Help with math problem please

Answer:

$2,500 or 25%

Step-by-step explanation:

Let's use the same example:

Cost of Equipment = $10,000

Useful Life = 4 years

Depreciation Rate = (Annual Depreciation / Cost of Equipment) * 100

Annual Depreciation = Cost of Equipment / Useful Life

Annual Depreciation = $10,000 / 4 years

Annual Depreciation = $2,500

Depreciation Rate = ($2,500 / $10,000) * 100

Depreciation Rate = 0.25 * 100

Depreciation Rate = 25%

The six terms of the series 247 are 2, 2, 2, 2, 2, and 2.

To find the six terms of a series, we need to first understand what a series is. A series is defined as the sum of the terms of a sequence. A sequence is a set of numbers in a specific order.

Therefore, a series is a sum of terms from a sequence. There are different types of series, and they have different formulas for calculating their terms.In this case, we are required to find the six terms of the series 247. Since this is a finite series, we can use a formula to calculate the nth term of a finite series.

The formula is given as follows:Tn = a + (n - 1) dWhere Tn is the nth term of the series, a is the first term of the series, n is the number of terms in the series, and d is the common difference between the terms of the series.To find the six terms of the series 247, we need to know the value of a and d. In this case, we can see that the first term of the series is 2. Therefore, a = 2.

Since this is a constant series, we can see that the common difference between the terms is 0. Therefore, d = 0.Substituting these values in the formula, we get:T1 = a + (1 - 1) dT1 = 2 + 0T1 = 2T2 = a + (2 - 1) dT2 = 2 + 0T2 = 2T3 = a + (3 - 1) dT3 = 2 + 0T3 = 2T4 = a + (4 - 1) dT4 = 2 + 0T4 = 2T5 = a + (5 - 1) dT5 = 2 + 0T5 = 2T6 = a + (6 - 1) dT6 = 2 + 0T6 = 2.

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Dot plots are graphical displays of data that use dots to represent the frequency or count of a specific value in a data set. The ages of five children in each of two families are represented in the dot plots above.

Both dot plots have a similar range of ages, but family A has a higher median age and a smaller spread of ages than family B.

For family A, the median age is 10 years old because the middle dot in the plot is located at 10. Family A has a small range of ages because all the dots are concentrated in the 8-12 age range.

In contrast, family B has a wider range of ages, from 6 to 15 years old, and a larger spread because the dots are scattered across the plot.

The median age for family B is approximately 11 years old.

In conclusion, dot plots are useful for comparing data sets and analyzing their characteristics such as median age and spread.

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To construct a frequency histogram based on the given temperature data, we will use the temperature ranges as the x-axis and the number of days as the y-axis.

The temperature ranges and their corresponding frequencies are as follows:

30-39: 2 days

40-49: 26 days

50-59: 28 days

60-69: 8 days

70-79: 2 days

To create the histogram, we will represent each temperature range as a bar and the height of each bar will correspond to the frequency of days.

Using an appropriate scale, we can label the x-axis with the temperature ranges (30-39, 40-49, 50-59, 60-69, 70-79) and the y-axis with the frequency values.

Now, we can draw rectangles (bars) on the graph, with the base of each rectangle corresponding to the temperature range and the height representing the frequency of days. The height of each bar will be determined by the corresponding frequency value.

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Julio will be 30 years old when his father is twice his age.

To determine the age at which Julio will be twice his father's age, we need to find the age difference between them and then add that difference to Julio's current age.

Currently, Julio is 12 years old, and his father is 42 years old. The age difference between them is 42 - 12 = 30 years.

For Julio to be twice his father's age, the age difference needs to remain the same as it is currently. Therefore, when Julio is x years old, his father will be x + 30 years old.

Setting up an equation to solve for x:

x + 30 = 2x

Simplifying the equation, we subtract x from both sides:

30 = x

Thus, when Julio is 30 years old, his father will be 30 + 30 = 60 years old. At this point, Julio will indeed be twice his father's age.

Therefore, Julio will be 30 years old when his father is twice his age.

Note : the translated question is Julio is 12 years old and his father is 42 years old. How old will Julio be when his father is twice his age?

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The vertex (25, 2400) of the parabola reveals the optimal ticket price (25 dollars) that maximizes the profit (2400 dollars) for the theater during the comedy show.

To find the vertex of the parabola, we can use the formula:

x = -b / (2a)

In this case, the function is p(d) = -4d² + 200d - 100, which can be rewritten in the form of ax² + bx + c.

Comparing it with the standard form ax² + bx + c, we can see that a = -4, b = 200, and c = -100.

Now, let's substitute these values into the formula to find the vertex:

d = -200 / (2 * -4)

d = -200 / -8

d = 25

The x-coordinate of the vertex is 25. To find the corresponding y-coordinate, we substitute this value back into the original function:

p(25) = -4(25)² + 200(25) - 100

p(25) = -4(625) + 5000 - 100

p(25) = -2500 + 5000 - 100

p(25) = 2400

The y-coordinate of the vertex is 2400.

Therefore, the vertex of the parabola is (25, 2400).

The vertex reveals important information about the situation. In this case, it represents the optimal point for maximizing profit. The x-coordinate (25) represents the price at which the theater should set the tickets to maximize their profit. The y-coordinate (2400) represents the maximum profit achievable at that price.

Additionally, since the coefficient of the quadratic term (a) is negative (-4), it indicates that the parabola opens downwards, forming a concave shape. This means that as the ticket price increases or decreases from the optimal price, the profit will decrease. Therefore, setting the tickets at a price other than the one corresponding to the vertex would result in a lower profit for the theater.

In summary, the vertex (25, 2400) of the parabola reveals the optimal ticket price (25 dollars) that maximizes the profit (2400 dollars) for the theater during the comedy show.

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The slope is represented by the number 1473 in the equation T = 1473c + 5495. The slope denotes the rate of change, or the amount by which the overall cost (T) rises with each new class attended (c).

In this scenario, the total cost of the semester increases by $1473 for each extra class attended at Michigan State University. The slope reflects the linear relationship between the total cos and the number of classes taken.

The number 5495 represents the y-intercept in the equation. When the number of classes (c) is 0, the y-intercept is the value of T. The y-intercept of $5495 shows the total cost for a semester at Michigan State University when no classes are taken. It includes a one-time accommodation and board price.

The y-intercept can be understood as the university's fixed cost component, which is independent of the number of classes taken. It includes expenditures such as housing and food plans that are incurred regardless of how many classes a student enrolls in.

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[tex]\underline{\underline{\purple{\huge\sf || ꪖꪀᦓ᭙ꫀ᥅}}}[/tex]

The given polynomial is:

p(t) = -t²(3 - 5t)(t² + t + 4)

To find the degree of the polynomial, we need to determine the highest power of t in the expression. This is given by:

degree = 2 + 1 + 2 = 5

So the degree of the polynomial is 5.

The leading term of the polynomial is the term with the highest power of t. This is given by:

leading term = -t² * 5t² = -5t^4

So the leading term of the polynomial is -5t^4.

The leading coefficient of the polynomial is the coefficient of the leading term. This is given by:

leading coefficient = -5

So the leading coefficient of the polynomial is -5.

The constant term of the polynomial is the term that does not contain any powers of t. This is given by:

constant term = -t²(3)(4) = -12t²

So the constant term of the polynomial is -12t^2.

To find the end behavior of the polynomial, we need to determine what happens to the value of the polynomial as t approaches positive or negative infinity. Since the degree of the polynomial is odd, we know that the end behavior will be opposite for t approaching positive or negative infinity. We can use the leading term to determine the end behavior:

- as t approaches positive infinity, the leading term approaches negative infinity, so the end behavior is p(t) → -∞ as t → ∞

- as t approaches negative infinity, the leading term approaches positive infinity, so the end behavior is p(t) → ∞ as t → -∞

So the end behavior of the polynomial is:

- p(t) → -∞ as t → ∞

- p(t) → ∞ as t → -∞

Answer: The expression, 3v^4, means that the variable, v, is multiplied by itself 4 times (v * v * v * v) and then is multiplied after by the coefficient 3.

Step-by-step explanation:

Exponent indicates the number of times a number needs to be multiplied by itself.

For example if your given x^2

that simply means that the variable, x, is being multiplied by itself twice:

x * x

if your given 2z^9

that means that the variable, z, is first multiplied by itself nine times and after that it is multiplied by the coefficient, two.

Therefore, the range of the function g(x) = -1/2(x - 6) + 1 is all real numbers less than or equal to 1, written as (-∞, 1].

To determine the range of the function g(x) = -1/2(x - 6) + 1, we need to find the set of all possible output values or the range of the function.

The given function is in the form of a linear equation, specifically in slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept. In this case, the slope is -1/2 and the y-intercept is 1.

Since the slope is negative, it means the function is decreasing. The range of the function would be the set of all possible y-values that the function can produce. Since the function is decreasing, the maximum value it can reach is its initial y-intercept, which is 1.

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To determine how the mean changes when a value of 98° is added to the data, we need to calculate the mean before and after the addition.

Before adding 98°, the given data set has 12 values. We can calculate the mean by summing all the values and dividing by the total number of values:

Mean = (58 + 61 + 71 + 77 + 91 + 100 + 105 + 102 + 95 + 82 + 66 + 57) / 12

Mean ≈ 83.67°

After adding 98° to the data set, the total number of values becomes 13. To calculate the new mean, we sum all the values, including the added 98°, and divide by the total number of values:

New Mean = (58 + 61 + 71 + 77 + 91 + 100 + 105 + 102 + 95 + 82 + 66 + 57 + 98) / 13

New Mean ≈ 86.15°

Therefore, the mean increases by approximately 2.48° when a value of 98° is added to the data. None of the provided answer choices accurately reflects this change, as they all mention different values (8.2° and 1.4°) that do not correspond to the actual change in the mean.

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Answer:

Yes; TSU ~ PJM

Step-by-step explanation:

Because the angle in between the sides are the same, we can see if each corresponding side is proportional to each other. If they are, then they are similar. (This is SAS similarity theorem)

First we can divide 10 by 15 and see if 14 divided by 21 is the same:

10/15 = 0.6666....

14/21 = 0.6666....

Yes, the two triangles are similar

Answer:

Δ STU  [tex]\bold{\sim}[/tex]  ΔPJM similar

Step-by-step explanation:

Similar triangles are two or more triangles that have the same shape, but their sides are in proportion. In other words, if you divide the length of any side of one triangle by the corresponding side of another triangle, you will get the same number.

For Question:

In Δ STU and ΔPJM

TS:US =10:14=5:7

PJ: JM=15:21= 5:7

Since the length of any side of one triangle is by the corresponding side of another triangle, you will get the same number.

Therefore,

Δ STU  [tex]\bold{\sim}[/tex]  ΔPJM is similar.

Hence Proved:

The correct equation is sin h° = z ÷ x, which states that the sine of angle A is equal to the ratio of the length of side AB to the length of side AC.

The correct equation relating the angles and segments in triangle ACB depends on the specific trigonometric function and the angle we are considering.

In triangle ACB, angle C is a right angle, which means it measures 90 degrees. The other two angles, angle A and angle B, are complementary angles, meaning their sum is also 90 degrees. Therefore, angle A measures h degrees and angle B measures g degrees, where h + g = 90.

Now let's consider the segments in the triangle. Segment AC measures x, segment CB measures y, and segment AB measures z.

When it comes to trigonometric functions, sine (sin) and cosine (cos) are commonly used. These functions relate the angles and sides of a right triangle.

The correct equation involving the angles and segments can be determined based on the trigonometric function that relates the desired angle to the desired segment.

If we want to relate angle A (measuring h degrees) to the segment AB (measuring z), we can use the sine function. Therefore, the correct equation is:

sin h° = z ÷ x

This equation relates the sine of angle A to the ratio of the lengths of the side opposite angle A (segment AB) and the hypotenuse (segment AC).

In summary, the correct equation is sin h° = z ÷ x, which states that the sine of angle A is equal to the ratio of the length of side AB to the length of side AC.

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The equation of the lines expressed in point-slope form and the average rate of change of the functions indicates that we get;

The point-slope form of the equation of a line can be represented in the following form; y - y₁ = m(x - x₁), where; m is the slope of the line and (x₁, y₁) is a point on the line.

1. (a) The slope of the line parallel to the line; 5·x - 3·y = 10, can be obtained by writing the equation of the line in slope-intercept form as follows;

5·x - 3·y = 10, therefore; y = 5·x/3 - 10/3

The slope of the parallel line is therefore; 5/3

The point-slope form of the equation of the line is therefore;

y - 6 = (5/3)·(x - (-2)) = (5/3)·(x + 2)

y - 6 = (5/3)·(x + 2)

3·y - 18 = 5·x + 10

3·y - 5·x = 10 + 18 = 18

(b) The equation of the line, 2·x + 4·y - 12 = 0, indicates;

The slope of the line is; -2/4 = -1/2

The slope of the perpendicular line = -1/(-1/2) = 2

The equation of the perpendicular line passing through the point (2, -7) in point-slope form is therefore;

y - (-7) = 2·(x - 2)

y + 7 = 2·(x - 2)

(c) The line passing through point (-2, -4), and parallel to y = -3 is the line y = -4

(d) The line passing through point (4, -1), and perpendicular to x = 0 is the line y = -1

2. (a) f(-1) = 2 × (-1)² + (-1) - 1 = 0

f(3) = 2 × (3)² + (3) - 1 = 20

The average is; (f(3) - f(-1))/(3 - (-1)) = 20/4 = 5

(b) f(1) = 150, f(3) = 50

Average = (f(3) - f(1))/(3 - 1)

Average = (150 - 50)/(3 - 1) = 50

(c) f(4) = 4² - 4  

f(4 + h) = (4 + h)² - (4 + h) = (h + 3)·(h + 4)

Average = (f(4 + h) - f(4))/(4 + h - 4) = ((h + 3)·(h + 4) - (4² - 4))/(h)

((h + 3)·(h + 4) - (4² - 4))/(h) = (h² + 7·h + 12 - 12)/h = h + 7

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The rest of the amount, represented by R, is $2000 in this case.

If the total amount is $3000 and a certain amount, c, is deposited in one investment, we can represent the remaining amount in dollars by subtracting the deposited amount from the total.

Let's denote the remaining amount as R.

R = Total amount - Deposit amount

R = $3000 - c

Here, R represents the rest of the amount in dollars after the deposit of c dollars. By subtracting the deposit amount from the total amount, we can determine the value of the remaining amount.

For example, if $1000 is deposited in one investment (c = $1000), we can find the remaining amount as follows:

R = $3000 - $1000

R = $2000

Therefore, the rest of the amount, represented by R, is $2000 in this case.

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Answer: [tex]10x^3-19x^2+20x-21[/tex]

Step-by-step explanation:

This problem is a binomial being multiplied by a trinomial.

We can solve it by multiply each term in the first expression by each term in the second expression and combine like terms.

[tex](2x - 3)(5x^2 -2x + 7)[/tex]

First lets multiply the 2x to the trinomial, (5x^2 - 2x + 7)

2x * 5x^2 = 10x^3

2x * -2x = -4x^2

2x * 7 = 14x

Next lets multiply the -3 to the trinomial (5x^2 - 2x + 7)

-3 * 5x^2 = -15x^2

-3 * -2x = 6x

-3 * 7 = -21

Now put all in order by highest power (Exponent) to lowest (to lowest exponent/ or constant)

[tex]10x^3-4x^2-15x^2+14x+6x-21[/tex]

And lastly, combine like terms:

[tex]10x^3-19x^2+20x-21[/tex]

Your final answer is [tex]10x^3-19x^2+20x-21[/tex]

The situation in which you would add 9 is c) 9 more than a number.

Addition is one of the four basic mathematical operations, including subtraction, multiplication, and division.

Mathematical operations involve the use of mathematical operands on variables, numbers, constants, and values to create algebraic expressions, equations, and functions.

Various Cases:

a) decrease of 9: What is required here is the subtraction of 9 and not addition.

b) descend 9 feet: When an object or person descends 9 feet, the measurement calls for a subtraction operation.

c) 9 more than a number: Addition operation is required to make a number to become 9 more than it was previously.

d) product of a number and 9: The product of a number and 9 involves multiplication and will not make add 9 to the number.

Thus, the addition of 9 is required in case c).

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a) decrease of 9

b) descend 9 feet

c) 9 more than a number

d) product of a number and 9

The quadratic function with the solutions given in the problem is defined as follows:

x² + 3x + 3 = 0.

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

y = ax² + bx + c.

The solutions are given as follows:

[tex]x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}[/tex]

Comparing the standard solution to the solution given in this problem, the parameters a and b are given as follows:

The coefficient c is then obtained as follows:

b² - 4ac = -3.

9 - 4c = -3

4c = 12

c = 3.

Hence the function is:

x² + 3x + 3 = 0.

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To determine the minimum sample size required to estimate the proportion of students who bring laptops to campus with a 99% confidence level and a margin of error within five percentage points, we can use the formula:

[tex]n = \frac{(Z^2 \times p \times (1 - p))}{ E^2}[/tex]

Where:

n is the required sample size,

Z is the Z-score corresponding to the desired confidence level,

p is the estimated population proportion (since no prior estimate is available, we use 0.5 as a conservative estimate),

E is the margin of error.

For a 99% confidence level, the Z-score is approximately 2.58 (obtained from the z table).

Plugging in the values:

[tex]n = \frac{(2.58^2 \times 0.5 \times (1 - 0.5))} { (0.05^2)}[/tex]

Simplifying the equation:

n = 663.924

Rounding up to the nearest whole number, the minimum sample size required is approximately 664 students.

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The absolute extrema of the function f(x) = 6[tex]x^4[/tex] - 8[tex]x^3[/tex] on the closed interval (-∞, ∞) are: the minimum value of -2 at x = 1 and no maximum value.

To find the absolute extrema of the function f(x) = 6[tex]x^4[/tex]- 8[tex]x^3[/tex] on a closed interval, we need to evaluate the function at its critical points and endpoints and compare their values.

The critical points occur where the derivative of the function is equal to zero or undefined. Let's find the derivative of f(x) first. Taking the derivative of f(x) = 6[tex]x^4[/tex] - 8[tex]x^3[/tex], we get f'(x) = 24[tex]x^3[/tex] - 24[tex]x^2[/tex].

Setting f'(x) equal to zero, we have 24[tex]x^3[/tex] - 24[tex]x^2[/tex] = 0. Factoring out 24[tex]x^2[/tex]from the equation, we get 24[tex]x^2(x - 1)[/tex] = 0. This equation is satisfied when x = 0 or x = 1.

Now we evaluate the function at these critical points and the endpoints of the interval. The given interval is not specified, so we'll assume it as the entire real number line (-∞, ∞).

For x = 0, f(0) = 6[tex](0)^4[/tex] - 8[tex](0)^3[/tex] = 0.

For x = 1, f(1) = 6[tex](1)^4[/tex] - 8[tex](1)^3[/tex] = -2.

Since the function is a fourth-degree polynomial, it approaches negative infinity as x approaches negative infinity and approaches positive infinity as x approaches positive infinity.

Therefore, the absolute extrema of the function f(x) = 6[tex]x^4[/tex] - 8[tex]x^3[/tex] on the closed interval (-∞, ∞) are: the minimum value of -2 at x = 1 and no maximum value.

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Answer:

A ∩ B = {7, 8}

Step-by-step explanation:

A = {1, 2, 5, 7, 8}

B = {6, 7, 8, 9}

A ∩ B = set of elements in both A and B

A ∩ B = {7, 8}

The general solution of the partial differential equation is:

U(x, t) = Σ [Aₙ*sin((nπ/3)x)]*e^(-(nπ/3)²t)

The partial differential equation (PDE) is given as:

Uxx = (1/2)Ut with the boundary and initial conditions as 0< X <3 with U(0,t) = U(3, t)=0, U(0, t) = 5sin(4πx)

Assume that the solution can be written as a product of two functions:

U(x, t) = X(x)T(t)

Substituting this into the PDE, we have:

X''(x)T(t) = (1/2)X(x)T'(t)

Dividing both sides by X(x)T(t), we get:

(X''(x))/X(x) = (1/2)(T'(t))/T(t)

Since the left side only depends on x and the right side only depends on t, both sides must be equal to a constant, denoted as -λ²:

(X''(x))/X(x) = -λ²

(1/2)(T'(t))/T(t) = -λ²

Simplifying the second equation, we have:

T'(t)/T(t) = -2λ²

Solving the second equation, we find:

T(t) = Ce^(-2λ²t)

Applying the boundary condition U(0, t) = 0, we have:

U(0, t) = X(0)T(t) = 0

Since T(t) ≠ 0, we must have X(0) = 0.

Applying the boundary condition U(3, t) = 0, we have:

U(3, t) = X(3)T(t) = 0

Again, since T(t) ≠ 0, we must have X(3) = 0.

Therefore, we can conclude that X(x) must satisfy the following boundary value problem:

X''(x)/X(x) = -λ²

X(0) = 0

X(3) = 0

The general solution to this ordinary differential equation is given by:

X(x) = Asin(λx) + Bcos(λx)

Applying the initial condition U(x, 0) = 5*sin(4πx), we have:

U(x, 0) = X(x)T(0) = X(x)C

Comparing this with the given initial condition, we can conclude that T(0) = C = 5.

Therefore, the complete solution for U(x, t) is given by:

U(x, t) = Σ [Aₙsin(λₙx) + Bₙcos(λₙx)]*e^(-2(λₙ)²t)

where:

Σ represents the summation over all values of n

λₙ are the eigenvalues obtained from solving the boundary value problem for X(x).

To find the eigenvalues λₙ, we substitute the boundary conditions into the general solution for X(x):

X(0) = 0: Aₙsin(0) + Bₙcos(0) = 0

X(3) = 0: Aₙsin(3λₙ) + Bₙcos(3λₙ) = 0

From the first equation, we have Bₙ = 0.

From the second equation, we have Aₙ*sin(3λₙ) = 0. Since Aₙ ≠ 0, we must have sin(3λₙ) = 0.

This implies that 3λₙ = nπ, where n is an integer.

Therefore, λₙ = (nπ)/3.

Substituting the eigenvalues into the general solution, we have:

U(x, t) = Σ [Aₙ*sin((nπ/3)x)]*e^(-(nπ/3)²t)

where Aₙ are the coefficients that can be determined from the initial condition.

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The solutions to the equation 2 log(x) = log(2) + log(3x - 4) are x = 2 and x = 4

From the question, we have the following parameters that can be used in our computation:

2 log(x) = log(2) + log(3x - 4)

Apply the rule of logarithm

So, we have

log(x²) = log(2 * (3x - 4))

This gives

log(x²) = log(6x - 8)

Cancel out the logarithms

x² = 6x - 8

So, we have

x² - 6x + 8 = 0

Factorize

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

So, we have

x = 2 and x = 4

Hence, the solutions to the equation are x = 2 and x = 4

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The value of x in the expression given is -7/6

(6x + 9)/2 =

First step is to cross multiply

6x + 9 = 2

Collect like terms

6x = 2 - 9

6x = -7

divide both sides by 6 to isolate x

x = -7/6

Therefore the value of x in the expression given is -7/6

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Answer:

$2100

Step-by-step explanation:

40% of 1500 is 600

1500 + 600 = 2100

The selling price will be $2100

Option(A) is the correct answer is A. 1,335 birds.

To calculate the number of birds that will be left after 20 years, we need to consider the annual decline rate of 2%.

We can use the formula for exponential decay:

N = N₀ * (1 - r/100)^t

Where:

N is the final number of birds after t years

N₀ is the initial number of birds (2,000 in this case)

r is the annual decline rate (2% or 0.02)

t is the number of years (20 in this case)

Plugging in the values, we get:

N = 2,000 * (1 - 0.02)^20

N = 2,000 * (0.98)^20

N ≈ 2,000 * 0.672749

N ≈ 1,345.498

Rounded to the nearest whole number, the number of birds that will be left after 20 years is 1,345.

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Answer:

[tex]\displaystyle 64-32\sqrt{2}+\frac{32\sqrt{2}}{3}\approx3.66[/tex]

Step-by-step explanation:

[tex]\displaystyle \iint_R(3x-y)\,dA\\\\=\int^\frac{\pi}{2}_\frac{\pi}{4}\int^4_0(3r\cos\theta-r\sin\theta)\,r\,dr\,d\theta\\\\=\int^\frac{\pi}{2}_\frac{\pi}{4}\int^4_0(3r^2\cos\theta-r^2\sin\theta)\,dr\,d\theta\\\\=\int^\frac{\pi}{2}_\frac{\pi}{4}\int^4_0r^2(3\cos\theta-\sin\theta)\,dr\,d\theta\\\\=\int^\frac{\pi}{2}_\frac{\pi}{4}\frac{64}{3}(3\cos\theta-\sin\theta)\,d\theta\\\\=\int^\frac{\pi}{2}_\frac{\pi}{4}\biggr(64\cos\theta-\frac{64}{3}\sin\theta\biggr)\,d\theta[/tex]

[tex]\displaystyle =\biggr(64\sin\theta+\frac{64}{3}\cos\theta\biggr)\biggr|^\frac{\pi}{2}_\frac{\pi}{4}\\\\=\biggr(64\sin\frac{\pi}{2}+\frac{64}{3}\cos\frac{\pi}{2}\biggr)-\biggr(64\sin\frac{\pi}{4}+\frac{64}{3}\cos\frac{\pi}{4}\biggr)\\\\=64-\biggr(64\cdot{\frac{\sqrt{2}}{2}}+\frac{64}{3}\cdot{\frac{\sqrt{2}}{2}}\biggr)\\\\=64-32\sqrt{2}+\frac{32\sqrt{2}}{3}\biggr\\\\\approx3.66[/tex]

Answer:

Domain: x [tex]\geq[/tex] -5

Range: y [tex]\geq[/tex] -3

Step-by-step explanation:

The domain is all the possible inputs or x value.  x will be greater than -5.

The range is all of the possible outputs or y value.   y will be greater than -3

Answer:

Domain:  [-5, ∞)

Range:  [-3, ∞)

Step-by-step explanation:

The given graph shows a continuous curve with a closed circle at the left endpoint (-5, -3) and an arrow at the right endpoint.

A closed circle indicates the value is included in the interval.

An arrow shows that the function continues indefinitely in that direction.

The domain of a function is the set of all possible input values (x-values).

As the leftmost x-value of the curve is x = -5, and it continues indefinitely in the positive direction, the domain of the  graphed function is:

The range of a function is the set of all possible output values (y-values).

From observation, it appears that the minimum y-value of the curve is y = -3. The curve continues indefinitely in the positive direction in quadrant I. Therefore, the range of the graphed function is:

The distance between A and B based on the diagram is approximately 1,685.2 cm.

Hypotenuse = 1,920 cm

Adjacent = 920 cm

A and B = opposite

Hypotenuse² = adjacent² + opposite²

1920² = 920² + AB²

3,686,400 = 846,400 + AB²

3,686,400 - 846,400 = AB²

2,840,000 = AB²

find the square root of both sides

AB = √2,840,000

AB = 1685.229954635271

Approximately,

AB = 1,685.2 cm

Hence, distance between A and B is 1,685.2 cm.

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To calculate the interest earned in each case, we can use the formula for compound interest:

A = P(1 + r/n)^(nt) - P

where:

A is the final amount (including principal and interest)

P is the principal amount (initial deposit)

r is the interest rate (in decimal form)

n is the number of times interest is compounded per year

t is the number of years

Given:

P = $21,000

r = 4% = 0.04

(a) If interest is compounded semiannually, n = 2 (twice a year), and t = 2 years. Plugging these values into the formula, we get:

A = 21000(1 + 0.04/2)^(2*2) - 21000

Calculate A to find the final amount and subtract the principal to get the interest earned.

(b) If interest is compounded quarterly, n = 4 (four times a year), and t = 2 years. Plugging these values into the formula, we get:

A = 21000(1 + 0.04/4)^(4*2) - 21000

Calculate A to find the final amount and subtract the principal to get the interest earned.

(c) To find the difference in interest earned when compounded quarterly, subtract the interest earned in part (a) from the interest earned in part (b).

By calculating the interest earned and the difference in interest, we can determine the specific amounts.

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