Solve the separable differential equation d y d x = − 8 y , and find the particular solution satisfying the initial condition y ( 0 ) = 2 . y ( 0 ) =2

The x and y components of acceleration are:-  [tex]ax = (1/k)(c(dy^2/dt^2)),- ay = c(d^2y/dt^2)[/tex]

To find the x-component of acceleration, we need to take the second derivative of the position function with respect to time:

2 = 4kx
2v = 4kx'  (taking the derivative with respect to time)
2a = 4kx''  (taking the derivative of v with respect to time)

So the x-component of acceleration is a_x = 2kx''.

To find the y-component of acceleration, we can use the given information about the velocity along the y-axis:

v_y = 1y ct

Taking the derivative with respect to time, we get:

a_y = c

So the y-component of acceleration is simply a_y = c.
To determine the x and y components of acceleration for the given path of a particle and the component of velocity along the y-axis, we'll use the given equations and find the second derivatives with respect to time.

The path of the particle is defined by [tex]y^2[/tex] = 4kx. First, differentiate this equation with respect to time (t) to find the relation between the x and y components of velocity:

(1) 2y(dy/dt) = 4k(dx/dt)

The component of velocity along the y-axis is given as vy = dy/dt = cy. Substituting this into equation (1):

(2) 2y(cy) = 4k(dx/dt)

Now, solve for dx/dt (vx):

[tex]vx = (1/k)(cy^2/2)[/tex]

Next, differentiate both vx and vy with respect to time to find the x and y components of acceleration:

[tex]ax = d^2x/dt^2 = (1/k)(c(dy^2/dt^2))[/tex]
[tex]ay = d^2y/dt^2 = c(d^2y/dt^2)[/tex]

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Part A :f has relative maxima at x = 1 and x = 3, and a relative minimum at x = 8, on the interval [0, 9].

Part B: the graph of f is concave up on the intervals (−∞, 1) and (8, ∞) and is concave down on the interval (1, 8).

Part C: x = 1 and x = 8 these are the points of inflection of f.

To find relative extrema, we want to find the basic places of the capability and afterward decide if they are greatest or least by really looking at the indication of the subsequent subordinate.

The point is a relative minimum when the second derivative is positive, and it is a relative maximum when the second derivative is negative. To determine the nature of the critical point, we must employ alternative strategies if the second derivative is zero.

Part A:

The values of x at which f'(x) is zero or undefined are the critical points of f. From the table, we see that f'(x) = 0 at x = 1 and x = 8, and f'(x) is unclear at x = 3.

At each critical point, we must now examine the sign of the second derivative to determine whether it is a maximum or a minimum.

Negative f''(1) = -2 is the case when x = 1. As a result, the relative maximum is the critical point at x = 1.

Because f''(3) is not defined, the second derivative test cannot be used to determine the nature of the critical point at x = 3. However, as the table demonstrates, f shifts from increasing to decreasing at x = 3, indicating that x = 3 is the relative maximum.

We have a positive value of f'(8) = 2 when x = 8. As a result, the relative minimum at x = 8 is the critical point.

Therefore, f has relative maxima at x = 1 and x = 3, and a relative minimum at x = 8, on the interval [0, 9].

Part B:

To determine where the graph of f is concave up and where it is concave down, we need to find the intervals where f''(x) > 0 and where f''(x) < 0, respectively.

From the table, we see that f''(x) > 0 for x < 1 and for x > 8, and f''(x) < 0 for 1 < x < 8. Therefore, the graph of f is concave up on the intervals (−∞, 1) and (8, ∞) and is concave down on the interval (1, 8).

Part C:

We must determine the values of x at the points on the graph where the concavity changes in order to locate any points of inflection of f. From part B, we see that the concavity changes at x = 1 and x = 8. Accordingly, these are the marks of expression of f.

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Answer:

8x+48>280

Step-by-step explanation:

first do your mom then do your dad

The slope for this linear relationship is -2, and it means that the amount of bait decreases by 2 pieces each hour in this situation.

The slope for this linear relationship is -2, which means that the amount of bait in the bucket decreases by 2 pieces for every hour of fishing. This slope indicates a constant rate of change in the amount of bait over time.

To understand this further, we can interpret the slope as the rate of consumption of bait per hour. Since the fisherman uses 2 pieces of bait every hour, the amount of bait in the bucket decreases by 2 pieces every hour. This linear relationship can be represented by the equation:

y = -2x + 20

where y is the amount of bait remaining in the bucket after x hours of fishing.

Therefore, the slope of -2 in this context is an important characteristic of the linear relationship between the amount of bait remaining and the time spent fishing. It tells us the rate at which the bait is being consumed, and helps the fisherman predict how much bait he will have left after a certain amount of time.

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The limit of the sequence as n approaches infinity is 4.

Therefore, lim n→[infinity] an = 4.

To determine whether the sequence converges or diverges, we need to find the limit of the sequence as n approaches infinity.

We can simplify the expression for the nth term of the sequence as follows:

an = 4 + 2n^2 / (2n + 3n^2)

= 4 + n / (1.5 + n)

As n approaches infinity, the denominator of the fraction approaches infinity much faster than the numerator. Therefore, the fraction approaches zero and the value of the nth term approaches 4.

In other words, as n becomes very large, the terms of the sequence become arbitrarily close to 4. Therefore, the sequence converges to 4.

Hence, the limit of the sequence as n approaches infinity is 4.

Therefore, lim n→[infinity] an = 4.

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The domain of g(x,y) = 1/xy is all real numbers except for x=0 and y=0.

This is because division by zero is undefined and would result in an error or undefined value. In other words, x and y cannot be zero in order for the function to have a valid output.

To understand this further, we can think about the function graphically. The function g(x,y) represents a three-dimensional surface where the height of the surface at any point (x,y) is given by 1/xy. However, we can see that the function is undefined at the points where x=0 or y=0.

At these points, the surface would have a "hole" or a "break" since the height of the surface is undefined. Therefore, the domain of the function is all real numbers except for x=0 and y=0.

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The solutions are given below.

1) Variables:

- Speed of the kayaker (unknown, let's call it x)

- Speed of the current = 3 mph (given)

- Distance kayaked one way = 1 mile (given)

- Total distance covered (round trip) = 2 miles (given)

- Total time of the trip = 3 hours 20 minutes = 3.33 hours (converted to hours for convenience)

Table:

Photo attached.

2) The equation to model the problem is:

distance = rate × time

Using this equation for each kayaking portion, we get:

1 = (x - 3) t

1 = (x + 3) t

We also know that the total time of the trip is 3.33 hours:

t + t = 3.33

2t = 3.33

t = 1.665

3) Now we can solve for x by substituting t = 1.665 in either of the above equations:

1 = (x - 3) (1.665)

x - 3 = 0.599

x = 3.599

Thus, the kayakers are paddling at a speed of 3.599 miles per hour.

4) The kayakers are paddling at a speed of 3.599 miles per hour. This solution is obtained by calculating the average speed of the kayakers over the entire trip, taking into account the opposing speed of the river current. The kayakers are traveling faster downstream (with the current) than upstream (against the current).

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The volume of the solid obtained by rotating the region bounded by the curves y = x^2, x = 0, and y = 1 about the y-axis is π/2 cubic units.

The solid obtained by rotating the region bounded by the curves y = x^2, x = 0, and y = 1 about the y-axis can be found using the method of disks or washers.

To use the method of disks, we can imagine dividing the region into thin vertical strips, each with width dx. The volume of each disk is then πy^2dx, where y is the distance from the strip to the axis of rotation (which is the y-axis in this case). We can find y in terms of x by solving y = x^2 for x, which gives x = sqrt(y).

Thus, the volume of each disk is π(sqrt(y))^2dx = [tex]\pi ydx[/tex]. Integrating from y = 0 to y = 1, we get the total volume of the solid as π∫(0 to 1) [tex]ydx[/tex].

Using the power rule of integration, this simplifies to π[x^2/2] from 0 to 1, which equals π/2.

To use the method of washers, we can imagine dividing the region into thin horizontal strips, each with height [tex]dy[/tex]. The volume of each washer is then π(R^2 - r^2)[tex]dy[/tex], where R is the outer radius (which is 1 in this case) and r is the inner radius. The inner radius is simply the distance from the strip to the y-axis, which is x.

Using the equation y = x^2, we can solve for x in terms of y to get x = sqrt(y).

Thus, the inner radius is sqrt(y) and the volume of each washer is π(1^2 - (sqrt(y))^2)[tex]dy[/tex] = π(1 - y)[tex]dy[/tex]. Integrating from y = 0 to y = 1, we get the total volume of the solid as π∫(0 to 1) (1 - y)[tex]dy[/tex].

Using the power rule of integration, this simplifies to π[y - y^2/2] from 0 to 1, which equals π/2.

Therefore, the volume of the solid obtained by rotating the region bounded by the curves y = x^2, x = 0, and y = 1 about the y-axis is π/2 cubic units, regardless of which method we use.

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The quadratic function with the given features is defined as follows:

y = 0.86x² - 5.86x + 5.

The standard definition of a quadratic function is given as follows:

y = ax² + bx + c.

When x = 0, y = 5, hence the coefficient c is given as follows:

c = 5.

Hence:

y = ax² + bx + 5.

When x = 1, y = 0, hence:

a + b + 5 = 0

a + b = -5.

The discriminant is given as follows:

D = b² - 4ac.

Hence:

D = b² - 20a

The minimum value is of -4, hence:

-D/4a = -5

(b² - 20a)/4a = -5

b² - 20a = 20a

b² = 40a

Since a = -5 - b, we have that the value of b is obtained as follows:

b² = 40(-5 - b)

b² + 40b + 200 = 0.

b = -5.86.

Hence the value of a is of:

a = -5 + 5.86

a = 0.86.

Then the equation is:

y = 0.86x² - 5.86x + 5.

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Answer:

B

Step-by-step explanation:

If a figure is enlarged by a scale factor of 1, the new figure will have the same size and shape as the original figure.

If a figure is enlarged by a negative scale factor, the new figure will be on the other side of the centre of enlargement and will be inverted (the figure appears upside down).

Therefore, if you enlarge shape X by a scale factor of -1, it will have the same size and shape as shape X, but it will be upside down.

Further information

A is a reflection across a vertical line.

C is a reflection across a horizontal line.

D is a rotation of 90° clockwise.

E is a rotation of 90° anticlockwise.

F is an enlargement of scale factor 1.

The required answer is the solid D1 in spherical coordinates is defined by: - sqrt(3) ≤ ρ ≤ sqrt(7) - 0 ≤ θ ≤ π/2 - 0 ≤ φ ≤ π/2

To express the solid D1 in spherical coordinates, we need to first understand the given conditions and the equations of the spheres.

D1 is in the octant where x≥0, y≤0, and z≥0. The two spheres have the equations x^2+y^2+z^2=3 and x^2+y^2+z^2=7.

Now, let's convert these equations into spherical coordinates. Recall that the spherical coordinates (ρ, θ, φ) are related to the Cartesian coordinates (x, y, z) as follows:

x = ρ * sin(φ) * cos(θ)
y = ρ * sin(φ) * sin(θ)
z = ρ * cos(φ)

Since x^2+y^2+z^2=ρ^2, the given spheres can be represented in spherical coordinates as:

1. ρ^2 = 3, so ρ = sqrt(3)
2. ρ^2 = 7, so ρ = sqrt(7)
A spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuthal angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane.

Now, we need to determine the range for θ and φ that corresponds to the specified octant. Since x≥0, y≤0, and z≥0, we have the following constraints on the angles:

- 0 ≤ θ ≤ π/2 (due to x≥0 and y≤0)
- 0 ≤ φ ≤ π/2 (due to z≥0)

So, the solid D1 in spherical coordinates is defined by:

- sqrt(3) ≤ ρ ≤ sqrt(7)
- 0 ≤ θ ≤ π/2
- 0 ≤ φ ≤ π/2

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Based on her results, the statement that describes the relationship between m, the speed of the car in miles per hour, and k, the speed of the car in kilometers per hour is A.. The relationship is proportional because the ratio of m to k is constant.

To ascertain whether the relationship is indeed proportional, we shall evaluate if the ratio of m to k remains consistent.

For each specified speed:

m = 11.0 mph, k = 17.699 kph

Ratio = 11.0 / 17.699 ≈ 0.621

m = 26.0 mph, k = 41.834 kph

Relation = 26.0 / 41.834 ≈ 0.622

m = 34.0 mph, k = 54.706 kph

Relation = 34.0 / 54.706 ≈ 0.621

The ratios - which are almost identical (0.621, 0.622, and 0.621) - lead us to believe that the correlation between m and k is a proportional one.

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By rigid transformations, the images of the parent function are, respectively:

Case 1: f(x) = - |x| + 3

Case 2: f(x) = |x - 2| - 2

Case 3: f(x) = |x + 15|

Case 4: f(x) = - |x + 3| + 3

In this problem we must derive the absolute value functions by means of rigid transformations used on parent function f(x) = |x|. We proceed to apply any of the following rigid transformations:

Horizontal translation

f(x) → f(x - k), where k > 0 for a right translation.

Vertical translation

f(x) → f(x) + k, where k > 0 for a upward translation.

Reflection around x-axis

f(x) → - f(x)

Case 1 (Reflection - Vertical translation)

f(x) = - |x| + 3

Case 2 (Horizontal translation - Vertical translation)

f(x) = |x - 2| - 2

Case 3 (Horizontal translation)

f(x) = |x + 15|

Case 4 (Reflection - Horizontal translation - Vertical translation)

f(x) = - |x + 3| + 3

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Answer:

  H  (8, -3)

Step-by-step explanation:

You want a possible vertex of rectangle KLMN if the midpoint of the diagonals is (2, 1) and one of the vertices is (-4, 5).

The diagonals of a rectangle are congruent and bisect each other, so the given point of intersection is the midpoint of the diagonals. If one of the diagonal end points is K = (-4, 5), then the other end of that diagonal is ...

  X = (K+M)/2

  M = 2X -K = 2(2, 1) -(-4, 5) = (2·2+4, 2·1-5)

  M = (8, -3)

Another vertex on the rectangle is (8, -3).

__

Additional comment

Segment KM will be the diameter of the circumcircle of the rectangle. Other possible vertices will lie on that circle. As it happens, none of the offered choices is the same distance from X as point K is.

The attached figure shows the given diagonal and one of an infinite number of possible rectangles KLMN.

The choice of naming the given vertex K is arbitrary.

The quantity of ounces for each bag would be as follows:

walnut= 8.82

almonds = 4.29 and

cashew = 11.89 ounces.

The quantity of walnut used = 21 ounces

The quantity of almonds used = 10.2 ounces

The quantity of cashew used = 28.3 ounces

Total = 59.5ounces.

The total ounces makes up = 25 bags

The ounce of each bag;

walnut = 21/59.5×25/1

= 8.82

almonds= 10.2/59.5 × 25/1

= 4.29

cashew = 28.3/59.5 × 25/1

= 11.89

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The equation is separable. The solution to the initial value problem is y(t) = e^(4 * e^t) , therefore option A is correct.

To determine whether the given equation is separable and solve the initial value problem:

We will follow these steps:

STEP 1: Identify the given differential equation: y'(t) = 4y * e^t
STEP 2: Rewrite the equation in terms of dy/dt: dy/dt = 4y * e^t
STEP 3:Rearrange the terms to separate variables: (1/y) dy = 4 * e^t dt
STEP 4:Integrate both sides: ∫(1/y) dy = ∫4 * e^t dt
STEP 5:Evaluate the integrals: ln|y| = 4 * e^t + C1 (we can drop the absolute value since y > 0)
STEP 6:Solve for y: y = e^(4 * e^t + C1)
STEP 7:Apply the initial condition y(0) = 1: 1 = e^(4 * e^0 + C1)
STEP 8:Solve for C1: C1 = 0
STEP 9:Substitute C1 back into the solution: y(t) = e^(4 * e^t)

The equation is separable. The solution to the initial value problem is y(t) = e^(4 * e^t) (Type an exact answer in terms of e).

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The partial fraction decomposition of the integrand for the integral of x² - 2x - 1 / (x - 3)²(x² + 9) dx is of the form: A / (x - 3) + B / (x - 3)² + (Cx + D) / (x² + 9), where A, B, C, and D are constants.

First, factorize the denominator: (x - 3)²(x² + 9). This gives us three distinct linear factors: (x - 3), (x - 3), and (x² + 9).

Write the partial fraction decomposition using the distinct linear factors as denominators. In this case, we have two (x - 3) terms and one (x² + 9) term:

x² - 2x - 1 / (x - 3)²(x² + 9) = A / (x - 3) + B / (x - 3)² + (Cx + D) / (x² + 9).

Multiply both sides of the equation by the common denominator (x - 3)²(x² + 9) to eliminate the denominators.

Simplify the numerators and equate the corresponding coefficients on both sides of the equation.

Solve the resulting system of equations to find the values of A, B, C, and D.

Substitute the values of A, B, C, and D back into the original partial fraction decomposition.

Therefore, the partial fraction decomposition of the integrand for the given integral is: A / (x - 3) + B / (x - 3)² + (Cx + D) / (x² + 9), where A, B, C, and D are constants

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Answer:

x = 2

Step-by-step explanation:

y = 3x - 7

y = 7 - x

следовательно,

3x - 7 = 7 - x

=> 2x = 14

=> x = 14/7

=> x = 2

The length of the side of the triangle BC is 0. 97 cm

It is important to note that their are different trigonometric identities. These identities are;

From the information given, we have that;

cos θ = adjacent/hypotenuse

sin θ = opposite/hypotenuse

tan θ = opposite/adjacent

Then, we have;

sin 17 = BC/√11

cross multiply the values, we have;

BC = sin 17 (√11)

BC = 0. 97 cm

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Miss Edwards paid about $17.88 for the gasoline.

Price is the amount of money or other valuable consideration that a buyer is willing to pay to acquire a good or service from a seller. In a market economy, prices are determined by the forces of supply and demand.

To estimate how much Miss Edwards paid for the gasoline, we can round the price per gallon to the nearest cent and multiply it by the number of gallons purchased.

The price per gallon is $1.49 9/10, which is closest to $1.50 when rounded to the nearest cent.

So, we can estimate the cost of 1 gallon of gasoline as $1.50.

Therefore, the cost of 11.92 gallons of gasoline would be approximately:

11.92 x $1.50 = $17.88

So, we can estimate that Miss Edwards paid about $17.88 for the gasoline.

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A is invertible if and only if both B and C are invertible.

To prove the converse statement, we need to show that if both B and C are invertible, then A is also invertible.

Assume that B and C are invertible. We need to find the inverse of A.

Let D = BC. Since B and C are invertible, D is also invertible. Moreover, we have:

A = [D 0; 0 I]

where 0 is a square matrix of the same size as B and I is the identity matrix of the same size as C.

To find the inverse of A, we need to find a matrix A-1 such that:

AA-1 = A-1A = I

Let:

A-1 = [E F; G H]

where E, F, G, and H are matrices of appropriate sizes.

Then we have:

AA-1 = [D 0; 0 I][E F; G H] = [DE GF; DG+HI]

and

A-1A = [E F; G H][D 0; 0 I] = [ED FG; GH+I]

Setting these equal to I and equating corresponding entries, we get:

DE = ED = I (since D is invertible)

GF = 0

DG + HI = 0

ED = DE = I (since D is invertible)

FG = 0

GH + I = 0

From the first equation, we have E = D-1. From the second equation, we have F = 0. From the third equation, we have G = -D-1H. Substituting these values into the fourth and sixth equations, we get:

-HD-1H + I = 0

which implies that HD-1H = D-1.

Since D = BC and both B and C are invertible, we have D-1 = C-1B-1. Substituting this into the above equation, we get:

HC-1B-1H = I

which implies that H(BH)-1(C-1H)-1 = I. Since B and C are invertible, BH and C-1H are also invertible. Therefore, H is invertible and we have:

A-1 = [D-1 0; -D-1HC-1B-1 D-1]

Thus, A is invertible if and only if both B and C are invertible.

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(a) A man, a woman, a boy, and a girl can be selected in 312 ways.

(b) A man or a girl can be selected in 57 ways.

(c) One person can be selected in 25 ways.


(a) To select a man, a woman, a boy, and a girl, use the multiplication principle. There are 13 men, 6 women, 2 boys, and 4 girls, so the number of ways is 13 * 6 * 2 * 4 = 312 ways.

(b) To select a man or a girl, there are 13 men and 4 girls, so the number of ways is 13 + 4 = 17 ways.

(c) To select one person, there are a total of 13 men + 6 women + 2 boys + 4 girls = 25 people, so there are 25 ways to choose one person.

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The kayakers are traveling faster downstream (with the current) than upstream (against the current).

1) Variables:

- Speed of the kayaker (unknown, let's call it x)

- Speed of the current = 3 mph (given)

- Distance kayaked one way = 1 mile (given)

- Total distance covered (round trip) = 2 miles (given)

- Total time of the trip = 3 hours 20 minutes = 3.33 hours (converted to hours for convenience)

Table:

Photo attached.

2) The equation to model the problem is:

distance = rate × time

Using this equation for each kayaking portion, we get:

1 = (x - 3) t

1 = (x + 3) t

We also know that the total time of the trip is 3.33 hours:

t + t = 3.33

2t = 3.33

t = 1.665

3) Now we can solve for x by substituting t = 1.665 in either of the above equations:

1 = (x - 3) (1.665)

x - 3 = 0.599

x = 3.599

Thus, the kayakers are paddling at a speed of 3.599 miles per hour.

4) The kayakers are paddling at a speed of 3.599 miles per hour. This solution is obtained by calculating the average speed of the kayakers over the entire trip, taking into account the opposing speed of the river current.

The kayakers are traveling faster downstream (with the current) than upstream (against the current).

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We need to check if a = 1 satisfies the original condition that AB = CB. If a = 1, then CB = √[(1 - 3[tex])^2[/tex] + (3 - 1[tex])^2[/tex]] = √[4 + 4] = 2√(2), which is equal to AB. Therefore, the solution is a = 1 or 5.

First, we need to find the length of AB and CB using the distance formula:

AB = √(1 - (-1)[tex])^2[/tex] + (3 - 1[tex])^2[/tex]] = √[4 + 4] = 2√(2)

CB = √(a - 3[tex])^2[/tex] + (3 - 1[tex])^2[/tex]]

Since AB = CB, we can set the two expressions equal to each other:

2√(2) = √[(a - 3[tex])^2[/tex] + 4]

Squaring both sides, we get:

8 = (a - 3[tex])^2[/tex] + 4

Subtracting 4 from both sides, we get:

4 = (a - 3[tex])^2[/tex]

Taking the square root of both sides, we get:

2 = a - 3 or -2 = a - 3

Solving for a, we get:

a = 5 or a = 1

However, we need to check if a = 1 satisfies the original condition that AB = CB. If a = 1, then CB = √[(1 - 3[tex])^2[/tex] + (3 - 1[tex])^2[/tex]] = √[4 + 4] = 2√(2), which is equal to AB. Therefore, the solution is a = 1 or 5.

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Answer:

Cost of item = $2500 + $200

Cost of item = $2700

Step-by-step explanation:

0_0

The answer of the given question based on the differential equation,  the answer is (C) 20,885.

A differential equation is equation that involves derivatives or differentials of unknown function. These equations describe how aquantity changes over time, space or some other variable. They are used to model a wide variety of phenomena in physics, engineering, biology, economics, and many other fields.

We are given the differential equation:

dy/dt = 0.4y

Utilising the separation of variables, we can resolve this.

dy/y = 0.4 dt

Integrating both sides:

ln|y| = 0.4t + C

where C is the constant of integration. To find C, we use the initial condition that the colony initially had 2000 bacteria, so when t = 0, y = 2000.

ln|2000| = 0 + C

C = ln|2000|

So the equation becomes:

ln|y| = 0.4t + ln|2000|

Simplifying, we get:

|y| = [tex]e^{(0.4t+ln|2000|)}[/tex] = [tex]2000e^{(0.4t)}[/tex]

Since the population of bacteria cannot be negative, we can drop the absolute value signs.

So, the solution to the differential equation is:

[tex]y = 2000e^{(0.4t)}[/tex]

To find the approximate number of bacteria at time t=7, we substitute t=7 into the equation:

[tex]y = 2000e^{(0.4(7)) }[/tex] = 20,885

Consequently, 20,885 bacteria are present at time t=7, which is a rough estimate.

So, the answer is (C) 20,885.

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The characteristic equation for the differential equation y′′ − y = 0 is r^2 - 1 = 0. This has roots r = ±1. Therefore, the general solution to the differential equation is y(t) = c1 e^t + c2 e^(-t).

Using the initial conditions, we can solve for the values of c1 and c2:

y(0) = 5/4 = c1 + c2

y'(0) = -3/4 = c1 - c2

Solving these equations simultaneously, we get c1 = 1/2 and c2 = 3/4. Therefore, the solution to the initial value problem is:

y(t) = 1/2 e^t + 3/4 e^(-t)

To find the minimum value of the solution, we can take the derivative of y(t) and set it equal to 0:

y'(t) = 1/2 e^t - 3/4 e^(-t)

y'(t) = 0 when e^t = (3/2) e^(-t)

e^(2t) = 3/2

t = ln(sqrt(3/2))

Substituting this value of t back into the original equation, we get the minimum value of the solution:

y(ln(sqrt(3/2))) = 1/2 e^(ln(sqrt(3/2))) + 3/4 e^(-ln(sqrt(3/2)))

y(ln(sqrt(3/2))) = sqrt(3)/4 + 3/(4sqrt(3))

y(ln(sqrt(3/2))) = (4sqrt(3) + 3)/(4sqrt(3))

Therefore, the minimum value of the solution is (4sqrt(3) + 3)/(4sqrt(3)) which is approximately 1.068.

To plot the solution for 0 ≤ t ≤ 2, we can use a graphing calculator or software to graph y(t) = 1/2 e^t + 3/4 e^(-t) and then set the viewing window to show the interval [0, 2].

The value of x , given the angle and the lines drawn , would be 142 degrees .

The fact that a straight line was drawn such that x degrees and 38 degrees make up the angles of that line , means that the total angular measure would be 180 degrees because this is the angle of a straight line.

This therefore means that to find the value of x, the formula would be :

x = 180 - 38

x  = 180 - 38

x = 142 degrees

In conclusion, the value of x would be 142 degrees.

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Okay, let's solve this step-by-step:

* Points J, K, L, M have polar coordinates:

J (6, 130°)

K (6, 160°)

L (6, 200°)

M (6, 250°)

* To find the angle between two points, we subtract their theta values (polar angle coordinates).

* Angle KJM = 160° - 130° = 30°

* Angle KLM = 250° - 200° = 50°

* Sum of angles KJM and KLM = 30° + 50° = 80°

* Since the sum of angles in any triangle is 180°, and we have 80°, the remaining angle must be 180° - 80° = 100°.

* Therefore, the sum of angles KJM and KLM is 180°.

Does this make sense? Let me know if you have any other questions!