Find the measures of angle A and B. Round to the nearest degree.

The solution of the quadratic equation  4e² - 15e = -4 in the exact form is e = (15 + √161)/8 or e = (15 - √161)/8

To solve the quadratic equation 4e² - 15e = -4, we can rearrange it into standard form as follows,

4e² - 15e + 4 = 0. We can then use the quadratic formula, which states that for an equation in the form ax² + bx + c = 0, the solutions are given by,

x = (-b ± √(b² - 4ac)) / 2a

Applying this formula to our equation, we have,

e = (-(-15) ± √((-15)² - 4(4)(4))) / 2(4)

Simplifying this expression, we get,

e = (15 ± √(225 - 64)) / 8

e = (15 ± √161) / 8

Therefore, the solutions to the equation 4e² - 15e = -4 are:

e = (15 + √161) / 8 or e = (15 - √161) / 8

These are exact solutions in radical form.

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Answer:

To find the vertices and name two points on the minor axis of the ellipse represented by the equation 9x^2+y^2-18x-6y+9=0, we need to first put it in standard form by completing the square for both x and y terms.

Starting with the x terms:

9x^2 - 18x = 0

9(x^2 - 2x) = 0

We need to add and subtract (2/2)^2 = 1 to complete the square inside the parentheses:

9(x^2 - 2x + 1 - 1) = 0

9((x-1)^2 - 1) = 0

9(x-1)^2 - 9 = 0

9(x-1)^2 = 9

(x-1)^2 = 1

x-1 = ±1

x = 2 or 0

Now we can do the same for the y terms:

y^2 - 6y = 0

y^2 - 6y + 9 - 9 = 0

(y-3)^2 - 9 = 0

(y-3)^2 = 9

y-3 = ±3

y = 6 or 0

So the center of the ellipse is (1, 3), the major axis is along the x-axis with a length of 2a = 2√(9/1) = 6, and the minor axis is along the y-axis with a length of 2b = 2√(1/9) = 2/3.

The vertices are the points on the major axis that are farthest from the center. Since the major axis is along the x-axis, the vertices will be (1±3, 3), or (4, 3) and (-2, 3).

To find two points on the minor axis, we can use the center and the length of the minor axis. Since the minor axis is along the y-axis, we can add or subtract the length of the minor axis from the y-coordinate of the center to find the two points. Therefore, the two points on the minor axis are (1, 3±1/3), or approximately (1, 10/3) and (1, 8/3).

Step-by-step explanation:

The critical point (4,-8) of given function is a saddle point. Therefore, the answer is D.

To find the critical point(s) of the function, we need to find where the gradient of the function is zero or undefined.

The gradient of f(x,y) is:

∇f(x,y) = (2x+4y-24, 2y+4x)

To find the critical points, we need to solve for ∇f(x,y) = 0:

2x+4y-24 = 0 (1)

2y+4x = 0 (2)

From equation (2), we can solve for y in terms of x:

y = -2x

Substituting this into equation (1), we get:

2x + 4(-2x) - 24 = 0

Simplifying, we get:

x = 4

Substituting x = 4 into equation (2), we get:

y = -8

Therefore, the only critical point of f(x,y) is (4,-8).

To determine whether this critical point is a local minimum, local maximum, or saddle point, we need to use the Second Derivative Test.

The Hessian matrix of f(x,y) is:

H = [2 4]

[4 2]

The determinant of H is:

det(H) = (2)(2) - (4)(4) = -12

Since det(H) is negative, the critical point (4,-8) is a saddle point. Therefore, the answer is D.

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The correct answer is: It is necessary to apply both the decomposition and the augmentation rules to F in order to form a minimal cover.

To form a minimal cover of a set of functional dependencies, we need to apply the decomposition rule, which involves breaking down each dependency in F into its simplest form, and the augmentation rule, which involves adding any missing attributes to the right-hand side of each dependency. In this case, we need to apply both rules to F to obtain a minimal cover.

For example, applying the decomposition rule to UVX->UW yields two dependencies: UV->UW and UX->UW. Applying the augmentation rule to UX->ZV yields UX->ZVY. Continuing in this way, we can obtain a minimal cover for F, which is:

UV->UW
UX->ZVY
VU->Y
V->Y
W->VY
a. It is necessary and sufficient to remove a dependency W->Y from F to form a minimal cover.

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The decision complexity advantage for 10 decisions with two alternatives compared to one decision with 20 alternatives is 3.31. This can be answered by the concept of Log.

To calculate the decision complexity advantage, we need to first plug in the given values for a and b into the formula RT = a + b log2(N), where N is the number of alternatives.

For 10 decisions with two alternatives each, N = 2¹⁰ = 1024. Thus, RT = 1 + 2 log2(1024) = 22 seconds.

For one decision with 20 alternatives, N = 20. Thus, RT = 1 + 2 log2(20) = 6.64 seconds.

The decision complexity advantage is calculated by taking the ratio of the RT values: 22/6.64 = 3.31. This means that making 10 decisions with two alternatives each is 3.31 times faster than making one decision with 20 alternatives.

Therefore, the decision complexity advantage for 10 decisions with two alternatives compared to one decision with 20 alternatives is 3.31.

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A quadratic equation is a polynomial equation of the second degree, meaning it contains at least one variable that is squared but no variables that are raised to a higher power. The general form of a quadratic equation in one variable (usually represented by x) is:

�

�

2

+

�

�

+

�

=

0

,

ax

2

+bx+c=0,

where a, b, and c are constants (numbers) and a is not equal to zero. The term ax^2 is called the quadratic term, bx is the linear term, and c is the constant term.

To solve a quadratic equation, we can use the quadratic formula:

�

=

−

�

±

�

2

−

4

�

�

2

�

.

x=

2a

−b±

b

2

−4ac

​

​

.

This formula gives us the solutions (values of x) for any quadratic equation in the standard form. The expression under the square root, b^2 - 4ac, is called the discriminant of the quadratic equation.

The discriminant can tell us a lot about the nature of the solutions of the quadratic equation. If the discriminant is positive, then the quadratic equation has two distinct real solutions. If the discriminant is zero, then the quadratic equation has one real solution, called a double root or a repeated root. If the discriminant is negative, then the quadratic equation has two complex (non-real) solutions, which are conjugates of each other.

The next three terms in the given geometric sequence are 24, 48, and 96

To find the next terms in the geometric sequence 3, 6, 12, we need to find the common ratio (r) first.

r = 6/3 = 2

Now we can use the formula for the nth term of a geometric sequence:

an = a1 * r^(n-1)

where n+1 is the nth term, a1 is the first term, r is the common ratio, and n is the term number.

Using this formula, we can find:

a4 = 24 (3 * 2^3)
a5 = 48 (3 * 2^4)
a6 = 96 (3 * 2^5)

Therefore, the next three terms in the sequence are 24, 48, and 96.

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The values of the missing sides and angles using trigonometric ratios are:

b = 7.06

c = 3.76

C = 28°

The six trigonometric ratios are sine, cosine, tangent, cosecant, secant, and cotangent.

The symbols used for them are:

sine: sin

cosine: cos

tangent: tan

cosecant: csc

secant: sec

cotangent: cot

The trigonometric ratios are defined as the ratio of the sides in right triangles.

Using trigonometric ratios, we have:

b/8 = sin 62

b = 8 * sin 62

b = 7.06

Similarly:

c/8 = cos 62

c = 8 * cos 62

c = 3.76

Sum of angles in a triangle is 180 degrees. Thus:

C = 180 - (90 + 62)

C = 28°

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1. The standard matrix A for r

A = [0 -1]
     [1  0]

2. The standard matrix B for S

B = [0 1]
     [1 0]

C = BA

3. To find the standard matrix C for T

C = [1  0]
     [0 -1]

4. The correct answer is E. C=BA.

Let's address each part of the question step by step:

1. The standard matrix A for r (counterclockwise rotation by π/2 radians) can be found using the following formula:

A = [cos(π/2) -sin(π/2)]
     [sin(π/2) cos(π/2)]

A = [0 -1]
     [1  0]

2. The standard matrix B for S (reflection about the line y=x) can be found by transforming the standard basis vectors:

B = [0 1]
     [1 0]

3. To find the standard matrix C for T (counterclockwise rotation by π/2 radians followed by reflection about the line y=x), we can compute the product of the matrices A and B:

C = BA
C = [0 1] [0 -1]
     [1 0] [1  0]

C = [1  0]
     [0 -1]

4. Since I is equal to the composition S∘R, we can obtain C from A and B using the following equation:

C = BA

So, the correct answer is E. C=BA.

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The solution to Ax=b using the LU factorization is: x = [27 -23/4 -30]

To find the solution to the system Ax=b using the LU factorization:

We need to first decompose the matrix A into its lower and upper triangular matrices L and U respectively, such that A = LU.

Using the given LU factorization of A, we can write:

L = [1 0 0] [1 0 0] [-1 3 1]

U = [2 3 4] [0 -4 3] [0 0 -1]

Next, we need to solve the system LY=b. We can substitute L and Y with their corresponding matrices and variables respectively:

[1 0 0] [1 0 0] [-1 3 1] [y1 y2 y3] = [4 17 43]

Simplifying this system, we get:

y1 = 4

y2 = 17

-y1 + 3y2 + y3 = 43

Solving for y3, we get:

y3 = 30

Now that we have the values for Y, we can solve the system Ux=Y to get the solution to Ax=b.

We can substitute U and X with their corresponding matrices and variables respectively:

[2 3 4] [0 -4 3] [0 0 -1] [x1 x2 x3] = [y1 y2 y3]

Simplifying this system, we get:

2x1 + 3x2 + 4x3 = 4

-4x2 + 3x3 = 17

-x3 = 30

Solving for x3, we get:

x3 = -30

Substituting x3 into the second equation, we get:

-4x2 + 3(-30) = 17

Solving for x2, we get:

x2 = -23/4

Substituting x2 and x3 into the first equation, we get:

2x1 + 3(-23/4) + 4(-30) = 4

Solving for x1, we get:

x1 = 27

Therefore, the solution to Ax=b using the LU factorization is:

x = [27 -23/4 -30]

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When the unit rate of the horse and the fox is compared, the statement that is true about them will be that The fox was 0.3 miles/minute faster than the horse. That is option C.

From the graph,

30 miles distance covered by the horse = 60 mins

But 60 mins = 1 hours

Therefore, the rate of distance covered by the horse = 30 miles/hr.

But the rate of distance covered in miles/ min = 5/10 = 0.5 miles/min.

If the fox covers 0.8miles/min then the difference between it and the horse = 0.8-0.5 = 0.3miles/min.

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Your answer: The elasticity of the demand function 2p 3q = 90 at the price p = 15 is -0.5.

To find the elasticity of the demand function, we need to use the following formula:

Elasticity = (dq/dp) * (p/q)

where dq/dp is the derivative of q with respect to p, and (p/q) is the ratio of the two variables at a given point.

First, we need to solve the demand function for q in terms of p:

2p + 3q = 90

3q = 90 - 2p

q = (90 - 2p)/3

Next, we need to find the derivative of q with respect to p:

dq/dp = (-2/3)

Finally, we can plug in the values for p and q to find the elasticity at p = 15:

q = (90 - 2(15))/3 = 20

(p/q) = 15/20 = 0.75

Elasticity = (-2/3) * (15/20) = -0.5

Therefore, the elasticity of the demand function 2p + 3q = 90 at the price p = 15 is -0.5. This means that a 1% increase in price would lead to a 0.5% decrease in quantity demanded.

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Answer:

x = 225

Step-by-step explanation:

Pythagorean Theorem: a^2 + b^2 = c^2, where a and b are the side lengths and c is the hypotenuse.

Substituting in the values:

(x - 3)^2 + 9^2 = x^2

Then, we isolate x:

(x - 3)^2 + 81 = (x - 3)(x - 3) + 81 = x^2 - 6x + 9 + 81 = x^2 - 6x + 90 = x^2

(Subtract x^2 from both sides)

- 6x + 90 = 0

(Add 6x to both sides, I also flipped the equation to put x on the left side)

6x = 90

(Divide both sides by 6)

x = 15

To double-check, substitute x with 15:

(15 - 3)^2 + 9^2 = 15^2

Simplify:

144 + 81 = 225 (true)

Answer:

One possible equation that fits the given domain and range is:

y = (x - 3)^2

This is a quadratic function that opens upwards and has its vertex at the point (3,0). It is defined for all real numbers except x = 0, and takes on only non-negative values, which means its range is [0,infinity).

The correct answer is d. None of these.

a. {L_L, LLU, LUL, LUU, ULL, ULU, UUL, UUU} represents the sample space of possible outcomes, where L represents a locked door and U represents an unlocked door. Each outcome represents a possible combination of locked and unlocked doors that Nanette may encounter.

b. {LLU, LUL, ULL, UUL, ULL, LUU} is not a complete sample space, as it is missing some possible outcomes. For example, the outcome where all doors are locked (LLL) is not included.

c. {LLL, UUU} is also not a complete sample space, as it only includes two possible outcomes. There are other possible combinations of locked and unlocked doors that are not represented.

Therefore, the correct answer is d. None of these. The complete sample space would include all possible combinations of locked and unlocked doors for the three doors that Nanette must pass through.

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The rule that describes a 90° clockwise rotation is (x, y) → (-y, x).

A 90° clockwise rotation in a two-dimensional Cartesian coordinate system involves rotating points 90 degrees in the clockwise direction around the origin (0,0) on the x-y plane.

In this rotation, the new x-coordinate becomes the negative of the original y-coordinate, and the new y-coordinate becomes the original x-coordinate.

So for the given option, we can see clearly that the rule that describes a 90° clockwise rotation is (x, y) → (-y, x).

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The simplified form of the expression (3d) (-5d²) (6d)⁴ is -19440d⁷.

Given the expression in the question:

(3d) (-5d²) (6d)⁴

To simplify the expression (3d)(-5d²)(6d)⁴, we need to expand the brackets and perform the multiplication of the terms.

(6d)⁴ = ( 6⁴ d⁴) = 1296d⁴

Hence, we have:

(3d)(-5d²)(1296d⁴)

Next , we can multiply the coefficients 3, -5, and 1296, to get -90:

-19440

Next, we can multiply the variables d, d², and d⁴, to get:

d⁷

So putting it all together, we get:
-19440d⁷

Therefore, the simplified form is -19440d⁷.

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Answer:

[tex]\left \{ {{a_1=1} \atop {a_n=a_{n-1}-5}} \right.[/tex]

Step-by-step explanation:

The recursive formula of an arithmetic sequence is[tex]\left \{ {{a_1=x} \atop {a_n=a_{n-1}+d}} \right.[/tex]. Plugging in each value ([tex]a_1 = 1, d=-5[/tex]) gives us the recursive formula  [tex]\left \{ {{a_1=1} \atop {a_n=a_{n-1}-5}} \right.[/tex].

You can be​ 90% confident that the population mean price of the stock is between the bounds of the​ 90% confidence​ interval, and​ 95% confident for the​ 95% interval.

Given data ,

The problem states that a random sample of 43 business days was taken, and the mean closing price of the stock in that sample was $112.15. The population standard deviation is assumed to be $9.95. Based on this information, a confidence interval can be calculated for the population mean.

Now , A wider interval results from a greater confidence level since it calls for more assurance.

If you compare the offered alternatives, option C accurately indicates that you can have a 90% confidence interval for the population mean price of the stock being inside the boundaries, and a 95% confidence interval. Because a 90% confidence interval demands more assurance than a 95% confidence interval, it is smaller. As a result, the population mean is more likely to fall inside the 90% confidence interval's boundaries.

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Answer:

The value of y is -13.

Step-by-step explanation:

Here, we will use the below following steps to find a solution using the transposition method:

[tex]\begin{gathered} \end{gathered}[/tex]

[tex]\longrightarrow\sf{2y + 6 = y - 7}[/tex]

[tex]\longrightarrow\sf{2y - y = - 7 - 6}[/tex]

[tex]\longrightarrow\sf{\underline{\underline{y = - 13}}}[/tex]

Hence, the value of y is -13.

[tex]\begin{gathered} \end{gathered}[/tex]

[tex]\longrightarrow\sf{2y + 6 = y - 7}[/tex]

Substituting the value of y.

[tex]\longrightarrow\sf{2 \times - 13 + 6 = - 13 - 7}[/tex]

[tex]\longrightarrow\sf{ - 26+ 6 = - 20}[/tex]

[tex]\longrightarrow\sf{ - 20 = - 20}[/tex]

[tex]\longrightarrow\sf{\underline{\underline{LHS = RHS}}}[/tex]

Hence, verified!

Given that x is a continuous variable with the probability density function f(x) = c(10-x)^2 for 0 < x < 10, we need to find the value of c.

Step 1: Understand that for a probability density function, the total area under the curve must equal 1. Mathematically, this is expressed as:

∫[f(x)] dx = 1, with integration limits from 0 to 10.

Step 2: Substitute f(x) with the given function and integrate:

∫[c(10-x)^2] dx from 0 to 10 = 1

Step 3: Perform the integration:

c ∫[(10-x)^2] dx from 0 to 10 = 1

Step 4: Apply the power rule for integration:

c[(10-x)^3 / -3] from 0 to 10 = 1

Step 5: Substitute the integration limits:

c[(-1000)/-3 - (0)/-3] = 1

Step 6: Solve for c:

(1000/3)c = 1

c = 3/1000

c = 0.003

So the probability density function f(x) = 0.003(10-x)^2 for 0 < x < 10.

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The conclusion that is true about f(x) and g(x) based on the table of values is: The function f(x) and g(x) are reflections over the y-axis.    

We know that the rule that describes the reflection over the y-axis is:

       (x,y) → (-x,y)

Hence, if we have a function f(x) as:

f(x) = 2ˣ

Then it's reflection over the y-axis is:

f(-x) = 2⁻ˣ

f(-x) = (¹/₂)⁻ˣ

Thus:

g(x) = (¹/₂)⁻ˣ

Hence, they are reflection over the y-axis.

Also, we know that the exponential function of the type:

y = abˣ                

where a > 0 is an increasing function if b>1 and is a decreasing function if: 0<b<1

Hence, f(x) is a increasing function and g(x) is a decreasing function.

Also, the initial value of a function is the value of function when x=0

when x=0 we see that both f(x)=g(x)=1

i.e. Both f(x) and g(x) have same initial value.

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The conclusion that is true about f(x) and g(x) based on the table of values is: The function f(x) and g(x) are reflections over the y-axis.    

We know that the rule that describes the reflection over the y-axis is:

       (x,y) → (-x,y)

Hence, if we have a function f(x) as:

f(x) = 2ˣ

Then it's reflection over the y-axis is:

f(-x) = 2⁻ˣ

f(-x) = (¹/₂)⁻ˣ

Thus:

g(x) = (¹/₂)⁻ˣ

Hence, they are reflection over the y-axis.

Also, we know that the exponential function of the type:

y = abˣ                

where a > 0 is an increasing function if b>1 and is a decreasing function if: 0<b<1

Hence, f(x) is a increasing function and g(x) is a decreasing function.

Also, the initial value of a function is the value of function when x=0

when x=0 we see that both f(x)=g(x)=1

i.e. Both f(x) and g(x) have same initial value.

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The boat that Claire bought 21 years ago, which is now worth £2,980 and depreciated at a rate of 1.25% per year was bought for £3,880. 94.

The depreciated value is the original cost less the accumulated depreciation.

Given the depreciated value and the annual depreciation rate, we can determine the original cost as follows:

The depreciation period = 21 years

Annual depreciation rate = 1.25%

The depreciated value of the boat = £2,980

Depreciation factor = (100 - 1.25)^21

= 0.9875^21

= 0.7678549

Proportionately, £2,980 = 0.7678549, while the original purchase price = £3,880. 94 (£2,980 ÷ 0.7678549)

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Since your retirement fund earns a 4.03% annual interest rate compounded continuously, we'll need to use the continuous compounding formula: A(t) = P * e^(rt)

where:
A(t) = value of the account after t years
P = principal amount (initial investment)
e = the base of the natural logarithm, approximately 2.718
r = interest rate (as a decimal)
t = number of years

Given that you've invested $25,000 (P) at a 4.03% interest rate (r = 0.0403), we'll find the value of the account after 10 years (t = 10).
A(10) = 25000 * e^(0.0403 * 10)

Now, calculate the value:
A(10) = 25000 * e^0.403
A(10) = 25000 * 1.4963 (rounded to 4 decimal places)

Finally, find the total value:
A(10) = 37357.50

After 10 years, the value of the account will be $37,357.50 (rounded to 2 decimal places).

Note that there is no indication of fraud in this scenario, and the interest rate used is 4.03%.

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The correct graph of equation on the coordinate plane is shown in figure.

We know that;

The equation of line with slope m and y intercept at point b is given as;

y = mx + b

Here, The equation is,

y = 2/3x + 1

Hence, Slope of equation is, 2/3

And, Y - intercept of the equation is, 1

Thus, The correct graph of equation on the coordinate plane is shown in figure.

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The equation 2x^2 - 9x^2y^2z^2 = 3 in cylindrical coordinates is 2r^2 * cos^2(θ) - 9r^4 * cos^2(θ) * sin^2(θ) * z^2 = 3 and the equation z = 7x^2 - 7y^2 in cylindrical coordinates is z = 7r^2 * cos^2(θ) - 7r^2 * sin^2(θ).

The cylindrical coordinate system uses three parameters: radius (r), azimuthal angle (θ), and height (z). To convert from Cartesian coordinates (x, y, z) to cylindrical coordinates (r, θ, z), we use the following relations:

x = r * cos(θ)
y = r * sin(θ)
z = z

(a) 2x^2 - 9x^2y^2z^2 = 3
Replace x and y with their cylindrical counterparts:

2(r * cos(θ))^2 - 9(r * cos(θ))^2(r * sin(θ))^2z^2 = 3

Simplify the equation:

2r^2 * cos^2(θ) - 9r^4 * cos^2(θ) * sin^2(θ) * z^2 = 3

This is the equation 2x^2 - 9x^2y^2z^2 = 3 in cylindrical coordinates.

(b) z = 7x^2 - 7y^2
Replace x and y with their cylindrical counterparts:

z = 7(r * cos(θ))^2 - 7(r * sin(θ))^2

Simplify the equation:

z = 7r^2 * cos^2(θ) - 7r^2 * sin^2(θ)

This is the equation z = 7x^2 - 7y^2 in cylindrical coordinates.

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The triangular prism has 3 cylindrical holes with a diameter of 4 cm. The volume of each hole is approximately 60π cubic centimeters, so the total volume of all three holes is about 180π cubic centimeters.

To find the volume of cylindrical holes in the triangular prism, we need to calculate the volume of one cylinder and then multiply it by three (since there are three cylindrical holes).

Volume of one cylinder = πr²h, where r is the radius of the cylinder and h is the height.

Given the diameter of the cylindrical hole is 4 cm, we can find the radius by dividing it by 2

radius (r) = 4 cm ÷ 2 = 2 cm

The height of the cylinder is the same as the length of the prism, which is 15 cm.

Volume of one cylinder = π(2 cm)² × 15 cm

= 60π cm³

Since there are three cylindrical holes, the total volume of the holes is

Total volume of cylindrical holes = 3 × 60π cm³

= 180π cm³

Therefore, the volume of the three cylindrical holes is 180π cubic centimeters.

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

" Pls help (part 1)

Find the volume of 3 cylindrical holes.

Give step by step explanation! "--

The expression for the temperature of the coffee at time t is:

T(t) = 20 ± (T0 - 20) [tex]e^{(-kt) }[/tex]

What is Algebraic expression ?

An algebraic expression is a combination of variables, numbers, and mathematical operations, such as addition, subtraction, multiplication, division, and exponentiation. Algebraic expressions can be used to represent a wide range of mathematical relationships and formulas in a concise and flexible manner.

The differential equation we discussed in Exercise 9.2.28 is:

dT÷dt = -k(T-20)

where T is the temperature of the coffee in Celsius, t is time in minutes, and k is a constant that depends on the properties of the coffee cup and the room.

To solve this differential equation, we need to separate the variables and integrate both sides.

dT  ÷ (T-20) = -k dt

Integrating both sides:

ln|T-20| = -kt + C

where C is an arbitrary constant of integration.

To solve for T, we exponentiate both sides:

|T-20| =[tex]e^{(-kt + C) }[/tex]

Using the property of absolute values, we can write:

T-20 = ± [tex]e^{(-kt + C) }[/tex]

or

T = 20 ± [tex]e^{(-kt + C) }[/tex]

We can determine the sign of the exponential term by specifying the initial temperature of the coffee. If the initial temperature is above 20°C, then the temperature of the coffee will decay towards 20°C, and we take the negative sign in the exponential term. If the initial temperature is below 20°C, then the temperature of the coffee will increase towards 20°C, and we take the positive sign in the exponential term.

To determine the value of the constant C, we use the initial temperature of the coffee. If the initial temperature is T0, then we have:

T(t=0) = T0 = 20 ± [tex]e^{C }[/tex]

Solving for C, we get:

C = ln(T0 - 20) if we took the negative sign in the exponential term

or

C = ln(T0 - 20) if we took the positive sign in the exponential term.

Therefore, the expression for the temperature of the coffee at time t is:

T(t) = 20 ± (T0 - 20) [tex]e^{(-kt) }[/tex]

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Answer:

Lim
x - 1

Step-by-step explanation:

(x-1)
(|x-1|)

(a) The fundamental system of differential equation is X(t) = c1 [tex]e^\((\lambda 1t)[/tex]v1 + c2 [tex]e^\((\lambda 1t)[/tex]v2

(b) The second-order linear differential equation with constant coefficients is d²x/dt² = (ad - bc)dx/dt + ([tex]a^2d + b^2c[/tex])x

(a) The homogeneous system of two linear differential equations with constant coefficients can be written as:

where a, b, c, and d are constants.

We can write this system as X' = AX, where X =[tex][x, y]^T[/tex] and A is the 2x2 matrix:

To find a fundamental system of differential equations, we need to find the eigenvalues and eigenvectors of A.

The characteristic equation of A is:

The eigenvalues of A are the roots of the characteristic equation:

λ1,2 = (a+d ± [tex]\sqrt^(a+d)^2[/tex] - 4(ad-bc))) / 2

The eigenvectors of A are the solutions to the equation (A - λI)v = 0, where v is a non-zero vector.

If λ1 and λ2 are distinct eigenvalues, then the eigenvectors corresponding to each eigenvalue form a fundamental system of differential equations. Specifically, if v1 and v2 are eigenvectors corresponding to λ1 and λ2, respectively, then the solutions to the differential equation X' = AX are given by:

X(t) = c1 [tex]e^\((\lambda 1t)[/tex] v1 + c2 [tex]e^\((\lambda 1t)[/tex] v2

where c1 and c2 are constants determined by the initial conditions.

If λ1 and λ2 are not distinct (i.e., they are repeated), then we need to find a set of linearly independent eigenvectors to form a fundamental system of differential equations. In this case, we use the method of generalized eigenvectors.

(b) To rewrite the system of differential equations as one 2nd order linear differential equation, we can differentiate the first equation with respect to t to obtain:

Substituting the second equation into the last expression, we get:

d²x/dt² = (a² + bc)x + (ab + bd)(-cx + d(dx/dt))

Simplifying, we obtain:

d²x/dt² = (ad - bc)dx/dt + (a²d + b²c)x

This is a second-order linear differential equation with constant coefficients.

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