Explicit formulas for compositions of functions. The domain and target set of functions f, g, and h are Z. The functions are defined as: . . f(x) = 2x + 3 g(x) = 5x + 7 h(x) = x2 + 1 = . Give an explicit formula for each function given below. (a) fog (b) gof (C) foh (d) hof

The function f(x) is: [tex]f(x) = x^9 + 30[/tex] and the cost function is: C(x) = 31x

A) We can find f(x) by integrating f '(x):

[tex]f(x) = ∫f '(x) dx = ∫3x^8 dx = x^9 + C[/tex]

We can determine the value of the constant C using the initial condition f(1) = 31:

[tex]31 = 1^9 + C[/tex]

C = 30

Therefore, the function f(x) is:

[tex]f(x) = x^9 + 30[/tex]

B) The marginal cost to produce one cup is the derivative of the cost function:

m(x) = C'(x) = 31

To find the cost function, we integrate the marginal cost:

C(x) = ∫m(x) dx = ∫31 dx = 31x + C

We can determine the value of the constant C using the fact that the cost of producing one cup is $31:

C(1) = 31

31 = 31(1) + C

C = 0

Therefore, the cost function is:

C(x) = 31x

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AE is greater than -13. However, without more information or specific constraints, we cannot determine the exact value of AE.

Based on the information given, we have a line AC with point B lying on it. Additionally, we have the lengths CD, BD, and BE.

Using the information CD = 7 and BD = 6, we can determine the length of BC. Since BC is the difference between CD and BD, we have:

BC = CD - BD

BC = 7 - 6

BC = 1

Now, we can focus on triangle BCE. We know the lengths of BC and BE, and we need to find the length of AE.

To find AE, we can use the fact that the sum of the lengths of the two sides of a triangle is always greater than the length of the third side. In other words, the triangle inequality states that:

BE + AE > BA

Substituting the given lengths:21 + AE > BA

We also know that BA is equal to BC + CD:

BA = BC + CD

BA = 1 + 7

BA = 8

Now, we can substitute the values into the inequality:

21 + AE > 8

Subtracting 21 from both sides:

AE > 8 - 21

AE > -13

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Therefore, the function y as a function of x is: y(x) = c1 e^(-x) - (1/2) e^x - (7/2) e^(3x) where c1 is a constant determined by the initial conditions.

We are given the differential equation:

y′′′ − 3y′′ − y′ + 3y = 0

To solve this equation, we can first find the characteristic equation by assuming that y = e^(rt), where r is a constant:

r^3 e^(rt) - 3r^2 e^(rt) - r e^(rt) + 3e^(rt) = 0

Simplifying and factoring out e^(rt), we get:

e^(rt) (r^3 - 3r^2 - r + 3) = 0

This equation has three roots, which we can find using numerical methods or by making educated guesses. We find that the roots are r = -1, r = 1, and r = 3.

Therefore, the general solution to the differential equation is:

y(t) = c1 e^(-t) + c2 e^t + c3 e^(3t)

where c1, c2, and c3 are constants that we need to determine.

Using the initial conditions, we can find these constants:

y(0) = c1 + c2 + c3 = -4

y′(0) = -c1 + c2 + 3c3 = -6

y′′(0) = c1 + c2 + 9c3 = -20

We can solve these equations simultaneously to find c1, c2, and c3. One way to do this is to subtract the first equation from the second and third equations, respectively:

c2 + 4c3 = -2

c2 + 8c3 = -16

Subtracting these two equations, we get:

4c3 = -14

Solving for c3, we get:

c3 = -14/4 = -7/2

Substituting this value of c3 into one of the earlier equations, we can solve for c2:

c2 + 8(-7/2) = -16

c2 = -1/2

Finally, we can use these values of c1, c2, and c3 to write the solution to the differential equation as:

y(t) = c1 e^(-t) - (1/2) e^t - (7/2) e^(3t)

Substituting x for t, we get:

y(x) = c1 e^(-x) - (1/2) e^x - (7/2) e^(3x)

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There are approximately 4 bright interference fringes on either side of the central maximum, for a total of 6 + 4 + 4 = 14 bright interference fringes in the whole pattern.

When light of wavelength 570 nm passes through a pair of slits that are 18 µm wide and 180 µm apart, we can use the formula for the position of the bright fringes in the interference pattern:

y = (mλL)/d

where y is the distance from the central maximum to the m-th bright fringe, λ is the wavelength of the light, L is the distance from the slits to the screen, d is the distance between the slits, and m is the order of the fringe.

For the central maximum, m = 0, so we have:

y_0 = (0.570 × 10^-6 m)(1 m)/(180 × 10^-6 m) = 3.17 × 10^-3 m

To find the number of bright interference fringes in the central maximum, we need to divide the width of the slits by the distance between adjacent fringes:

n_0 = 18 × 10^-6 m / 3.17 × 10^-3 m = 5.67

So there are approximately 6 bright interference fringes in the central maximum.

For the whole pattern, we need to find the number of bright fringes on either side of the central maximum. Since the distance between adjacent fringes decreases as we move away from the central maximum, we need to take this into account. We can use the formula:

y_m = (mλL)/d

to find the distance from the central maximum to the m-th bright fringe on either side. Setting this equal to half the distance between adjacent fringes, we get:

(m + 1/2)λL/d = Δy

where Δy is the distance between adjacent fringes. Solving for m, we get:

m = Δy d/λL - 1/2

Plugging in the values, we get:

m = (1.570 × 10^-6 m)(1 m)/(180 × 10^-6 m) - 1/2 = 4.43

So there are approximately 4 bright interference fringes on either side of the central maximum, for a total of 6 + 4 + 4 = 14 bright interference fringes in the whole pattern.

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The smallest positive integer a such that the Intermediate Value Theorem guarantees a zero exists between 0 and a is 2.

Intermediate Value Theorem (IVT) states that if a function f(x) is continuous on a closed interval [a, b], then for any value c between f(a) and f(b), there exists at least one value x = k, where a [tex]\leq[/tex] k [tex]\leq[/tex] b, such that f(k) = c.

From our question, we want to find the smallest positive integer a such that there exists a zero of the polynomial f(x) between 0 and a.

Since f(x) is a polynomial, it is continuous for all values of x. Therefore, the IVT guarantees that if f(0) and f(a) have opposite signs, then there must be at least one zero of f(x) between 0 and a.

We can evaluate f(0) and f(a) as follows:

f(x)=−2x³ + x² + 4x + 4

f(0) = -2(0)³ + (0)² + 4(0) + 4 = 4

f(a) = -2a³ + a² + 4a + 4

We want to find the smallest positive integer a such that f(0) and f(a) have opposite signs. Since f(0) is positive, we need to find the smallest positive integer a such that f(a) is negative.

We can try different values of a until we find the one that works.

Let's start with a = 1:

f(1) = -2(1)³ + (1)² + 4(1) + 4 = -2 + 1 + 4 + 4 = 7 (≠ 0)

f(2) = -2(2)³ + (2)² + 4(2) + 4 = -16 + 4 + 8 + 4 = 0

Since f(2) is zero, we know that f(x) has a zero between 0 and 2. Therefore, the smallest positive integer a such that the Intermediate Value Theorem guarantees a zero of f(x) between 0 and a is a = 2.

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The coordinates of point P could be approximately,

⇒ (0.0345, 0.9994).

Now, the possible coordinates of point P on the unit circle, we need to use,

tan(y) = opposite/adjacent.

Since the radius of the unit circle is 1, we can simplify this to;

= opposite/1  

= opposite.

We can also use the Pythagorean theorem to find the adjacent side.

Since the radius is 1, we have:

opposite² + adjacent² = 1

adjacent² = 1 - opposite²

adjacent = √(1 - opposite)

Now that we have expressions for both the opposite and adjacent sides, we can use the given value of tan(y) to solve for the opposite side:

tan(y) = opposite/adjacent

opposite = tan(y) adjacent

opposite = tan(y) √(1 - opposite)

Substituting the given value of tan(y) into this equation, we get:

opposite = 5.34  √(1 - opposite)

Squaring both sides and rearranging, we get:

opposite = (5.34)² (1 - opposite)

= opposite (5.34) (5.34) - (5.34)

opposite = opposite ((5.34) - 1)

opposite = (5.34) / ((5.34) - 1)

opposite ≈ 0.9994

Now that we know the opposite side, we can use the Pythagorean theorem to find the adjacent side:

adjacent = 1 - opposite

adjacent ≈ 0.0345

Therefore, the coordinates of point P could be approximately,

⇒ (0.0345, 0.9994).

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There are 123,480 ways for Alex to choose his selection.

To determine the number of ways Alex can choose his selection, we need to use the multiplication principle of counting.

The number of ways to choose 4 shirts from 10 is given by the number of combinations of 10 items taken 4 at a time:

10C4 = (10!)/(4!(10-4)!) = 210

Similarly, the number of ways to choose 2 pairs of jeans from 7 is given by the number of combinations of 7 items taken 2 at a time:

7C2 = (7!)/(2!(7-2)!) = 21

Finally, the number of ways to choose 6 pairs of socks from 8 is given by the number of combinations of 8 items taken 6 at a time:

8C6 = (8!)/(6!(8-6)!) = 28

To obtain the total number of ways for Alex to choose his selection, we need to multiply these three quantities together:

210 × 21 × 28 = 123,480

Therefore, there are 123,480 ways for Alex to choose his selection.

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The probability of committing a Type I error when the null hypothesis is true as an equality is d. The level of significance.

The level of significance, also known as alpha, is the threshold value that is used to determine if a result is statistically significant or not. It is the maximum probability of committing a Type I error that researchers are willing to accept.

                             A lower level of significance will decrease the probability of committing a Type I error, but it will increase the probability of committing a Type II error (failing to reject a false null hypothesis). It is important to carefully select an appropriate level of significance in order to balance these two types of errors.

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The solution to the differential equation is y ( t ) = 100 1 + 19 e 0.09 t

This is a separable differential equation, which we can solve using separation of variables:

d y d t = 0.09 y ( 1 − y 100 )

d y 0.09 y ( 1 − y 100 ) = d t

Integrating both sides, we get:

ln | y | − 0.01 ln | 100 − y | = 0.09 t + C

where C is the constant of integration. We can solve for C using the initial condition y(0) = 5:

ln | 5 | − 0.01 ln | 100 − 5 | = 0.09 ( 0 ) + C

C = ln | 5 | − 0.01 ln | 95 |

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

ln | y | − 0.01 ln | 100 − y | = 0.09 t + ln | 5 | − 0.01 ln | 95 |

Simplifying, we get:

ln | y ( t ) | 100 − y ( t ) = 0.09 t + ln 5 95

To solve for y(t), we can take the exponential of both sides:

| y ( t ) | 100 − y ( t ) = e 0.09 t e ln 5 95

| y ( t ) | 100 − y ( t ) = e 0.09 t 5 95

y ( t ) 100 − y ( t ) = ± e 0.09 t 5 95

Solving for y(t), we get:

y ( t ) = 100 e 0.09 t 5 95 ± e 0.09 t 5 95

Using the initial condition y(0) = 5, we can determine that the sign in the solution should be positive, so we have:

y ( t ) = 100 e 0.09 t 5 95 + e 0.09 t 5 95

Simplifying, we get:

y ( t ) = 100 1 + 19 e 0.09 t

Therefore, the solution to the differential equation is:

y ( t ) = 100 1 + 19 e 0.09 t

where y(0) = 5.

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(a) P (9,5) = 15,120

(b) P (9,9) = 362,880

(c) P (9,4) = 6,120

(d) P (9,1) = 9

(a) P (9,5) means choosing 5 objects from a total of 9 and arranging them in a specific order. Therefore, we have 9 options for the first object, 8 options for the second object, 7 options for the third object, 6 options for the fourth object, and 5 options for the fifth object. Multiplying these options together gives us P (9,5) = 9 x 8 x 7 x 6 x 5 = 15,120.

(b) P (9,9) means choosing all 9 objects from a total of 9 and arranging them in a specific order. This is simply 9! = 362,880, as there are 9 options for the first object, 8 options for the second, and so on until there is only one option for the last object.

(c) P (9,4) means choosing 4 objects from a total of 9 and arranging them in a specific order. This is calculated as 9 x 8 x 7 x 6 = 6,120.

(d) P (9,1) means choosing 1 object from a total of 9 and arranging it in a specific order. Since there is only 1 object and no other objects to arrange with it, there is only 1 way to arrange it, giving us P (9,1) = 9 x 1 = 9.

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The particular solution is:

y = -1 - e^(3x)

We have the differential equation:

y' - 3y = 3

To find the integrating factor, we multiply both sides by e^(-3x):

e^(-3x)y' - 3e^(-3x)y = 3e^(-3x)

Notice that the left-hand side is the product rule of (e^(-3x)y), so we can write:

d/dx (e^(-3x)y) = 3e^(-3x)

Integrating both sides with respect to x, we get:

e^(-3x)y = ∫ 3e^(-3x) dx + C

e^(-3x)y = -e^(-3x) + C

y = -1 + Ce^(3x)

Using the initial condition y(0) = -1, we can find the value of C:

-1 = -1 + Ce^(3*0)

C = -1

So the particular solution is:

y = -1 - e^(3x)

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Let's denote:

x = the number of pounds of electronics recycled by Solomon

y = the number of pounds of appliances recycled by Marcus

The equation representing how much Solomon would earn by recycling electronics is:

Earned amount by Solomon = ($1.50 * x) - $27

The first term represents the amount earned per pound of electronics, and the second term is the fixed hazardous waste fee.

The equation representing how much Marcus would earn from recycling appliances is:

Earned amount by Marcus = $0.60 * y

The term $0.60 represents the amount earned per pound of appliances.

To find out how many pounds they would need to recycle to earn the same amount of money, we can set the two equations equal to each other:

($1.50 * x) - $27 = $0.60 * y

Simplifying the equation further, we get:

$1.50 * x = $0.60 * y + $27

Now, to find the values of x and y, we need additional information or an additional equation relating the two variables. Without that information, we cannot determine the specific values for x and y to make their earnings equal.

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Answer:

To find a unit vector that is orthogonal to both u = <-8, -6, 4> and v = <17, -18, -1>, we can use the cross product of u and v, which will give us a vector that is orthogonal to both u and v. Then, we can divide this vector by its magnitude to obtain a unit vector.

The cross product of u and v can be computed as follows:

u x v = |i j k |

|-8 -6 4 |

|17 -18 -1 |

where i, j, and k are the unit vectors along the x, y, and z axes, respectively. Using the formula for the cross product, we have:

u x v = (6 x (-1) - 4 x (-18))i - (-8 x (-1) - 4 x 17)j + (-8 x (-18) - (-6) x 17)k

= -102i - 68j - 222k

To obtain a unit vector that is orthogonal to both u and v, we need to divide this vector by its magnitude:

|u x v| = sqrt((-102)^2 + (-68)^2 + (-222)^2) = 262

So, a unit vector that is orthogonal to both u and v is:

(-102i - 68j - 222k) / 262

Dividing each component of the vector by 262, we get:

(-102/262)i - (68/262)j - (222/262)k

which simplifies to:

(-51/131)i - (34/131)j - (111/131)k

Therefore, a unit vector that is orthogonal to both u and v is:

< -51/131, -34/131, -111/131 >.

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The speed of a car on a New York tollway during rush hour traffic is a continuous random variable.

The speed of a car on a New York tollway during rush hour traffic is a continuous random variable. This is because the speed can take on any value within a given range and is not limited to specific, separate values like a discrete random variable would be.

A random variable is a mathematical concept used in probability theory and statistics to represent a numerical quantity that can take on different values based on the outcomes of a random event or experiment.

Random variables can be classified into two types: discrete random variables and continuous random variables.

Discrete random variables are those that take on a countable number of distinct values, such as the number of heads in multiple coin flips.

Continuous random variables are those that can take on any value within a certain range or interval, such as the weight or height of a person.

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The flow rate through the upper hemisphere of the sphere is zero.

We can use the divergence theorem to calculate the flow rate of the fluid through the upper hemisphere of the sphere. The divergence theorem states that the flux through a closed surface is equal to the volume integral of the divergence of the vector field over the enclosed volume.

First, we need to calculate the divergence of the velocity field:

div(v) = ∂u/∂x + ∂v/∂y + ∂w/∂z

Since the velocity field is given as v = (5, 0, 0), the partial derivatives are:

∂u/∂x = 5, ∂v/∂y = 0, ∂w/∂z = 0

Therefore, the divergence of v is:

div(v) = ∂u/∂x + ∂v/∂y + ∂w/∂z = 5

Now, we can use the divergence theorem to calculate the flow rate through the upper hemisphere of the sphere with radius 4:

Φ = ∫∫S v · dS = ∭V div(v) dV

where S is the surface of the upper hemisphere and V is the enclosed volume.

Since the sphere is symmetric, we can integrate over the upper hemisphere only, which has area A = 2πr² and volume V = (2/3)πr³:

Therefore, the flow rate through the upper hemisphere of the sphere is zero. This makes sense since the velocity field is constant in the x-direction and does not change as we move along the surface of the sphere.

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Booker has a room to store 120 - 85 = 35 video games more on his shelves. Therefore, he has room for 35 more games.

Given that,

Booker owns 85 video games.

He has 3 shelves to put the games on.

Each shelve can hold 40 games.

Using these given values,

let's calculate the games that Booker can store in all the 3 shelves.

Each shelf can store 40 video games.

So, 3 shelves can store = 3 x 40 = 120 video games.

Therefore, Booker has a room to store 120 video games.

How many more games does he has room for:

Booker has 85 video games.

The three shelves he has can accommodate a total of 120 games (40 games each).

So, he has a room to store 120 - 85 = 35 video games more on his shelves.

Therefore, he has room for 35 more games.

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the statement "there are arbitrarily many different mathematical functions that interpolate a given set of data points" is false.

Interpolation is the process of constructing a mathematical function that passes through a given set of data points. However, not every set of data points can be interpolated by a unique function. For example, if we have two data points (x1, y1) and (x2, y2) where x1 ≠ x2, then there exists a unique linear function f(x) = mx + b that passes through these two points.

However, if we have three or more data points, there may be multiple functions that interpolate the data. Nevertheless, there are some conditions that can guarantee the uniqueness of the interpolating function, such as if the data points are the values of a polynomial of degree n or less, then there exists a unique polynomial of degree n or less that interpolates the data.

Therefore, the statement "there are arbitrarily many different mathematical functions that interpolate a given set of data points" is false. The number of possible interpolating functions depends on the properties of the data points and the type of function used for interpolation.

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The relationship between height and volume is not linear because the volume increase is inconsistent. The graph of the equation V = 12πh of a cone with a radius of 6 units is shown.

The graph of the equation V = 12πh of a cone with a radius of 6 units is shown below. The relationship between the height and volume of a cone with a radius of 6 units is not linear.

A linear relationship is when a change in one variable produces an equal and consistent change in another.

In the case of a cone with a radius of 6 units, the relationship between height and volume is not linear because a change in height produces an increase in volume, but the increase in volume is not consistent.

Therefore, the relationship between height and volume is not linear because the increase in volume is not consistent. The graph of the equation V = 12πh of a cone with a radius of 6 units is shown.

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Answer: 31

Step-by-step explanation: 775 divided by 25 = 31

The set of points at which the function f(x, y) = xy/(8ex − y) is continuous is the set of all points (x, y) such that 8ex ≠ y.

To determine the set of points at which the function is continuous, we need to check if the limit of the function exists and is equal to the value of the function at that point.

Taking the limit of the function as (x,y) approaches (a,b) gives:

lim_(x,y)→(a,b) f(x,y) = lim_(x,y)→(a,b) xy/8ex-y

Using L'Hopital's rule, we can find that the limit is equal to [tex]ab/8e^(b-a)[/tex].

The function is continuous for all points (a,b) in [tex]R^2[/tex].

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8π radians corresponds to 4 rotations around the unit circle.

One rotation around the unit circle corresponds to an angle of 2π radians (or 360 degrees), since the circumference of the circle is 2π times its radius (which is 1). Therefore, 4 rotations around the unit circle correspond to an angle of:

4 rotations × 2π radians/rotation = 8π radians

So, 8π radians corresponds to 4 rotations around the unit circle.

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Answer:

1/4

Step-by-step explanation:

Think of 4 written like this:

[tex] \frac{4}{1} [/tex]

and now flip it upside down for the reciprocal and it's 1/4.

The probability of obtaining exactly 4 heads (or 4 tails) when tossing 16 coins simultaneously is approximately 0.0984, or 9.84%.

When tossing 16 coins simultaneously, the probability of getting 4 heads (or tails, as the probability is the same for both outcomes) can be calculated using the concept of binomial probability.

The formula for binomial probability is given by:

P(X=k) = (nCk) * p^k * q^(n-k)

Where:

P(X=k) is the probability of getting exactly k successes,

n is the total number of trials (in this case, the number of coins tossed),

k is the number of successful outcomes (in this case, 4 heads or 4 tails),

p is the probability of a single success (getting a head or a tail, which is 1/2 in this case),

q is the probability of a single failure (1 - p, which is also 1/2 in this case), and

nCk represents the number of combinations of n items taken k at a time.

Applying the formula to our scenario:

P(X=4) = (16C4) * (1/2)^4 * (1/2)^(16-4)

Using the binomial coefficient calculation:

(16C4) = 16! / (4! * (16-4)!)

= (16 * 15 * 14 * 13) / (4 * 3 * 2 * 1)

= 1820

Now, substituting the values into the formula:

P(X=4) = 1820 * (1/2)^4 * (1/2)^12

= 1820 * (1/2)^16

≈ 0.0984

Therefore, the probability of obtaining exactly 4 heads (or 4 tails) when tossing 16 coins simultaneously is approximately 0.0984, or 9.84%.

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The given equation can be rewritten as an exponential equation like:

4x + 8 = exp(x + 5)

Remember that the exponential equation is the inverse of the natural logarithm, this means that:

exp( ln(x) ) = x

ln( exp(x) ) = x

Here we have the equation:

ln(4x + 8) = x + 5

If we apply the exponential in both sides, we will get:

exp( ln(4x + 8)) = exp(x + 5)

4x + 8 = exp(x + 5)

Now the equation is exponential.

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With long-term financing covering all capital assets, permanent current assets, and half of the temporary current assets, Lear's earnings before interest and taxes of $200,000 will be subject to a 30% tax rate.

Therefore, the company's earnings after taxes can be calculated.

To determine Lear's earnings after taxes, we need to apply the tax rate of 30% to the earnings before interest and taxes (EBIT) of $200,000. The tax rate represents the portion of EBIT that is paid as taxes, leaving the remaining portion as earnings after taxes.

To calculate the earnings after taxes, we multiply the EBIT by (1 - tax rate). In this case, the calculation would be:

Earnings after taxes = EBIT * (1 - tax rate)

= $200,000 * (1 - 0.30)

= $200,000 * 0.70

= $140,000

Therefore, Lear's earnings after taxes would amount to $140,000. This calculation reflects the portion of earnings remaining after accounting for the 30% tax rate applied to the EBIT.

This calculation assumes no other factors, such as deductions or credits, that may affect the final tax liability.

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I'm sorry, but there is no problem statement provided for problem 21-30. Please provide the full problem statement for me to assist you with the solution.

The t critical values are:

(a) 2.571, (b) 2.306, (c) 3.169, (d) 3.250, (e) 2.831, (f) 2.750

We have,

(a) Using a t-table or calculator,

The t critical value for a two-sided confidence interval at a 95% confidence level with df = 5 is 2.571.

(b)

Using a t-table or calculator,

The t critical value for a two-sided confidence interval at a 95% confidence level with df = 10 is 2.228.

(c)

Using a t-table or calculator,

The t critical value for a two-sided confidence interval at a 99% confidence level with df = 10 is 3.169.

(d)

Using a t-table or calculator,

The t critical value for a two-sided confidence interval at a 99% confidence level with n = 10 is 3.250.

(e)

Using a t-table or calculator,

The t critical value for a two-sided confidence interval at a 98% confidence level with df = 21 is 2.518.

(f)

Using a t-table or calculator,

The t critical value for a two-sided confidence interval at a 99% confidence level with n = 36 is 2.718.

Thus,

The critical values are:

(a) 2.571, (b) 2.306, (c) 3.169, (d) 3.250, (e) 2.831, (f) 2.750

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The lower sum is 1.357 and the upper sum is 7.699.

To find the lower sum, we need to evaluate f(x) at the left endpoint of each subinterval and multiply by the width of each subinterval:

L(f, P) = [(6-0) x f(0)] + [(4-6) x f(6)] + [(3-4) x f(4)] + [(2-3) x f(3)] + [(2-0) x f(2)] = [(6-0) x 0] + [(4-6) x 0.994] + [(3-4) x 0.951] + [(2-3) x 0.141] + [(2-0) x 0.412] = 0.412

To find the upper sum, we need to evaluate f(x) at the right endpoint of each subinterval and multiply by the width of each subinterval:

U(f, P) = [(6-0) x f(6)] + [(4-6) x f(4)] + [(3-4) x f(3)] + [(2-3) x f(2)] + [(2-0) x f(2)] = [(6-0) x 0.994] + [(4-6) x 0.951] + [(3-4) x 0.141] + [(2-3) x 0.412] + [(2-0) x 0.412] = 3.764

Therefore, the lower sum is approximately 0.412 and the upper sum is approximately 3.764.

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The value of g(0) is 3, which we can obtain by substituting 0 for x in the function g(x) and simplifying.

To find the value of g(0), we substitute 0 for x in the function g(x) and simplify:

g(0) = 4^0 - 5 + 7

= 1 - 5 + 7

= 3

Therefore, g(0) = 3.

We can also explain this result in more detail by understanding the properties of exponential functions. The function g(x) is an exponential function with base 4. This means that as x increases, the value of g(x) increases rapidly.

When we substitute 0 for x, we get:

g(0) = 4^0 - 5 + 7

Since any number raised to the power of 0 is 1, we can simplify this expression to:

g(0) = 1 - 5 + 7

Combining like terms, we get:

g(0) = 3

Therefore, the value of g(0) is 3.

We can also verify this result by graphing the function g(x) using a graphing calculator or software. When we plot the graph of g(x) for values of x ranging from -5 to 5, we can see that the function takes the value of 3 when x is equal to 0.

We can also explain this result by understanding the properties of exponential functions and verifying it using a graph.

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The solution to the initial-value problem y'' + x²y = 0, y(0) = 0, y'(0) = 0 is y(x) = 0.

This is because the given differential equation is a homogeneous linear second-order differential equation with constant coefficients, and its characteristic equation has roots of i and -i.

Since the roots are purely imaginary, the solution is of the form y(x) = c1*cos(x) + c2*sin(x), where c1 and c2 are constants determined by the initial conditions.

Plugging in y(0) = 0 and y'(0) = 0 yields c1 = 0 and c2 = 0, hence the only solution is y(x) = 0.

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