Find the function value, if possible. (If an answer is undefined, enter UNDEFINED.)
f(x) = 3x − 6
a)f(-3)
(c) f(x + 3)

The values of cos, tan, and sec cannot be determined as they involve taking the square root of a negative number, resulting in undefined values.

If sin(θ) = 45, where 0 ≤ θ ≤ π/2, we can use the Pythagorean identity to find the values of other trigonometric functions.

cos(θ) = √(1 - sin²(θ)) = √(1 - 45²) = √(1 - 2025) = √(-2024) (The value is undefined as the square root of a negative number is not a real number)

tan(θ) = sin(θ) / cos(θ) = 45 / √(-2024) (The value is undefined as the denominator is not a real number)

sec(θ) = 1 / cos(θ) = 1 / √(-2024) (The value is undefined as the denominator is not a real number)

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The viable solutions are the values of "Gallons" that are greater than zero.

To determine the viable solutions, we need to consider the relationship between the amount of gas (in gallons) in John's vehicle's tank and the corresponding number of miles he can travel.

that John's vehicle can drive 28 miles for every gallon of gas, the relationship can be represented by the equation:

Miles = 28 * Gallons

In this equation, "Miles" represents the number of miles John can travel, and "Gallons" represents the amount of gas in the tank.

Based on this equation, we can deduce the following:

1. Viable solutions: Any positive value for "Gallons" will result in a positive number of miles. Therefore, solutions where the number of gallons is greater than zero are viable.

2. Non-viable solution: A non-viable solution would be when the number of gallons is zero or negative. Since you cannot have negative or zero gallons of gas, these solutions do not make sense in this context.

Therefore, To determine the viable solutions, we need to consider the relationship between the amount of gas (in gallons) in John's vehicle's tank and the corresponding number of miles he can travel.

that John's vehicle can drive 28 miles for every gallon of gas, the relationship can be represented by the equation:

Miles = 28 * Gallons

In this equation, "Miles" represents the number of miles John can travel, and "Gallons" represents the amount of gas in the tank.

Based on this equation, we can deduce the following:

1. Viable solutions: Any positive value for "Gallons" will result in a positive number of miles. Therefore, solutions where the number of gallons is greater than zero are viable.

2. Non-viable solution: A non-viable solution would be when the number of gallons is zero or negative. Since you cannot have negative or zero gallons of gas, these solutions do not make sense in this context.

Therefore, the viable solutions are the values of "Gallons" that are greater than zero.

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You paid approximately $0.0053 per foot of twine.

To determine the price per foot of twine, you can divide the total cost by the length of the twine.

Price per foot = Total cost / Length of twine

Given that you paid $24 for 4500 feet of twine, you can calculate the price per foot as follows:

Price per foot = $24 / 4500 feet

Price per foot = $0.0053 (rounded to four decimal places)

Therefore, you paid approximately $0.0053 per foot of twine.

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The end of the 48-inch long treadmill is raised approximately 8.36 inches.

The incline of the treadmill is given as 10 degrees.

We can use trigonometry to calculate the height of the end of the treadmill.

The height (h) can be found using the formula h = l * sin(θ), where l is the length of the treadmill and θ is the angle of inclination.

Substitute the values into the formula:

h = 48 inches * sin(10 degrees)

Calculate the sine of 10 degrees using a calculator:

sin(10 degrees) ≈ 0.1736

Multiply the length of the treadmill by the sine of the angle:

h = 48 inches * 0.1736 ≈ 8.36 inches

The end of the 48-inch long treadmill is raised approximately 8.36 inches when set at an incline of 10 degrees.

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The inverse demand function for the given demand function D(p) = 20 - p/4 is 5 - q/4.

To find the inverse demand function, we need to solve the demand function for the quantity (q) in terms of the price (p).

Given the demand function D(p) = 20 - p/4, we can rewrite it as:

q = 20 - p/4

Now, rearranging the equation to solve for p, we get:

p/4 = 20 - q

p = 4(20 - q)

Therefore, the inverse demand function is p = 4(20 - q), which simplifies to p = 80 - 4q. By dividing both sides of the equation by 4, we obtain the inverse demand function in the standard form:

p = 5 - q/4

In summary, the inverse demand function for the given demand function D(p) = 20 - p/4 is 5 - q/4.

32 points: The elasticity of demand for wheat does not depend on the unit of measurement (bushels or gallons). Therefore, if the elasticity of demand for wheat is -0.50 when measured in bushels, it will still be -0.50 when measured in gallons. None of the other answer choices are correct.

41 points: The own-price elasticity of demand can be calculated using the formula:

E = (ΔQ/ΔP) * (P/Q)

Given the market demand function D(p) = 10 - p, we can substitute p = 6 into the equation and calculate the slope of the demand function at that price.

ΔQ = D(p2) - D(p1) = (10 - p2) - (10 - p1) = (10 - 6) - (10 - 6) = 4 - 4 = 0

ΔP = p2 - p1 = 6 - 6 = 0

Using these values in the elasticity formula, we get:

E = (0/0) * (6/10) = 0

Therefore, the own-price elasticity of demand at a price of p = 6 is 0.

Income elasticity of demand measures the responsiveness of quantity demanded to changes in income. The income elasticity of demand for roses can be calculated using the formula:

E = (ΔQ/ΔM) * (M/Q)

Given Q = 12 - 3p + (1/100)M, we substitute M = 100 and p = 1 into the equation:

Q = 12 - 3(1) + (1/100)(100)

Q = 12 - 3 + 1

Q = 10

ΔQ = Q2 - Q1 = 10 - 10 = 0

ΔM = M2 - M1 = 100 - 100 = 0

Using these values in the elasticity formula, we get:

E = (0/0) * (100/10) = 0

Therefore, the income elasticity of demand for roses when M = 100 and p = 1 is 0.

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The probability of an event is a measure of the likelihood that the event will occur. In this case, the event is "randomly selecting a gold coin from the box."

If the probability of selecting a gold coin, denoted as P(gold coin), is 50%, it means that if you were to randomly select an item from the box, there's a 50% chance that the item you select will be a gold coin.

In other words, if you were to repeat the process of randomly selecting an item from the box many times (under the same conditions), about half of the time you would expect to draw a gold coin. This is assuming that after each draw, the item is replaced back into the box, so the conditions remain the same for each draw.

Remember that probability is a theoretical measure, and actual results can vary, especially if the number of draws is small. But as the number of draws increases, the proportion of times you draw a gold coin should get closer and closer to 50%.

The unknown sides are a ≈ 48.35 and b ≈ 23.61.

A triangle ABC such that c=75, A= 43°, and B=20°.

To find the unknown values of the triangle,First we know that the sum of all angles of a triangle is 180°. Hence we can find C.

Using the angle sum property we have,

C = 180° - A - B= 180° - 43° - 20°= 117°

Now we know, A, B, and C,

we can find the unknown side lengths of the triangle using the Law of Sines.

The Law of Sines is given by: $$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$$

Thus, we have:

$$\frac{a}{\sin 43°}=\frac{b}{\sin 20°}=\frac{75}{\sin 117°}$$

Solving for a and b, we have;

$$a = 75 \cdot \frac{\sin 43°}{\sin 117°} ≈ 48.35$$ $$b = 75 \cdot \frac{\sin 20°}{\sin 117°} ≈ 23.61$$.

Hence, the unknown sides are a ≈ 48.35 and b ≈ 23.61.

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The solutions to the quadratic equation p^2 - 8p - 50 = 24, after completing the square, are p = 4 + 3√10 and p = 4 - 3√10.

The quadratic equation p^2 - 8p - 50 = 24 can be solved by completing the square. The equation after completing the square, but before taking the square root, is (p - 4)^2 = 98.

To solve the equation, we follow these steps:

⇒ Move the constant term to the right side of the equation:

p^2 - 8p - 50 - 24 = 0

p^2 - 8p - 74 = 0

⇒ Add the square of half the coefficient of p to both sides of the equation:

p^2 - 8p + (-8/2)^2 = 74 + (-8/2)^2

p^2 - 8p + 16 = 74 + 16

p^2 - 8p + 16 = 90

⇒ Rewrite the left side of the equation as a perfect square:

(p - 4)^2 = 90

Therefore, the equation after completing the square is (p - 4)^2 = 90.

To find the solutions, we take the square root of both sides of the equation:

p - 4 = ±√90

Simplifying the square root of 90:

p - 4 = ±3√10

Now, we isolate p by adding 4 to both sides of the equation:

p = 4 ± 3√10

Hence, the solutions to the equation p^2 - 8p - 50 = 24, in simplest radical form, are p = 4 + 3√10 and p = 4 - 3√10.

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The difference in the present values if you are paid these annual amounts of money at the beginning of each year rather than at the end of each year is $9,470.

Let's start with the present value when the payments are made at the end of each year.PV1 = ($9,700 x [1 - (1 / (1 + 0.09)22)]) / 0.09= $119,737.13

Now let's find the present value when the payments are made at the beginning of each year. To do this, we need to adjust the interest rate and the number of periods.

The interest rate needs to be adjusted because it's an annual rate and the payments are being made at the beginning of each year, so we need to compound the interest for the year before we receive the payment.

The number of periods needs to be adjusted because we're receiving the payments one year earlier.

PV2 = ($9,700 x [1 - (1 / (1 + (0.09/1))21)]) / (0.09/1)= $129,207.86

Finally, we can find the difference between these two present values:

PV2 - PV1= $129,207.86 - $119,737.13= $9,470.73

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The exact value of the expression is -3/4.

To find the exact value of the expression sin 120° tan 60° cos (4π/3) sin (5π/3), we can make use of the double-angle identity for sine. The double-angle identity states that sin(2θ) = 2sin(θ)cos(θ).

First, let's simplify the expression using the double-angle identity for sine:

sin 120° = sin(2 * 60°) = 2sin(60°)cos(60°) = 2(√3/2)(1/2) = √3/2

Next, we simplify the expression sin (5π/3) using the double-angle identity:

sin (5π/3) = sin (2 * (2π/3) + (π/3)) = sin (2π/3)cos(π/3) + cos (2π/3)sin (π/3)

Since sin(2π/3) = sin(π/3) = √3/2 and cos(π/3) = 1/2, the expression becomes:

sin (5π/3) = (√3/2)(1/2) + (1/2)(√3/2) = √3/4 + √3/4 = √3/2

Now, we have:

sin 120° tan 60° cos (4π/3) sin (5π/3) = (√3/2)(1)(-1/2)(√3/2) = -3/4

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The slope of the graphed line is 2.

Here is a table showing three pairs of numbers for the function y = 2x - 2:

x y

0 -2

1 0

2 2

To graph these points on a coordinate plane, plot the x-value on the horizontal axis and the corresponding y-value on the vertical axis. The graphed points will be (0, -2), (1, 0), and (2, 2).

Next, draw a line through these points:

     |

     |

     |

     |       x (2, 2)

     |      /

     |     /

     |    /

     |   /

     |  /

     | /

     |/

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

     |      

     |      

     |       x (1, 0)

     |      

     |      

     |      

     |      

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

     |      

     |      

     |      

     |      

     |       x (0, -2)

     |      

     |      

The slope of the graphed line can be determined by calculating the change in y divided by the change in x. Taking any two points on the line, let's choose (0, -2) and (2, 2):

Change in y = 2 - (-2) = 4

Change in x = 2 - 0 = 2

Slope = Change in y / Change in x = 4 / 2 = 2

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The quadratic function in standard form is:

g(x) = x^2 + 4x + 4

To graph this quadratic function, we can use the vertex form of a quadratic equation:

g(x) = a(x - h)^2 + k

where (h, k) represents the vertex of the parabola. By comparing the given function with the vertex form, we can determine the values of a, h, and k.

In the given function, we have:

a = 1

h = -2 (since -h = 4)

k = 4

Therefore, the vertex of the parabola is (-2, 4).

To sketch the graph, we plot the vertex (-2, 4) on the coordinate plane. Since the coefficient of x^2 is positive (a = 1), the parabola opens upward. From the vertex, we can identify additional points on the graph.

By substituting x = -3, -1, 0, and 1 into the function, we can calculate the corresponding y-values:

When x = -3, g(x) = (-3)^2 + 4(-3) + 4 = 9 - 12 + 4 = 1

When x = -1, g(x) = (-1)^2 + 4(-1) + 4 = 1 - 4 + 4 = 1

When x = 0, g(x) = (0)^2 + 4(0) + 4 = 0 + 0 + 4 = 4

When x = 1, g(x) = (1)^2 + 4(1) + 4 = 1 + 4 + 4 = 9

We can plot these points on the graph and connect them smoothly to form a parabolic shape. The resulting graph is a U-shaped parabola with the vertex at (-2, 4).

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[tex]g(a+h)=a^{2} + h^{2}+2ah+2[/tex]

To formulate and solve a linear programming model for the resource-allocation problem in Excel, you will need to use the appropriate formulas in each cell. Additionally, you can utilize Solver to optimize the solution.

In Excel, you can use the SUMPRODUCT formula to calculate the contribution per unit by multiplying the profit per unit of each activity by the corresponding allocation. For example, if the profit per unit of activity A is in cell B2 and the allocation for activity A is in cell C2, you can use the formula "=B2*C2" to calculate the contribution per unit for activity A.

To formulate the linear programming model, you will need to define the objective function and the constraints. The objective function represents the goal you want to maximize or minimize, while the constraints specify the limitations on the resources or activities.

To set up Solver, go to the Data tab in Excel and click on Solver. In the Solver Parameters dialog box, you need to specify the objective cell (the cell containing the formula for the total contribution), select the optimization type (maximize or minimize), and add the constraints. The constraints can be added by clicking on the Add button and specifying the cell references for the constraints.

Once Solver is set up, you can click Solve to find the optimal solution that maximizes or minimizes the objective function while satisfying the constraints.

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Answer:

A triangle is a right triangle if and only if the sum of the squares of the two shorter sides is equal to the square of the longest side. This is known as the Pythagorean theorem.

Let's check each triangle to see if it satisfies this condition:

Triangle P:

Shortest side = 12

Middle side = 24

Longest side = 30

12^2 + 24^2 = 144 + 576 = 720

30^2 = 900

720 is not equal to 900, so triangle P is not a right triangle.

Triangle Q:

Shortest side = 9

Middle side = 40

Longest side = 41

9^2 + 40^2 = 81 + 1600 = 1681

41^2 = 1681

1681 is equal to 1681, so triangle Q is a right triangle.

Triangle R:

Shortest side = 5

Middle side = 18

Longest side = 21

5^2 + 18^2 = 25 + 324 = 349

21^2 = 441

349 is not equal to 441, so triangle R is not a right triangle.

Therefore, only triangle Q is a right triangle.

Answer:

Triangle Q

Step-by-step explanation:

To know if a triangle is a right triangle given all 3 side lengths, we can use pythagorean theorem, which states:

[tex]a^{2} +b^{2} =c^{2}[/tex]

Triangle P: 12,24,30

[tex]12^{2} +24^{2} =30^{2} \\144+576=900\\720 = 900[/tex]

FALSE

Triangle Q: 9, 40, 41

[tex]9^{2} +40^{2} =41^{2} \\81+1,600=1,681\\1,681=1,681\\[/tex]

TRUE

Triangle R: 5,18,21

[tex]5^{2} +18^{2} =21^{2} \\25+324=441\\349=441[/tex]

FALSE

So, Triangle Q is a right triangle.

Hope this helps! :)

Answer:

15

Step-by-step explanation:

Given:

p=h+5

with h being hours

If Pamela spends 10 hours writing, substitute in 10 for h:

p=10+5

p=15

So, 15 pages will be written.

Hope this helps! :)

For the function f(x) = -x^2 + 5x - 4, the relative maximum (x, y) is: (2.5, 2.25) and relative minimum (x, y): DNE (Does Not Exist).

By observing the graph, we can see that the function is a downward-opening parabola, which means it has a single relative maximum and no relative minimum (since it goes to negative infinity on both sides).

The vertex of the parabola represents the relative maximum.

Approximating the coordinates of the vertex:

The x-coordinate of the vertex can be found using the formula:

x = -b / 2a, where a = -1 and b = 5.

x = -5 / (2 * (-1)) = 5/2 = 2.5

To obtain the y-coordinate, substitute the x-coordinate back into the function:

f(2.5) = -(2.5)^2 + 5(2.5) - 4 = -6.25 + 12.5 - 4 = 2.25

So, the relative maximum (x, y) is approximately (2.5, 2.25).

Relative minimum (x, y): DNE (Does Not Exist) since the function is a downward-opening parabola and does not have a relative minimum.

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The equation provided is:d³x/dt³ - d²x/dt² - 4 dx/dt + 4 x = 0.Here's how to find the general solution of this equation:Solution of the given third order differential equation:Let's find the characteristic equation by considering x(t) = eλt, where λ can be any constant.   Substitute x(t) = eλt in the given differential equation to get the following characteristic equation:λ³eλt - λ²eλt - 4λeλt + 4eλt = 0.On simplification we get the following cubic equation:λ³ - λ² - 4λ + 4 = 0.Now let's solve this cubic equation to get the value of λ as follows:λ³ - λ² - 4λ + 4 = 0On observation we find λ = 1 to be one of the roots of the cubic equation.  So, divide the cubic equation by λ - 1 using long division method to get a quadratic equation.λ³ - λ² - 4λ + 4 = 0λ² (λ - 1) - 4(λ - 1) = 0On simplification, we get:λ² - 4 = 0On solving this quadratic equation, we get two distinct roots,λ = -2, λ = 2Thus the roots of the cubic equation are given as follows: λ = 1, λ = -2, λ = 2.The three possible solutions of the given differential equation are:x(t) = et, x(t) = e-2t, x(t) = e2t.Therefore, the general solution of the given differential equation is:x(t) = c1et + c2e-2t + c3e2t, where c1, c2, c3 are arbitrary constants.The equation (2) of first order system isdx/dt = 4x - y, dy/dt = x + 4yLet's rewrite this equation (2) of first order system as a matrix equation as follows:Write x = [x1, x2]T and x' = [x1', x2']T. Then equation (2) can be written as follows:x' = Ax, where x = [x1, x2]T is the vector and A is the matrix of the form [4, -1; 1, 4].The general solution of x' = Ax is given by the formula,x = c1x1 + c2x2, where c1 and c2 are arbitrary constants and x1 and x2 are the eigenvectors of the matrix A.For finding the eigenvectors of the matrix A, we use the following steps:First, let's find the eigenvalues of the matrix A by solving the characteristic equation of A, which is given by |A - λI| = 0, where I is the identity matrix. On solving this equation, we get the eigenvalues as λ1 = 3, λ2 = 5.Now let's find the eigenvectors of the matrix A corresponding to the eigenvalues λ1 and λ2. For λ1 = 3, we get the eigenvector as x1 = [1, -1]T and for λ2 = 5, we get the eigenvector as x2 = [1, 1]T.Therefore, the general solution of x' = Ax is given by,x = c1[1, -1]Te3t + c2[1, 1]Te5t.Which is the quicker method for finding the general solution?The quicker method for finding the general solution is by the direct method of solving the differential equation and finding its roots to get the general solution. The method of finding the general solution of the first order system by finding the eigenvectors and eigenvalues of the matrix A is a little complicated method.

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As sin∅=3/4  tan ∅=3√7/7 according to calculation.

The equation sin(θ) = 3/4 represents a relationship between the angle θ and the sine of that angle. It states that the sine of θ is equal to 3/4. The sine function relates the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle.

To understand the condition tan(θ) < 0, we need to consider the tangent function. The tangent of an angle is defined as the ratio of the sine of the angle to the cosine of the angle. In terms of trigonometric ratios in a right triangle, it can also be expressed as the ratio of the length of the side opposite the angle to the length of the adjacent side.

If tan(θ) < 0, it means that the tangent of the angle is negative. This occurs when the side opposite the angle and the side adjacent to the angle are in opposite directions in the coordinate plane.

In summary, the equation sin(θ) = 3/4 implies that the sine of θ is equal to 3/4, while the condition tan(θ) < 0 indicates that the tangent of θ is negative, which suggests that the angle θ lies in the third or fourth quadrant of the coordinate plane.

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1. The amount of manufacturing overhead cost that would have been applied to the Koopers job is $9,840.

2. Using departmental predetermined overhead rates, the total manufacturing cost of the Koopers job is $47,040.

3. The bid price would have been $70,560.

The plantwide predetermined overhead rate for the current year is $2.40 per direct labour dollar ($4,032,000 ÷ $1,680,000).

The amount of manufacturing overhead cost that would have been applied to the Koopers job is $10,200 ($30,000 × 0.34).  

The predetermined overhead rate for each department is:

Department A = $0.60 per direct labour dollar ($672,000 ÷ 1,120,000)

Department B = $0.96 per direct labour dollar ($1,344,000 ÷ $1,400,000)

Department C = $0.72 per direct labour dollar ($1,008,000 ÷ $1,400,000)

The amount of manufacturing overhead cost that would have been applied to the Koopers job is $9,840 ($48,000 × 0.205 + $84,000 × 0.172 + $54,000 × 0.144).

Using a plantwide predetermined overhead rate, the total manufacturing cost of the Koopers job is $48,500 ($18,000 + $20,000 + $10,200).

Thus, the bid price is $72,750 (150% × $48,500). Using departmental predetermined overhead rates, the total manufacturing cost of the Koopers job is $47,040 ($18,000 × 0.205 + $20,000 × 0.172 + $10,000 × 0.144 + $18,000 + $20,000 + $10,000).

Thus, the bid price would have been $70,560 (150% × $47,040).

Hence, the solution to the problem is as follows: The company's plantwide predetermined overhead rate for the current year is $2.40 per direct labour dollar. The amount of manufacturing overhead cost that would have been applied to the Koopers job is $10,200.

Suppose that instead of using a plantwide predetermined overhead rate, the company had used departmental predetermined overhead rates based on direct labour cost. The predetermined overhead rate for each department for the current year is:

Department A = $0.60 per direct labour dollar

Department B = $0.96 per direct labour dollar

Department C = $0.72 per direct labour dollar

The amount of manufacturing overhead cost that would have been applied to the Koopers job is $9,840.Using a plantwide predetermined overhead rate, the total manufacturing cost of the Koopers job is $48,500.

Thus, the bid price is $72,750.Using departmental predetermined overhead rates, the total manufacturing cost of the Koopers job is $47,040. Thus, the bid price would have been $70,560.

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The graph of the logarithmic function represented by the table would be a curve that passes through the given points. The domain of the function is all positive values of x, and the range is all real numbers.

The graph of the logarithmic function will be a curve passing through the points given in the table. The domain of the function is all positive values of x, and the range is all real numbers.

The table represents a logarithmic function f(x). The x-values in the table are the inputs to the function, and the y-values are the outputs.

Looking at the table, we can see that as x increases, y also increases, indicating that the function is increasing.

To graph the function, we plot the given points and connect them with a smooth curve. The curve will approach the y-axis but will never touch it, as the logarithm of 0 is undefined. The graph extends infinitely in the positive x-direction.

The domain of the function is all positive values of x since the logarithm is only defined for positive inputs. In inequality notation, we can express the domain as x > 0. In interval notation, the domain is (0, ∞).

The range of the function is all real numbers. As x approaches infinity, y also approaches infinity. As x approaches 0, y approaches negative infinity. Therefore, the range is (-∞, ∞) in interval notation. In set-builder notation, we can express the range as {y | y ∈ ℝ}.

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Using substitution, the (w∘g)(1) = 68 and (g∘w)(1) = -4.

To find (w∘g)(1), we first evaluate g(1) and then substitute that result into w(x). Similarly, to find (g∘w)(1), we evaluate w(1) and substitute the result into g(x).

Given w(x) = 2x² + 4x - 2 and g(x) = 2x + 3, we have:

g(1) = 2(1) + 3 = 2 + 3 = 5

Substituting g(1) into w(x), we get:

(w∘g)(1) = w(g(1)) = w(5) = 2(5)² + 4(5) - 2 = 50 + 20 - 2 = 68

Similarly, substituting w(1) into g(x), we have:

(g∘w)(1) = g(w(1)) = g(-1) = 7(-1) + 3 = -7 + 3 = -4

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Answer:

To find the area of the triangle, we can use the formula:

Area = (1/2) * a * c * sin(B)

where "a" and "c" are the lengths of the sides of the triangle, and "B" is the angle between them.

Plugging in the values we have:

Area = (1/2) * 7.3ft * 3.2ft * sin(37°)

Area = 8.57 square feet (rounded to two decimal places)

Therefore, the area of the triangle is 8.57 square feet.

For the function f(x) = -3x^(2) - 10x + 3, the value of A) f(-1) + f(1) is 0, and the value of B) f(-1) - f(1) is 20.

A) f(-1) + f(1):

Step 1: Compute f(-1):

f(-1) = -3(-1)^(2) - 10(-1) + 3 = -3 + 10 + 3 = 10

Step 2: Compute f(1):

f(1) = -3(1)^(2) - 10(1) + 3 = -3 - 10 + 3 = -10

Step 3: Add f(-1) and f(1):

f(-1) + f(1) = 10 + (-10) = 0

So, f(-1) + f(1) = 0.

B) f(-1) - f(1):

Step 1: Compute f(-1) (already computed above):

f(-1) = 10

Step 2: Compute f(1) (already computed above):

f(1) = -10

Step 3: Subtract f(1) from f(-1):

f(-1) - f(1) = 10 - (-10) = 10 + 10 = 20

So, The computed value f(-1) - f(1) = 20.

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When x₁ < 2x₂, the marginal rate of substitution (MRS) function for u(x₁, x₂) = min{x₁, 2x₂} is given by: MRS = ∂u/∂x₁ / ∂u/∂x₂

To find the partial derivatives, we consider two cases:

1. When x₁ < 2x₂:

In this case, the utility function u(x₁, x₂) = x₁.

∂u/∂x₁ = 1

∂u/∂x₂ = 0

Therefore, the MRS function is:

MRS = 1/0 = undefined (since ∂u/∂x₂ = 0)

2. When x₁ > 2x₂:

In this case, the utility function u(x₁, x₂) = 2x₂.

∂u/∂x₁ = 0

∂u/∂x₂ = 2

Therefore, the MRS function is: MRS = 0/2 = 0

To summarize:

- When x₁ < 2x₂, the MRS function is undefined.

- When x₁ > 2x₂, the MRS function is 0.

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To determine how tall you could be and still be completely in the shadow of the tree, we need to consider the concept of similar triangles.

The ratio of the height of the tree to the length of its shadow is the same as the ratio of your height to the length of your shadow.

Given that the tree is 40 feet tall and its shadow is 40 feet long, the ratio is 40/40, which simplifies to 1.

Now, let's determine the length of your shadow. Since you are standing 35 feet away from the tree, your shadow would also be 35 feet long.

To find your maximum height, we can set up a proportion:

1/35 = x/40

By cross-multiplying, we get:

x = (1/35) * 40

Simplifying, we find that x, the maximum height you could be and still be completely in the shadow of the tree, is approximately 1.14 feet.

In conclusion, you could be up to 1.14 feet tall and still be completely in the shadow of the tree.

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Answer: The lowest possible product would be -6724 given the numbers 82 and -82.

We can find this by setting the first number as x + 164. The other number would have to be simply x since it has to have a 164 difference.

Next we'll multiply the numbers together.

x(x+164)

x^2 + 164x

Now we want to minimize this as much as possible, so we'll find the vertex of this quadratic graph. You can do this by finding the x value as -b/2a, where b is the number attached to x and a is the number attached to x^2

-b/2a = -164/2(1) = -164/2 = -82

So we know one of the values is -82. We can plug that into the equation to find the second.

x + 164

-82 + 164

82

Step-by-step explanation: Hope this helps.

Answer:

250 lunches

Step-by-step explanation:

If 24% order lunch, that means that 76% bring their lunch (100% - 24% = 76%)

We are trying to find 76% of what number is 190.  Let x stand for the total number of lunches

.76x = 190  Divide both sides by .76  76% as a decimal is .76

[tex]\frac{.76x}{.76}[/tex] = [tex]\frac{190}{.76}[/tex]

x = 250

Helping in the name of Jesus.

The value of x is 4log5[(1/81)(B^4)(h^4)]. This is derived by equating the volume formula of a pyramid (V = (1/3)Bh) to the expression 5^(x/4) and simplifying the equation.

The volume of a pyramid is given by the formula V=(1/3)Bh, where B is the area of the base and h is the height. To find the value of x if the volume is equivalent to 5^(x/4) cm^3, we need to equate the two expressions.

First, let's rewrite 5^(x/4) in terms of its base and exponent. Using the property of exponentiation, we have (5^x)^(1/4). Now, we can equate the two expressions: (1/3)Bh = (5^x)^(1/4). To simplify further, we can raise both sides of the equation to the power of 4, resulting in [(1/3)Bh]^4 = 5^x.

Next, we can simplify the left side of the equation by expanding it: (1/81)(B^4)(h^4) = 5^x. Since the equation is now in terms of x, we can see that x = 4log5[(1/81)(B^4)(h^4)]. In summary, the value of x, when the volume of the pyramid is equivalent to 5^(x/4) cm^3, is x = 4log5[(1/81)(B^4)(h^4)].

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Equation shows that Tom's indirect utility function V(p, w) is convex in p, as desired.

V(λ[tex]p1[/tex] + (1-λ)[tex]p2[/tex], w) ≤ λV([tex]p1[/tex], w) + (1-λ)V([tex]p2[/tex], w)

To prove that Tom's indirect utility function V(p, w) is convex in p, where p is the price vector and w is the wealth, we need to show that for any two price vectors p1 and p2, and for any λ ∈ [0,1], the following inequality holds:

V(λ[tex]p1[/tex] + (1-λ)[tex]p2[/tex], w) ≤ λV([tex]p1[/tex], w) + (1-λ)V([tex]p2[/tex], w)

To prove this, we can use the concept of homothetic preferences.

Homothetic preferences imply that the utility function is homogeneous of degree zero, meaning that multiplying prices and income by the same positive constant does not affect the consumer's preferences.

Let's assume Tom's utility function is U(x), where x represents the consumption bundle.

Tom's indirect utility function can be defined as:

V(p, w) = max { U(x) | px ≤ w }

Now, consider two price vectors p1 and p2, and let x1 and x2 be the optimal consumption bundles for p1 and p2, respectively.

Since U(x) is homogeneous of degree zero, we have:

U(λ[tex]x1[/tex]+ (1-λ)[tex]x2[/tex]) = U(x1 + λ(x2 - x1)) = U(x1) [using homogeneity]

From the definition of the indirect utility function, we know that V(p, w) = U(x), where x is the consumption bundle that maximizes U(x) subject to the budget constraint.

Therefore, we have:

V(λp1 + (1-λ)p2, w) = U(x1) [since λx1 + (1-λ)x2 is the optimal consumption bundle for λp1 + (1-λ)p2]

Now, let's consider the right-hand side of the inequality:

λV(p1, w) + (1-λ)V(p2, w) = λU(x1) + (1-λ)U(x2) [using the definition of the indirect utility function]

Since U(x1) = U(x2) (as shown above), we can simplify the right-hand side:

λV(p1, w) + (1-λ)V(p2, w) = U(x1)

Therefore, we have:

V(λp1 + (1-λ)p2, w) ≤ λV(p1, w) + (1-λ)V(p2, w)

This shows that Tom's indirect utility function V(p, w) is convex in p, as desired.

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