/(t2 - t1), where h(t) is the height function.

Using the given height function, h(t) = 34t - 4.9t^2, we calculate the average velocities:
1. [3, 3.01]:
Average Velocity = (h(3.01) - h(3))/(3.01 - 3) ≈ -17.147 m/s
2. [3, 3.001]:
Average Velocity = (h(3.001) - h(3))/(3.001 - 3) ≈ -17.194 m/s
3. [3, 3.0001]:
Average Velocity = (h(3.0001) - h(3))/(3.0001 - 3) ≈ -17.199 m/s
4. [2.99, 3]:
Average Velocity = (h(3) - h(2.99))/(3 - 2.99) ≈ -17.243 m/s
5. [2.999, 3]:
Average Velocity = (h(3) - h(2.999))/(3 - 2.999) ≈ -17.205 m/s
6. [2.9999, 3]:
Average Velocity = (h(3) - h(2.9999))/(3 - 2.9999) ≈ -17.200 m/s
To estimate the instantaneous velocity at t=3, observe the average velocities as the time intervals approach t=3:
As the intervals get closer to t=3, the average velocities appear to approach -17.2 m/s. Thus, the estimated instantaneous velocity at t=3 is:
V ≈ -17.2 m/s

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Answer:

1. 2(x+3.60) = 19.40

Divide both sides by 2:

x+3.60 = 9.70

Subtract 3.60 from both sides:

x = 6.10

Answer: 6.10

2. 45.93 + 112 + (−61.24)

45.93 + 112 = 157.93

157.93 - 61.24 = 96.69

Answer: 96.69

3. 20x + 2 > −98

Subtract 2 from both sides:

20x > −100

Divide both sides by 20:

x > −5

Answer: x > −5

4. 2/5 (4x - 8)

= 8x/5 - 16/5

Answer: 8x/5 - 16/5

5. On a school field trip, the number of students (y) is always proportional to the number of adults (x). In one group there are 96 students and 8 adults. What is the constant of proportionality between this relationship?

The constant of proportionality is the number that, when multiplied by the number of adults, gives the number of students. In this case, the constant of proportionality is 96/8 = 12.

Answer: 12

Answer: The answer is B, y= 3x+5

Answer:

108 ft ^2

Step-by-step explanation:

12^2 - 6^2 = 108

The composite figure has an area of 24 square units.

In this question we find the representation of a composite figure formed by the combination of four figures, a triangle and three rectangles, whose area formulas are listed below:

Rectangle

A = b · h

Triangle

A = 0.5 · b · h

Where:

Now we proceed to determine the area of the composite figure:

A = 2 · 3 + 0.5 · 2 · 1 + 7 · 2 + 1 · 3

A = 6 + 1 + 14 + 3

A = 24

The area of the composite figure is equal to 24 square units.  

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At the given rate, the athlete could run 10.584 miles in 1.5 hours.

To determine how many miles the athlete could run in 1.5 hours at the given rate, follow these steps:

Step 1: Calculate the athlete's speed in miles per minute.

The athlete can run 6 miles in 51 minutes, so their speed is:

Speed = Distance ÷ Time = 6 miles ÷ 51 minutes ≈ 0.1176 miles per minute.

Step 2: Convert 1.5 hours to minutes.

1.5 hours = 1.5 × 60 = 90 minutes.

Step 3: Calculate the distance the athlete can run in 1.5 hours.

Distance = Speed × Time = 0.1176 miles per minute × 90 minutes ≈ 10.584 miles.

Therefore, at the given rate, the athlete could run approximately 10.584 miles in 1.5 hours.

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The mean (expected value) of the random variable x associated with the probability density function f(x) = (1/72)x^2 over the interval [0, 6] is 4.5.

To find the mean (expected value) of the random variable x associated with the probability density function f(x) = 1/72 x^2 over the interval [0, 6], we use the formula:

E(x) = ∫[0,6] x f(x) dx

= ∫[0,6] x (1/72 x^2) dx

= (1/72) ∫[0,6] x^3 dx

= (1/72) [(1/4) x^4] [0,6]

= (1/72) [(1/4) (6^4 - 0^4)]

= (1/72) (6^4/4)

= (1/72) (324)

= 4.5

To find the mean (expected value) of the random variable x associated with the probability density function f(x) = (1/72)x^2 over the interval [0, 6], we need to integrate the product of x and the probability density function over the given interval.

Mean (expected value) = E(x) = ∫(x * f(x)) dx, over the interval [0, 6]

E(x) = ∫(x * (1/72)x^2) dx from 0 to 6
E(x) = (1/72) * ∫(x^3) dx from 0 to 6

Now, integrate x^3 with respect to x:

E(x) = (1/72) * (x^4 / 4) | from 0 to 6

Now, evaluate the integral at the limits:

E(x) = (1/72) * ((6^4 / 4) - (0^4 / 4))
E(x) = (1/72) * (1296 / 4)
E(x) = (1/72) * 324

Finally, multiply the result:

E(x) = 4.5

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As the marginal cost and average cost are both equal to $8 at x = 5, we can conclude that the marginal cost and average cost are equal at this production level.

To find the production level at which the average cost is a minimum, we need to first find the average cost function. The average cost function is given by:

[tex]AC(x) = C(x)/x[/tex]

Substituting C(x) from the given equation, we get:

[tex]AC(x) = (x^3 - 10x^2 + 33x)/x[/tex]

Simplifying this, we get:

[tex]AC(x) = x^2 - 10x + 33[/tex]

To find the production level at which the average cost is a minimum, we need to find the value of x that minimizes the average cost function. We can do this by taking the derivative of the average cost function and setting it equal to zero:

[tex]d/dx (x^2 - 10x + 33) = 2x - 10 = 0[/tex]

Solving for x, we get:

x = 5

Therefore, the production level at which the average cost is a minimum is x = 5.

To show that the marginal cost and average cost are equal at this production level, we need to first find the marginal cost function. The marginal cost function is given by the derivative of the cost function:

[tex]MC(x) = d/dx (x^3 - 10x^2 + 33x) = 3x^2 - 20x + 33[/tex]

Substituting x = 5, we get:

[tex]MC(5) = 3(5)^2 - 20(5) + 33 = 8[/tex]

Therefore, the marginal cost at x = 5 is $8.

To find the average cost at x = 5, we can substitute x = 5 into the average cost function:

[tex]AC(5) = 5^2 - 10(5) + 33 = 8[/tex]

Therefore, the average cost at x = 5 is also $8.

Since the marginal cost and average cost are both equal to $8 at x = 5, we can conclude that the marginal cost and average cost are equal at this production level.

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The balance after 3 years on a Certificate of Deposit with an APR of 2.25% that compounds annually is $2,163.05.

Balance refers to the equality between two expressions or equations, where both sides have the same value. It is often used in solving equations or evaluating algebraic expressions.

A compound refers to a combination of two or more simple mathematical statements or propositions, connected by logical operators such as "and", "or", or "not". It is used in logic and boolean algebra.

To calculate the balance after 3 years on a Certificate of Deposit with an APR of 2.25% that compounds annually, we can use the formula:

A = P(1 + r/n)^{n*t}

Where:

A = the balance after t years,

P = the principal amount invested,

r = the annual interest rate as a decimal,

n = the number of times the interest is compounded per year,

t = the number of years

Plugging in the given values, we get:

P = $2,000r = 0.0225 (2.25% expressed as a decimal)

n = 1 (compounded annually)

t = 3 years,

[tex]A = 2,000(1 + 0.0225/1)^{1*3}[/tex]

[tex]A = 2,000(1 + 0.0225)^3[/tex]

[tex]A = 2,000(1.0225)^3[/tex]

A = $2,163.05 (rounded to the nearest cent)

Therefore, the balance after 3 years on a Certificate of Deposit with an APR of 2.25% that compounds annually is $2,163.05.

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The particle is at rest when the velocity is zero.

To find the values of t, you need to calculate the first derivatives of the parametric equations and set them equal to zero.

Main answer: The particle is at rest for t = 0 and t = 2.


1. Calculate the first derivatives of x(t) and y(t):
dx/dt = 3t² - 6t
dy/dt = 6t² - 6t - 12

2. Set the derivatives equal to zero and solve for t:
3t² - 6t = 0
6t² - 6t - 12 = 0

3. Factor the equations:
t(3t - 6) = 0
6(t² - t - 2) = 0

4. Solve for t:
t = 0, (3t - 6) = 0
t² - t - 2 = 0

5. From the first equation, t = 0 or t = 2.
From the second equation, use the quadratic formula:
t = (1 ± √(1 + 8))/2
t ≈ 1.41, -1.41

6. The particle is at rest for t = 0 and t = 2. The other values do not correspond to a stationary point.

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The numeric variable equivalent to equivalent to (x > 5) is, !(x < 5). The answer is D.

The original statement is "x > 5". The negation of this statement is "not (x > 5)", which is equivalent to "x <= 5". However, option A is the opposite of the correct answer since it says "x < 5", not "x <= 5". Option B says "not (x >= 5)", which is equivalent to "x < 5", but again, it is not the correct answer since it uses the "not greater than or equal to" symbol.

Option C says "not (x <= 5)", which is equivalent to "x > 5", but this is the opposite of the original statement. Therefore, the correct answer is D. !(x < 5), which is equivalent to "not (x is less than 5)", or "x is greater than or equal to 5". Hence, option D is correct.

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The length of the curve y = x^4 for 0≤ x ≤1 is approximately 1.082.

To find the length of the curve y = x^4 for 0≤ x ≤1, you'll need to use the arc length formula:

Arc length = ∫√(1 + (dy/dx)^2) dx from a to b, where a = 0 and b = 1.

First, find the derivative of y with respect to x:
y = x^4
dy/dx = 4x^3

Now, square the derivative and add 1:
(4x^3)^2 + 1 = 16x^6 + 1

Next, find the square root of the result:
√(16x^6 + 1)

Now, integrate the expression with respect to x from 0 to 1:
∫(√(16x^6 + 1)) dx from 0 to 1

Unfortunately, this integral doesn't have a closed-form solution, so we'll need to use numerical methods, such as Simpson's rule or a numerical integration calculator, to approximate the length.

Using a numerical integration calculator, the length of the curve y = x^4 for 0≤ x ≤1 is approximately 1.082.

Your answer: 1.082

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The length of the curve y = x^4 for 0≤ x ≤1 is approximately 1.082.

To find the length of the curve y = x^4 for 0≤ x ≤1, you'll need to use the arc length formula:

Arc length = ∫√(1 + (dy/dx)^2) dx from a to b, where a = 0 and b = 1.

First, find the derivative of y with respect to x:
y = x^4
dy/dx = 4x^3

Now, square the derivative and add 1:
(4x^3)^2 + 1 = 16x^6 + 1

Next, find the square root of the result:
√(16x^6 + 1)

Now, integrate the expression with respect to x from 0 to 1:
∫(√(16x^6 + 1)) dx from 0 to 1

Unfortunately, this integral doesn't have a closed-form solution, so we'll need to use numerical methods, such as Simpson's rule or a numerical integration calculator, to approximate the length.

Using a numerical integration calculator, the length of the curve y = x^4 for 0≤ x ≤1 is approximately 1.082.

Your answer: 1.082

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The solution of the initial-value problem of x'=[1/2 0; 1 -1/2] x is x(t) = [4/3 * e^(t/2); 5/3 * e^t + 8/3 * e^(t/2)].

To solve the given initial value problem x'=[1/2 0; 1 -1/2] x with x(0)=[4;9], we need to find the solution of the system of differential equations.

The characteristic equation of the matrix [1/2 0; 1 -1/2] is λ^2 - (3/2)λ + (1/4) = 0, which has two distinct roots, λ_1 = 1/2 and λ_2 = 1.

The general solution of the system is x(t) = c_1 * [1; 2] * e^(λ_1t) + c_2 * [0; 1] * e^(λ_2t), where c_1 and c_2 are constants to be determined using the initial condition x(0) = [4; 9].

Substituting the values of λ_1, λ_2, and x(0) in the above equation, we get c_1 = 4/3 and c_2 = 5/3.

Therefore, the solution of the initial-value problem is x(t) = [4/3 * e^(t/2); 5/3 * e^t + 8/3 * e^(t/2)].

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--The given question is incomplete, the complete question is given

" Solve the given initial-value problem x' is matrix of 2x2 form, x' = [1/2  0   1  −1/2] x,  x(0) = [4 9] of 2x1 matrix form. find x(t)"--

The factored form of polynomial P(x) is [tex]P(x) = 1(x + 2)(x^2 - x + 9).[/tex]

To factor [tex]P(x) = x^3 + 2x^2 + 9x + 18,[/tex]we need to first look for any common factors that we can factor out. In this case, we can factor out a 1, so:

[tex]P(x) = 1(x^3 + 2x^2 + 9x + 18)[/tex]

Next, we can try to find the roots of the polynomial by using the Rational Root Theorem, which states that if a polynomial has integer coefficients, then any rational root of the polynomial must have the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. In this case, the constant term is 18 and the leading coefficient is 1, so the possible rational roots are:

±1, ±2, ±3, ±6, ±9, ±18

We can try these roots by using synthetic division or long division to see if they are roots of the polynomial. After trying a few of these roots, we find that -2 is a root of the polynomial, so we can factor out (x + 2):

[tex]P(x) = 1(x^3 + 2x^2 + 9x + 18)\\     = 1(x + 2)(x^2 + ax + b)[/tex]

where a and b are coefficients that we need to find. To find a and b, we can use the fact that the coefficient of x^2 in the factored form should be equal to the coefficient of x^2 in the original polynomial. That is,

2 + 2a = 2

Solving for a, we get a = -1. Next, we can expand the factor (x^2 - x + b) and equate the coefficients of x and the constant term to the corresponding coefficients in the original polynomial. That is,

2a + b = 9
2b = 18

Solving for b, we get b = 9. Therefore, we have:

[tex]P(x) = 1(x + 2)(x^2 - x + 9)[/tex]

So the factored form of P(x) is [tex]P(x) = 1(x + 2)(x^2 - x + 9).[/tex]

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The factored form of polynomial P(x) is [tex]P(x) = 1(x + 2)(x^2 - x + 9).[/tex]

To factor [tex]P(x) = x^3 + 2x^2 + 9x + 18,[/tex]we need to first look for any common factors that we can factor out. In this case, we can factor out a 1, so:

[tex]P(x) = 1(x^3 + 2x^2 + 9x + 18)[/tex]

Next, we can try to find the roots of the polynomial by using the Rational Root Theorem, which states that if a polynomial has integer coefficients, then any rational root of the polynomial must have the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. In this case, the constant term is 18 and the leading coefficient is 1, so the possible rational roots are:

±1, ±2, ±3, ±6, ±9, ±18

We can try these roots by using synthetic division or long division to see if they are roots of the polynomial. After trying a few of these roots, we find that -2 is a root of the polynomial, so we can factor out (x + 2):

[tex]P(x) = 1(x^3 + 2x^2 + 9x + 18)\\     = 1(x + 2)(x^2 + ax + b)[/tex]

where a and b are coefficients that we need to find. To find a and b, we can use the fact that the coefficient of x^2 in the factored form should be equal to the coefficient of x^2 in the original polynomial. That is,

2 + 2a = 2

Solving for a, we get a = -1. Next, we can expand the factor (x^2 - x + b) and equate the coefficients of x and the constant term to the corresponding coefficients in the original polynomial. That is,

2a + b = 9
2b = 18

Solving for b, we get b = 9. Therefore, we have:

[tex]P(x) = 1(x + 2)(x^2 - x + 9)[/tex]

So the factored form of P(x) is [tex]P(x) = 1(x + 2)(x^2 - x + 9).[/tex]

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Required function g(h(x)) is 6 x - 9.

A function is a relationship between a set of outputs referred to as the range and a set of  inputs referred to as the domain, with the condition that each input is contain  to exactly one output. An input x corresponding to a function f output, which is represented by f(x).

We can combine two functions so that the outputs of one function become the inputs of the other if we have two functions is known as composite function . A composite function is defined by this action,that the function g f(x) = g(f(x)) is known as a composite function. This is occasionally referred to as a function of a function. g f can also be written as g o f instead.

We have, h(x)=3 x-5 and g(x)=2 x+1.

So, g(h(x)) = g(3 x - 5) = 2(3 x - 5) + 1 = 6 x - 9.

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Answer: 26 child tickets were sold that day.

Step-by-step explanation:

Let's say the number of child tickets sold is "x".

According to the problem, the number of adult tickets sold is four times the number of child tickets sold. So, the number of adult tickets sold would be 4x.

6.10x + 9.90(4x) = 1188.20

6.10x + 39.60x = 1188.20

45.70x = 1188.20

x = 26

By the rank-nullity theorem, we know that for any matrix A, the rank of A plus the dimension of the null space of A is equal to the number of columns of A. If the null space of a 7 times 9 matrix is 3-dimensional, Rank A = 6, Dim Row A = 6, Dim Col A = 6

we know that for any matrix A, the rank of A plus the dimension of the null space of A is equal to the number of columns of A. That is:

Rank A + Dim Null A = # of columns of A

In this case, we are given that the null space of the 7x9 matrix A is 3-dimensional. Therefore, we have:

Rank A + 3 = 9

Solving for Rank A, we get:

Rank A = 6

Now, we also know that the rank of a matrix is equal to the dimension of its row space and the dimension of its column space. That is:

Rank A = Dim Row A = Dim Col A

Therefore, we have:

Rank A = Dim Row A = Dim Col A = 6

So the correct option is: Rank A = 6, Dim Row A = 6, Dim Col A = 6

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Currently Jane is 24 years old. Let's start by using variables to represent the ages of Jane and Amy. Let j be Jane's current age and a be Amy's current age.

From the first sentence of the problem, we know that j = a + 8. Now, let's focus on the second sentence of the problem. It says that Amy is now twice as old as Jane was at one-third Jane's current age.

Let's break down this sentence into smaller pieces. "Jane was at one-third Jane's current age" means that Jane's age at that time was j/3. "Amy is now twice as old as Jane was at one-third Jane's current age" means that:

a = 2(j/3)

3/2 × a = j

Now we have two equations that relate the ages of Jane and Amy:

j = a + 8

3/2 × a = j

We can substitute the first equation into the second equation to get an equation that only has one variable:

1/2 × a = 8

a = 16

So Amy = 16 years old. We can use the first equation to find Jane's age:

j = a + 8

j = 16 + 8

j = 24

Currently Jane is 24 years old.

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The given differential equation y'' + 8y' + 16y = 0 represents a damped spring system with a damping coefficient of 8 and a spring constant of 16.

To describe the spring system represented by the differential equation y'' + 8y' + 16y = 0, we will be using the given terms.

1. Differential equation (DE): The given DE is a second-order linear homogeneous differential equation with constant coefficients. It represents the motion of a damped spring system, where y'' denotes the acceleration, y' denotes the velocity, and y denotes the displacement of the mass.

2. Damping: The term 8y' represents the damping in the spring system. It is proportional to the velocity (y') of the mass, and acts to oppose the motion, thus slowing down the oscillation.

3. Spring constant: The term 16y represents the restoring force exerted by the spring, which is proportional to the displacement (y) of the mass. The spring constant is 16.

4. Natural frequency: The natural frequency of the spring system can be found by considering the undamped case (i.e., without the 8y' term). In this case, the DE becomes y'' + 16y = 0. The natural frequency (ω_n) can be calculated as the square root of the spring constant divided by the mass (ω_n = √(k/m)). We don't have the mass value, so we can only state that ω_n = √(16/m).

5. Damping coefficient: The damping coefficient is the constant proportionality factor for the damping term. In this case, it is 8.

6. Damped frequency: Damped frequency (ω_d) is the frequency of oscillation when damping is present. It can be found using the natural frequency and the damping ratio (ζ). However, we do not have enough information to calculate the damping ratio or the damped frequency in this case.

In summary, the given differential equation y'' + 8y' + 16y = 0 represents a damped spring system with a damping coefficient of 8 and a spring constant of 16. The natural frequency depends on the mass, but the damped frequency cannot be calculated without additional information.

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The size of angle MWT is calculated to 1 d.p. to give

37.8 degrees

The size of angle MWT is solved using trigonometry tan

tan (angle MWT) = (distance midpoint of a to edge w) / b

Where distance midpoint of a to edge w is calculated using Pythagoras theorem

(distance midpoint of a to edge w)² = (1/2 a)² + c²

(distance midpoint of a to edge w)² = (1.35)² + 9.2²

distance midpoint of a to edge w = 9.3

tan (angle MWT) = 9.3 / 12

angle MWT = arc tan (9.3/12) = 37.776

angle MWT = 37.8 degrees to 1 d.p.

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If Z is between -1 and 1, then the percentage is within the 68% range. If Z is between -2 and 2, then the percentage is within the 95% range. If Z is between -3 and 3, then the percentage is within the 99.7% range.

To use the empirical rule to estimate the percentage of cold sufferers who experience symptoms for less than 9.9 days, we first need to know the mean (average) and the standard deviation of the data.

Let's assume that the mean (µ) is X days and the standard deviation (σ) is Y days. The empirical rule states that for a normal distribution:
- Approximately 68% of the data falls within 1 standard deviation (σ) of the mean (µ)
- Approximately 95% of the data falls within 2 standard deviations (σ) of the mean (µ)
- Approximately 99.7% of the data falls within 3 standard deviations (σ) of the mean (µ)

Now, we want to estimate the percentage of cold sufferers who experience symptoms for less than 9.9 days. We need to determine how many standard deviations away 9.9 days is from the mean.

To do this, use the formula:

Z = (Observed Value - Mean) / Standard Deviation
Z = (9.9 - X) / Y

Once you calculate the Z score, refer to the empirical rule:
- If Z is between -1 and 1, then the percentage is within the 68% range.
- If Z is between -2 and 2, then the percentage is within the 95% range.
- If Z is between -3 and 3, then the percentage is within the 99.7% range.

Finally, based on the Z score and the empirical rule, you can estimate the percentage of cold sufferers who experience symptoms for less than 9.9 days.

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Check the picture below.

[tex]\cfrac{1^2}{2^2}=\cfrac{110}{A}\implies \cfrac{1}{4}=\cfrac{110}{A}\implies A=440~in^2[/tex]

Answer:

3

Step-by-step explanation:

gd=14cm

dc=17cm

then,

gd-dc

14cm-17cm

0=14cm-17cm

0=-3

0+3

3

The value of b is given as follows:

b = 5.

The slope-intercept representation of a linear function is given by the equation presented as follows:

y = mx + b

The coefficients of the function and their meaning are described as follows:

When two lines are parallel, they have the same slope, hence:

4x + 5y = 1

5y = -4x + 1

y = -4x/5 + 1.

Hence:

y = -4x/5 + b.

When x = 4, y = 3, hence the intercept is given as follows:

3 = -16/5 + b

b = 31/5

Hence, in standard format, the equation will be given as follows:

y = -4x/5 + 31/5

4x/5 + y = 31/5

4x + 5y = 31

Meaning that the value of b is of 5.

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The shapes involved in the first figure is a triangle and a trapezium, with an area of 139.5. The shapes involved in the second figure is a triangle and a rectangle, with an area of 22 square units.

The first figure can be observed to be made up of a triangle and a trapezium. While the second is a triangle and a rectangle, so we shall calculate for the area and sum the results to get the total area of the composite figures as follows:

First figure:

area of the triangle = 1/2 × 9 × 6 = 27 square units

area of the trapezium = 1/2 × (6 + 9) × 15 = 112.5 square units

area of the first figure = 27 + 112.5 = 139.5 square units

Second figure:

area of the triangle = 1/2 × 4 × 2 = 4 square units

area of the rectangle = 9 × 2 = 18 square

area of the second figure = 4 + 18 = 22 square units.

Therefore, the shapes involved in the first figure is a triangle and a trapezium, with an area of 139.5. The shapes involved in the second figure is a triangle and a rectangle, with an area of 22 square units.

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The following parts can be answered by the concept of Differential equation.

(a) For the differential equation 4x²y'' + 4xy' - (64x² - 9)y = 0, we can rewrite it as:

y'' + (1/x)y' - (16 - 9/x²)y = 0

This is a Bessel's equation of order ν = 3. The general solution is given by:

y(x) = c_1 J_3(2√2x) + c_2 Y_3(2√2x)

where c_1 and c_2 are constants, J_3 is the Bessel function of the first kind of order 3, and Y_3 is the Bessel function of the second kind of order 3.

(b) For the differential equation x²y'' + xy' - (36x² - 9)y = 0, we can rewrite it as:

y'' + (1/x)y' - (36 - 9/x²)y = 0

This is also a Bessel's equation, but with order ν = 3/2. The general solution is given by:

y(x) = c_1 J_(3/2)(6x) + c_2 Y_(3/2)(6x)

where c_1 and c_2 are constants, J_(3/2) is the Bessel function of the first kind of order 3/2, and Y_(3/2) is the Bessel function of the second kind of order 3/2.

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The order of integration does not affect the final answer, but may affect the complexity of the integrals.

To calculate the volume of the region W using triple integrals, we need to determine the bounds for each variable.

First, we can see that the planes z=0 and y=1 bound the region in the z and y directions, respectively.

Next, to find the bounds for x, we need to find the intersection of the two cylinders. Solving for y in the equation [tex]z=1-y^2[/tex], we get y = ±sqrt(1-z). Substituting this into the equation [tex]y=x^2[/tex], we get [tex]x^2[/tex] = ±sqrt(1-z), or x = ±sqrt(sqrt(1-z)). So the bounds for x are -sqrt(sqrt(1-z)) to sqrt(sqrt(1-z)).

Now we can set up the triple integrals in the three orders:

Note that the order of integration does not affect the final answer, but may affect the complexity of the integrals.

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The derivative of f(x) at x = c does not exist.

To find the derivative of f(x) at x = c using the alternative form of the derivative, we first need to calculate the derivative of f(x) with respect to x.

Given that f(x) = x^3 - 2x^2 + 9, we can find the derivative of f(x) using the power rule and the constant multiple rule. The power rule states that the derivative of x^n, where n is a constant, is n*x^(n-1). The constant multiple rule states that the derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function.

Applying the power rule and constant multiple rule to f(x), we get:

f'(x) = 3x^2 - 4x

Now, we can evaluate f'(x) at x = c, which in this case is x = -2:

f'(-2) = 3(-2)^2 - 4(-2)

= 3(4) + 8

= 12 + 8

= 20

So, the derivative of f(x) at x = -2 is 20. However, we are asked to find the derivative at x = c = -2 using the alternative form of the derivative.

The alternative form of the derivative states that the derivative of a function at a specific point is equal to the limit of the difference quotient as x approaches the given point. In other words, the derivative at x = c is equal to the limit of (f(x) - f(c))/(x - c) as x approaches c.

Substituting c = -2 into the alternative form of the derivative, we get:

f'(-2) = lim(x->-2) (f(x) - f(-2))/(x - (-2))

However, if we try to evaluate this limit, we get an indeterminate form of 0/0. This means that the derivative of f(x) at x = -2 does not exist, as the limit of the difference quotient is undefined. Therefore, the main answer is that the derivative of f(x) at x = c does not exist.

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