Find a and b such that f is differentiable everywhere. f(x)={(4 cos(x) text(if ) x < 0,ax b text(if ) x >= 0)

The value of x is after solving the logarithm equation logx⁽⁵¹²⁾ = 3 is 8.

A logarithmic equation is an equation that involves the logarithm of an expression containing a variable.

To solve the logarithm equation, we follow the steps below.

Given:

Step 1:

Therefore,

Step 2:

Convert 512 to index form. (i.e 8³)

Comparing both side of equation 1,

Hence, the value of x is 8.

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2(x+3)=x-4

2x+6=x-4

+6 .     +6

2x=x+2

-x .  -x

x=2

4(5x-2)=2(9x+3)

20x-8=18x+6

+8 .      +8

20x=18x+14

-18x      -18x

2x=14

x=7

The number of variables you have recorded for each of the 243 used cars in an online car dealership is 4.

A variable is a characteristic or attribute of an object or a set of objects that can be measured or recorded. In this case, you have recorded the age, paint color, engine size, and cost of each of the 243 used cars. Each of these characteristics represents a separate variable.

For example, the age of a car is a variable that can be measured or recorded, and it provides information about the car's history and how long it has been in use. The paint color, engine size, and cost of each car are also variables that can be recorded, and they provide additional information about each car's unique characteristics.

In total, you have recorded 4 different variables for each of the 243 used cars, so the number of variables is 4.

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The equation is an identity. Mathias is correct.

Equations are mathematical statement that shows that two expressions are equal. Equality sign is compulsory for an equation.

as example, 2x + 5Y =12 is an equation that has two variables.

An identity is an equation that is considered as true in all the cases whether any value is substituted or not.

given, 2x² + 10x +6 = (x+3) (2x+1) + 3(x+1)

From the right-hand side, (x+3) (2x+1) +3 (x+1)

                                  we expand, {x × 2x + x×1 + 3×2x + 3×1 + 3 ×x + 3×1}

                                                        2x² + x +6x+ 3 +3x +3

                    we collect the same term, 2x²+ x+ 6x+3x + 3+3

                         we add the same term, 2x² +10x +6

                                                          = left hand side

So, the given equation is an identity.

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The dimensions the garden are:

Shorter side: 290 feet

Longer side: 290 feet

Greatest possible area: A = 84,100 square feet.

To maximize the area of a rectangular garden, we need to find the dimensions that use up the available fencing while keeping one side bordering the river.

Let's denote the shorter side of the garden as x feet. Since one side is already bordered by the river and doesn't require fencing, we only need to fence the other three sides.

The total length of the three sides that need fencing is x + x + y + y, where y represents the longer side of the garden.

According to the given information, the total length of fencing available is 1160 feet. Therefore, we can set up the equation:

2x + 2y = 1160

Now, we can solve for y in terms of x:

2y = 1160 - 2x

y = (1160 - 2x) / 2

y = 580 - x

The area of the rectangular garden is given by A = x * y. Substituting the expression for y, we get:

A = x * (580 - x)

A = 580x - \(x^2\)

To find the greatest possible area, we can take the derivative of A with respect to x and set it equal to zero:

dA/dx = 580 - 2x = 0

Solving for x, we find:

2x = 580

x = 290

Substituting this value of x back into the expression for y, we get:

y = 580 - 290

y = 290

Therefore, the dimensions that guarantee the garden has the greatest possible area are:

Shorter side: 290 feet

Longer side: 290 feet

Greatest possible area: A = 290 * 290 = 84,100 square feet.

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Tax Liability = Tax on 10% Bracket + Tax on 12% Bracket + Tax on 22% Bracket + Tax on 24% Bracket

To determine Fred and Ginger's tax liability for 2021, we will use the tax formula and consider the various items of income and expenses provided. Let's go through each category step by step:

Calculate Adjusted Gross Income (AGI):

AGI = (Fred's Salary) + (Ginger's Salary) + (Interest Income on State of Arizona Bonds) + (Interest Income on US Treasury Bonds) + (Qualified Cash Dividends) + (Share of Hollywood Corporation S Corporation Income) + (Short-term Capital Gains) + (15% Long-term Capital Gains) + (Share of RKO Partnership Loss) + (Life Insurance Proceeds)

AGI = $110,000 + $125,000 + $3,000 + $8,000 + $6,000 + $30,000 + $5,000 + $30,000 + (-$10,000) + $150,000

AGI = $547,000

Determine Itemized Deductions:

Itemized Deductions = (Home Mortgage Interest) + (Home Equity Loan Interest) + (Vacation Home Loan Interest) + (Car Loan Interest) + (Home Property Taxes) + (Vacation Home Property Taxes) + (Car Tags) + (Arizona Sales Taxes Paid) + (Medical Insurance Premiums) + (Unreimbursed Medical Bills) + (Charitable Contributions)

Itemized Deductions = $18,000 + $6,000 + $8,400 + $3,000 + $6,000 + $1,800 + $950 + $6,500 + $12,000 + $10,000 + $12,000

Itemized Deductions = $95,650

Calculate Taxable Income:

Taxable Income = AGI - Itemized Deductions

Taxable Income = $547,000 - $95,650

Taxable Income = $451,350

Determine Tax Liability using the Tax Table or Tax Formula:

Based on the provided information, we'll assume Fred and Ginger are filing as Married Filing Jointly for 2021. Using the tax brackets and rates for that filing status, we can calculate their tax liability. Please note that the tax rates and brackets are subject to change, so it's important to refer to the most recent tax regulations.

Tax Liability = (Tax on 10% Bracket) + (Tax on 12% Bracket) + (Tax on 22% Bracket) + (Tax on 24% Bracket)

The taxable income falls into multiple brackets, so we'll calculate the tax liability for each bracket separately:

Tax on 10% Bracket: $0 - $19,900 = $0

Tax on 12% Bracket: $19,901 - $81,050 = ($81,050 - $19,900) * 0.12

Tax on 22% Bracket: $81,051 - $172,750 = ($172,750 - $81,050) * 0.22

Tax on 24% Bracket: $172,751 - $451,350 = ($451,350 - $172,750) * 0.24

Calculate the total tax liability:

Tax Liability = Tax on 10% Bracket + Tax on 12% Bracket + Tax on 22% Bracket + Tax on 24% Bracket

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Answer: 4

Step-by-step explanation: x is multiplied by 4

Answer:

2 31/100

Step-by-step explanation:

Answer:

Slope: 0.1      Y-intercept: 25

Step-by-step explanation:

No matter how many miles you use, you will have to pay $25 right off the bat. To express this, your y-intercept would have to be 25.

For every mile you drive, you have to pay $0.10, or ten cents. This would make our slope 0.1.

Given :

An adults heart beats about 2100 times every 30 minutes.

A baby beats about 2600 times every 20 minutes.

To Find :

How many more heart beat baby have than adult in 60 min .

Solution :

Now , total number of heart beats of adult in 60 min , \(T_a=2100\times 2=4200\)

Also , total number of heart beats of adult in 60 min , \(T_b=2600\times 3=7800\)

Difference between them , \(7800-4200=3600\) beats .

So , baby have 3600 more heart beats than adult .

Hence , this the required solution .

Answer:220 miles

Step-by-step explanation: 88 divided by 4 is 22 then 22 multiplied by 10 is 220

The velocity of the projectile after 2 seconds is 90.4 m/s. The velocity of the projectile after 13 seconds is -68.4 m/s.

The given position function of a free-falling object is as follows: s(t) = -4.9t² + v₀t + s₀ Where, v₀ is the initial velocity s₀ is the initial position of the projectile, therefore, we need to determine the velocity of a projectile after 2 seconds and 13 seconds.

We will use the given formula to determine the velocity of a projectile using its position function as follows: Velocity of a projectile = ds(t)/dt Where, d/dt is the derivative of s(t) with respect to t. Hence, Velocity, v(t) = d/dt[-4.9t² + v₀t + s₀]Differentiating the given equation, we get, v(t) = -9.8t + v₀ We know that the initial velocity of the projectile is 110 m/sv₀ = 110 m/s

Thus, the velocity function of the projectile isv (t) = -9.8t + 110Let's find the velocity of the projectile after 2 seconds. Velocity after 2 seconds, v(2) = -9.8(2) + 110= 90.4 m/s Hence, the velocity of the projectile after 2 seconds is 90.4 m/s.

Velocity after 13 seconds, v(13) = -9.8(13) + 110= -68.4 m/s (rounded to one decimal place), Therefore, the velocity of the projectile after 13 seconds is -68.4 m/s.

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A circle has a center, radius, diameter and points in a plane. Depending on the context, the definition of a circle must contain at least one of the following terms:

Center

Radius

Diameter

Points in a plane

The complete question is as follows:-

James defines a circle as "the set of all the points equidistant from a given point." His statement is not precise enough

because he should specify that

A circle includes its diameter

The set of points is in a plane

A circle includes its radius

The set of points is collinear

A circle is a two-dimensional geometry on the plane having a center point and the circular line is drawn equidistant from the center point.

Based on James' definition of a circle, he needed to specify that the points are in a plane (option c).

As presented in his definition "the set of all the points" can be interpreted in several ways. Some of which are:

the set of points on a line

the set of points in a plane

the set of points in a region

Etc

Of these numerous possible interpretations, James should have specified that the set of points is in a plane

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The parametric equation for the line segment from (−4, 14, 32) to (10, −9, 46), using the parameter t is r(t) = (-4+14t, 14-23t, 32+14t), for 0<= t <=1.

The given points are (-4,14,32) and (10,-9,46).

To find the parametric equations for the line segment we need to follow the below steps.

Step 1: Calculate the direction vector of the line segment. We use the difference of the given points for this purpose.  Let d= direction vector = (10,-9,46)-(-4,14,32)  = (14,-23,14)

Step 2: Choose the initial point on the line segment. We can choose (-4,14,32).

Step 3:  Since there are infinite choices for parameter t, choose any parameter t. We choose t in the range 0<= t <=1, to get the parametric equation for the line segment.

Let the parametric equation for the line segment be r(t).

Then,  r(t) = (-4,14,32) + t(14,-23,14)    

= (-4+14t, 14-23t, 32+14t), for 0<= t <=1.

Thus, the parametric equation for the line segment is r(t) = (-4+14t, 14-23t, 32+14t), for 0<= t <=1.

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Answer:

x=25

Step-by-step explanation:

they are supplementary and adjacent, thus you do 4x-3+3x+8=180, simplify and get 7x+5=180, subtract 5 from 180 which is 175, then ending up with 7x=175 and 175/7 is 25.

Answer:

Step-by-step explanation:

See attachment for the graph of the reflection of f(x) across the y-axis

The function is given as:

f(x) = 1.5(0.5)^x

The rule of reflection across the y-axis is

(x, y) = (-x, y)

So, we have the following equation

f'(x) = 1.5(0.5)^-x

Rewrite the function as

g(x) = 1.5(0.5)^-x

So, we plot the graph of the function g(x)

See attachment for the graph of the reflection of f(x)

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step 1 :

g(x) = 1.5

step 2 :

g(0): 1.5

step 3 :

plot at (0,1.5)

step 4 :

g(1) = 3

g(-1) = .75

step 5 :

plot (1,3) & (-1,0.75)

step 6:

y=0

Answer:

  108°

Step-by-step explanation:

The two given angle are vertical angles, which means they are congruent.

  (x +81)° = 4x°

  81 = 3x . . . . . . . subtract x, divide by °

  27 = x . . . . . . . . divide by 3

  (27 +81)° = 108° . . . . find the angle measure

The two marked angles have measures of 108°.

The maximum value of z is 0, and the values of the decision variables are x1 = 0, x2 = 10, x3 = 0, x4 = 0.

maximize: z = c1x1 + c2x2 + ... + cnxn

subject to

a11x1 + a12x2 + ... + a1nxn ≤ b1

a21x1 + a22x2 + ... + a2nxn ≤ b2

am1x1 + am2x2 + ... + amnxn ≤ bmxi ≥ 0 for all i

In our case,

the given LP is:maximize: z = 36x1 + 30x2 - 3x3 - 4x

subject to:

x1 + x2 - x3 ≤ 5

6x1 + 5x2 - x4 ≤ 10

xi ≥ 0 for all i

We can rewrite the constraints as follows:

x1 + x2 - x3 + x5 = 5  (adding slack variable x5)

6x1 + 5x2 - x4 + x6 = 10  (adding slack variable x6)

Now, we introduce the non-negative variables x7, x8, x9, and x10 for the four decision variables:

x1 = x7

x2 = x8

x3 = x9

x4 = x10

The objective function becomes:

z = 36x7 + 30x8 - 3x9 - 4x10

Now we have the problem in standard form as:

maximize: z = 36x7 + 30x8 - 3x9 - 4x10

subject to:

x7 + x8 - x9 + x5 = 5

6x7 + 5x8 - x10 + x6 = 10

xi ≥ 0 for all i

To apply the simplex algorithm, we initialize the simplex tableau as follows:

  |  Cj   |   x5   |   x6   |   x7   |   x8   |   x9   |   x10  |    RHS  |

---------------------------------------------------------------------------

z |  0    |   0    |   0    |  36    |   30   |   -3   |   -4   |    0    |

---------------------------------------------------------------------------

x5|   0   |   1    |   0    |   1    |   1    |   -1   |   0    |    5    |

---------------------------------------------------------------------------

x6|   0   |   0    |   1    |   6    |   5    |   0    |   -1   |   10    |

---------------------------------------------------------------------------

Now, we can proceed with the simplex algorithm to find the optimal solution. I'll perform the iterations step by step:

Iteration 1:

1. Choose the most negative coefficient in the 'z' row, which is -4.

2. Choose the pivot column as 'x10' (corresponding to the most negative coefficient).

3. Calculate the ratios (RHS / pivot column coefficient) to find the pivot row. We select the row with the smallest non-negative ratio.

Ratios: 5/0 = undefined, 10/(-4) = -2.5

4. Pivot at the intersection of the pivot row and column. Divide the pivot row by the pivot element to make the pivot element 1.

5. Perform row operations to

make all other elements in the pivot column zero.

After performing these steps, we get the updated simplex tableau:

  |  Cj   |   x5   |   x6   |   x7   |   x8   |   x9   |   x10  |    RHS  |

---------------------------------------------------------------------------

z |  0    |   0    |  0.4   |  36    |   30   |   -3   |   0    |   12    |

---------------------------------------------------------------------------

x5|   0   |   1    |  -0.2  |   1    |   1    |   -1   |   0    |   5     |

---------------------------------------------------------------------------

x10|   0  |   0    |   0.2  |   1.2  |   1   |   0    |   1    |   2.5   |

---------------------------------------------------------------------------

Iteration 2:

1. Choose the most negative coefficient in the 'z' row, which is -3.

2. Choose the pivot column as 'x9' (corresponding to the most negative coefficient).

3. Calculate the ratios (RHS / pivot column coefficient) to find the pivot row. We select the row with the smallest non-negative ratio.

Ratios: 12/(-3) = -4, 5/(-0.2) = -25, 2.5/0.2 = 12.5

4. Pivot at the intersection of the pivot row and column. Divide the pivot row by the pivot element to make the pivot element 1.

5. Perform row operations to make all other elements in the pivot column zero.

After performing these steps, we get the updated simplex tableau:

  |  Cj   |   x5   |   x6   |   x7   |   x8   |   x9   |   x10  |    RHS  |

---------------------------------------------------------------------------

z |  0    |   0    |  0.8   |  34    |   30   |   0    |   4    |   0     |

---------------------------------------------------------------------------

x5|   0   |   1    |  -0.4  |   0.6  |   1    |   5   |   -2   |   10    |

---------------------------------------------------------------------------

x9|   0   |   0    |   1    |   6    |   5    |   0   |   -5   |   12.5  |

---------------------------------------------------------------------------

Iteration 3:

No negative coefficients in the 'z' row, so the optimal solution has been reached.The optimal solution is:

z = 0

x1 = x7 = 0

x2 = x8 = 10

x3 = x9 = 0

x4 = x10 = 0

x5 = 10

x6 = 0

Therefore, the maximum value of z is 0, and the values of the decision variables are x1 = 0, x2 = 10, x3 = 0, x4 = 0.

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Answer:

64,890

Step-by-step explanation:

105×3 =315

103×2= 206

315×206= 64,890

Answer:

6.489× 10⁴

Step-by-step explanation:

(3×105)×(2×103)

315× 206

64,890

6.489×10⁴

Answer:

y= 3/2 x -2

Step-by-step explanation:

I guess you want to find the y=mx+b form

So you do +3x on both sides which will gave you: 2y= 3x-4

Then, you will divide by two on both sides to find the y value: 2y/2 = (3x-4)/ 2

So 3/2 is the fraction like the answer above and -4/2 = -2

Hope that helps :)

Answer:

B x= -9

Step-by-step explanation:

if you graph x equals negative 9 you always have a vertical line

Answer: b) x = -9

a) gives you a diagonal line when you graph it

b) gives you a vertical line since you only have the x-value, and it passes through the point (-9,10) when you graph it

c) gives you a horizontal line when you graph it since you only have the y-value

A set of values for the decision variables that satisfy all the constraints and yields the best objective function value is a feasible solution that optimizes the objective function.

In optimization problems, decision variables are the quantities that we can control or adjust to achieve a desired outcome. Constraints are the limitations or conditions that these decision variables must satisfy. The objective function represents the goal or objective we want to optimize.

A feasible solution refers to a set of values for the decision variables that satisfy all the given constraints. This means that the solution meets all the specified requirements and does not violate any constraints. However, there can be multiple feasible solutions that meet the constraints.

Among these feasible solutions, the one that yields the best objective function value is the optimal solution. The objective function value is a measure of how well the solution aligns with the desired objective. The goal is typically to maximize or minimize this objective function value, depending on the problem.

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Answer:

A, B, E

Step-by-step explanation:

A) 40 + 40

B) 2/22 +20

E)22 +15) +22+5)

Answer:

width = 15 inches

length = 30 inches

Step-by-step explanation:

\(\sf \boxed{\text{\bf Perimeter of rectangle =2*length + 2*width}}\)

  Let width = x inches

     Length =  2*x = 2x inches

                 Perimeter = 90 inches

2 *length  + 2* width = 90

            2*2x  + 2*x     = 90

                   4x + 2x    = 90

Combine like terms,

                             6x = 90

Divide both sides by 6,

                                x = 90/6

                                x = 15

Dimensions:

Width = 15 inches

Length = 2*15 = 30 inches.

Answer:

p(x) = -4*x^4  - 2*x^3 + 3*x^2 + 1*x - 2

Step-by-step explanation:

We want to find a polynomial p(x), such that if we add that polynomial to:

f(x) = 4*x^4 + 2*x^3 - 2*x^2 + x - 1

we get:

g(x) = x^2 + 2*x - 3

This is:

f(x) + p(x) = g(x)

Notice that f(x) is a polynomial of degree 4 and g(x) is a polinomial of degree 2, so p(x) must be also a polynomial of degree 4.

p(x) = a*x^4 + b*x^3 + c*x^2 + d*x + e

Then we get:

(4*x^4 + 2*x^3 - 2*x^2 + x - 1) + (a*x^4 + b*x^3 + c*x^2 + d*x + e) = x^2 + 2*x - 3

We can simplify the left side to:

(4 + a)*x^4 + (2 + b)*x^3 + (-2 + c)*x^2 + (1 + d)*x + (-1 + e) =  x^2 + 2*x - 3

Because in the right side we do not have terms with exponent 4 and 3, we must have that:

4 + a = 0

2 + b = 0

and for the other exponents of x we just match the exponent in the left side with the correspondent one in the right side:

(-2 + c) = 1

(1 + d) = 2

(-1 + e) = -3

Solving the 5 equations we get:

a = -4

b = -2

c = 1 + 2 = 3

d = 2 - 1 = 1

e = -3 + 1 = -2

Then the equation for p(x) is:

p(x) = -4*x^4  - 2*x^3 + 3*x^2 + 1*x - 2

According to the given percentages, 18% of the students at the orientation are male arts majors.

The statement correctly calculates that 60% of the students at the orientation are men and 30% are arts majors.

To determine the percentage of students who are male arts majors, we multiply these two percentages together: 60% x 30% = 18%. Therefore, 18% of the students at the orientation are male arts majors.

This calculation follows the principles of probability, where the intersection of two events (being a male and being an arts major) is determined by multiplying the probabilities of each event occurring individually.

In this case, it results in 18% of the students meeting both criteria.


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Question - Sixty percent of the students at an orientation are men and 30% of the students at the orientation are arts majors. Therefore, 60% X 30% = 18% of the students at the orientation are male arts majors.