find the mass of a lamina bounded by y = 2 √ x , x = 1 , and y = 0 y=2x,x=1,and y=0 with the density function σ ( x , y ) = x 3 σ(x,y)=x 3 . round your answer to four decimal places.

Answer:

on what...like brush how do we help u if u don't have a question

Answer: look at the picture

Step-by-step explanation: Hope this help :D

Answer:

\({k = - \frac{363}{8} \: \: \: or \: \: - 45 \frac{3}{8} } \\ \)

Step-by-step explanation:

-8(k + 46) = -5

Expand the terms on the right

-8k - 368 = - 5

Add 368 to both sides of the equation

- 8k - 368 + 368 = - 5 + 368

- 8k = 363

Divide both sides by - 8

\( \frac{ - 8k}{ - 8} = - \frac{363}{8} \\\)

We have the final answer as

\(k = - \frac{363}{8} \: \: \: or \: \: - 45 \frac{3}{8} \\ \)

Hope this helps you

a = 4 and b = 5 are the answers to the system of equations.

Let's square both sides of the first equation to eliminate the square root:

√a + b = 7

(√a + b)² = 7²

a + 2√ab + b² = 49

a + b² = 49 - 2√ab ---(1)

Now, let's square both sides of the second equation:

√b + a = 11

(√b + a)² = 11²

b + 2√ab + a² = 121

a² + b + 2√ab = 121 ---(2)

We can use equation (1) to substitute for √ab in equation (2):

a + b² = 49 - 2√ab

√ab = (49 - a - b²)/2

Substituting for √ab in equation (2), we get:

a² + b + 2(49 - a - b²)/2 = 121

Simplifying and rearranging, we get:

a² - a + b² - b - 36 = 0

(a - 1/2)² + (b - 1/2)² = 37.25

This is the equation of a circle centered at (1/2, 1/2) with a radius √37.25. We need to find the points where this circle intersects the line defined by equation (1).

Substituting b = 49 - a - 2√(a(49 - a))/2 into equation (1), we get:

a + (49 - a - 2√(a(49 - a)))² = 49 - 2√a(49 - a)

Simplifying and rearranging, we get:

4a³ - 294a² + 2421a - 5929 = 0

Using a numerical solver or the rational root theorem, we can find that one solution of this cubic equation is a = 4.

Substituting this value back into equation (1), we can solve for b:

4 + b² = 49 - 2√(4b)

b² + 2√(4b) - 45 = 0

Using the quadratic formula, we get:

b = 5

Therefore, the solutions of the system of equations are a = 4 and b = 5.

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Complete question:

√a+b=7 and √b +a = 11 If a and b are real numbers that satisfy the equation above, what is the value of a and b respectively?

The inverse function for the given function F(x)=1/x² -3x +1  is given by

f⁻¹(x) =(3 ± √(9 + 4/(x - 1))) /2.

Function f(x) is equals to,

F(x)=1/x² -3x +1

Inverse of a function, we need to swap the positions of the x and y variables and then solve for y.

Let's start with the original function

f(x) = 1/x^2 - 3x + 1

Now we will swap x and y,

⇒ x = 1/y^2 - 3y + 1

Next, Solve for y in terms of x

⇒ x = 1/y^2 - 3y + 1

⇒ x - 1 = 1/y^2 - 3y

⇒ 1/(x - 1) = y^2 - 3y

⇒ 1/(x - 1) = y(y - 3)

⇒ y(y - 3) = 1/(x - 1)

⇒ y^2 - 3y - 1/(x - 1) = 0

Using the quadratic formula, solve for y to get inverse function we have,

y = (3 ± √(9 + 4/(x - 1))) / 2

Therefore, the inverse function of f(x) is equal to f⁻¹(x) =(3 ± √(9 + 4/(x - 1))) /2.

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The above question is incomplete, the complete question is:

F(x)=1/x squared -3x +1 then Find the inverse

Answer:

x = 12

Step-by-step explanation:

Theorem of Thales

\(\frac{28}{12} =\frac{x+(x-3)}{x-3}\)

\(\frac{28}{12} =\frac{2x-3}{x-3}\)

\(28(x-3)=12(2x-3)\)

\(28x-84=24x-36\)

\(28x-24x=-36+84\)

\(4x=48\)

\(x=48/4\)

\(x=12\)

Hope this helps

Answer:

The correct option is;

A dilation by a scale factor of Two-fifths and then a translation of 3 units up

Step-by-step explanation:

Given that the coordinates of the vertices of square S are

(0, 0), (5, 0), (5, -5), (0, -5)

Given that the coordinates of the vertices of square S' are

(0, 1), (0, 3), (2, 3), (2, 1)

We have;

Length of side, s, for square S is s = √((y₂ - y₁)² + (x₂ - x₁)²)

Where;

(x₁, y₁) and (x₂, y₂) are the coordinates of two consecutive vertices

When (x₁, y₁) = (0, 0) and (x₂, y₂) = (5, 0), we have;

s = √((y₂ - y₁)² + (x₂ - x₁)²) = s₁ = √((0 - 0)² + (5 - 0)²) = √(5)² = 5

For square S', where (x₁, y₁) = (0, 1) and (x₂, y₂) = (0, 3)

Length of side, s₂, for square S' is s₂ = √((3 - 1)² + (0 - 0)²) = √(2)² = 2

Therefore;

The transformation of square S to S' involves a dilation of s₂/s₁ = 2/5

The after the dilation (about the origin),  the coordinates of S becomes;

(0, 0) transformed to (remains at) (0, 0) ....center of dilation

(5, 0) transformed to (5×2/5, 0) = (2, 0)

(5, -5) transformed to (2, -2)

(0, -5) transformed to (0, -2)

Comparison of (0, 0), (2, 0), (2, -2), (0, -2) and (0, 1), (0, 3), (2, 3), (2, 1) shows that the orientation is the same;

For (0, 0), (2, 0), (2, -2), (0, -2) we have;

(0, 0), (2, 0) the same y-values, (∴parallel to the x-axis)

(2, -2), (0, -2) the same y-values, (∴parallel to the x-axis)

For (0, 1), (0, 3), (2, 3), (2, 1) we have;

(0, 3), (2, 3) the same y-values, (∴parallel to the x-axis)

(0, 1), (2, 1) the same y-values, (∴parallel to the x-axis)

Therefore, the lowermost point closest to the y-axis in (0, 0), (2, 0), (2, -2), (0, -2) which is (0, -2) is translated to the lowermost point closest to the y-axis in (0, 1), (0, 3), (2, 3), (2, 1) which is (0, 1)

That is (0, -2) is translated to (0, 1) which shows that the translation is T((0 - 0), (1 - (-2)) = T(0, 3) or 3 units up

The correct option is therefore a dilation by a scale factor of Two-fifths and then a translation of 3 units up.

Answer:

a

Step-by-step explanation:

Answer:

k=1/4

Step-by-step explanation:

x² - 5kx+ k=0  ....(1)

suppose 2 roots: a and 4a

(x-a)(x-4a)=0

x² - 5ax + 4a² = 0 ....(2)

compare (1) and (2): -5a = -5k              a=k

4a² = k                4a² = a   (substitute)

a(4a-1) = 0

a = 0 or a = 1/4        (if a=0   k=0 and x=0 is not the solution)

a = 1/4      k = 1/4

(a₂ = 1/4 * 4 = 1)

Answer:

k = 1/4.

Step-by-step explanation:

Let the roots be  r and 4r   and using the fact that sum of the roots = -b/a and product = a/c, we have:

r + 4r = -(-5k)

r*4r = k

5r = 5k

4r^2 = k

From the 3rd equation:

r = k, therefore:

4k^2 = k

4k = 1

k = 1/4.

The line integral ∫[C] (Pdx + Qdy) over the given curve C consisting of the arc of the parabola y = x² from (0,0) to (1, 1), and the line segment from (1,1) to (0,1) is equal to 2/5.

What is integral?

The value obtained after integrating or adding the terms of a function that is divided into an infinite number of terms is generally referred to as an integral value.

To evaluate the line integral using Green's theorem, we need to find a vector field F = (P, Q) such that ∇ × F = Qₓ - Pᵧ, where Qₓ represents the partial derivative of Q with respect to x, and Pᵧ represents the partial derivative of P with respect to y.

Let's consider F = (P, Q) = (x²y², xy).

Now, let's calculate the partial derivatives:

Qₓ = ∂Q/∂x = ∂(xy)/∂x = y

Pᵧ = ∂P/∂y = ∂(x²y²)/∂y = 2x²y

The curl of F is given by ∇ × F = Qₓ - Pᵧ = y - 2x²y = (1 - 2x²)y.

Now, let's find the line integral using Green's theorem:

∫[C] (Pdx + Qdy) = ∫∫[R] (1 - 2x²)y dA,

where [R] represents the region enclosed by the curve C.

To evaluate the line integral, we need to parameterize the curve C.

The arc of the parabola y = x² from (0, 0) to (1, 1) can be parameterized as r(t) = (t, t²) for t ∈ [0, 1].

The line segment from (1, 1) to (0, 1) can be parameterized as r(t) = (1 - t, 1) for t ∈ [0, 1].

Using these parameterizations, the region R is bounded by the curves r(t) = (t, t²) and r(t) = (1 - t, 1).

Now, let's calculate the line integral:

∫∫[R] (1 - 2x²)y dA = ∫[0,1] ∫[t²,1] (1 - 2t²)y dy dx + ∫[0,1] ∫[0,t²] (1 - 2t²)y dy dx.

Integrating with respect to y first:

∫[0,1] [(1 - 2t²)(1 - t²) - (1 - 2t²)t²] dt.

Simplifying:

∫[0,1] [1 - 3t² + 2t⁴] dt.

Integrating with respect to t:

[t - t³ + (2/5)t⁵]_[0,1] = 1 - 1 + (2/5) = 2/5.

Therefore, the line integral ∫[C] (Pdx + Qdy) over the given curve C consisting of the arc of the parabola y = x² from (0,0) to (1,1), and the line segment from (1,1) to (0,1) is equal to 2/5.

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Answer:

The answer would be 41.8

Step-by-step explanation:

All you have to do is multiply length times width to get the area of an object.

\({\Large \begin{array}{llll} y=ab^x \end{array}} \\\\[-0.35em] ~\dotfill\\\\ f(-4)=1\hspace{5em}1=ab^{-4}\implies 1=\cfrac{a}{b^4}\implies b^4=a \\\\[-0.35em] ~\dotfill\\\\ f(4)=69\hspace{5em}69=ab^4\implies \stackrel{\textit{substituting from the 1st equation}}{69=(b^4)b^4\implies 69=b^8} \\\\\\ \sqrt[8]{69}=b\implies (69)^{\frac{1}{8}}=b \\\\[-0.35em] ~\dotfill\)

\(y=\sqrt{69}\left( (69)^{\frac{1}{8}} \right)^x\implies \boxed{y=\sqrt{69}(69)^{\frac{x}{8}}} \\\\\\ \textit{when x=6.5, what is "y"?}\qquad y=\sqrt{69}(69)^{\frac{6.5}{8}}\implies y\approx 1522.73\)

Adriana applies the same arrangement of books on the new bookshelf, which is based on the classification of books.

She put the novels, short stories, and poetry books on the top shelf, history and biography books on the second shelf, textbooks and reference materials on the third shelf, and cookbooks and magazines on the bottom shelf. Adriana has already organized her books based on the genre or category before she buys a new bookshelf. The same classification will be applied to the new bookshelf that will be added to accommodate more books. It is an easy and convenient way of arranging and finding books

Adriana has an efficient way of arranging her books on her bookshelves based on genre or category. She applies the same arrangement of books on the new bookshelf that she bought to accommodate more books. It is a convenient and efficient way of organizing books that can help anyone easily find the book they need.

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Answer:

430 minutes or 7 hours and 10 minutes

Step-by-step explanation:

From 8:35 am to 9 am is 25 minutes. There are 60 minutes in an hour and 9 am to 3 pm is 6 hours. 3:00 to 3:45 pm is an additional 45 minutes.

60 minutes x 6 hours = 360 minutes

360 + 25 + 45 = 430 minutes

Answer:

7 hours with 10 minutes

Step-by-step explanation:

First, find how much minutes it takes to get to 9 a.m, which is 25 minutes.

Then, count up to 3 p.m, which is 6 hours.

Then, add the 45 minutes to 25, which will give you 70, but since you have more than 60 minutes, convert those 60 minutes to an hour adding that up to 7 hours and ten minutes.

Interior angles play an important role in geometry, particularly in proving theorems related to parallel lines and angles. Understanding this concept can help you solve problems involving parallel lines and transversals.

Interior angles that lie on opposite sides of a transversal are called alternate interior angles. Alternate interior angles are formed when a transversal intersects two parallel lines. Here's how you can identify alternate interior angles:

1. Look for a transversal that intersects two parallel lines.

2. Identify any pair of interior angles that are on opposite sides of the transversal.

3. These angles are called alternate interior angles.

Alternate interior angles are congruent, which means they have the same measure. This property allows us to solve for unknown angles or prove certain geometric relationships.

For example, if we have two parallel lines cut by a transversal, and we know the measure of one alternate interior angle, we can use that information to find the measure of another alternate interior angle.

In the diagram, if angle 1 is 60 degrees, then angle 2 will also be 60 degrees. This is because alternate interior angles are congruent. Alternate interior angles play an important role in geometry, particularly in proving theorems related to parallel lines and angles. Understanding this concept can help you solve problems involving parallel lines and transversals.

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\(\qquad \qquad\huge \underline{\boxed{\sf Answer}}\)

Therefore, the are of given triangle with Length " a " cm and breadth " b " cm can be depicted as :

\( \qquad \sf  \dashrightarrow \: a \times b\)

\( \qquad \sf  \dashrightarrow \: ab \: \: cm {}^{2} \)

Answer:

if i didnt get anything mixed up it would be A. dependent

Step-by-step explanation:

When the outcome affects the second outcome, which is what we called dependent events. Dependent events: Two events are dependent when the outcome of the first event influences the outcome of the second event.

The equation of the transformed exponential function g(x) is g(x) = 2^-x - 1

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

Parent function: y = 2^x

The graph of the transformed exponential function g(x) passes through the points  (-2,3), (-1,1), (0,0), (1,-0.5) and (2, -0.75)

So, we have the following transformation steps:

1st Transformation:

Reflect y = 2^x across the y-axis

So, we have

y = 2^-x

2nd Transformation:

Translate y = 2^-x down by 1 unit

So, we have

y = 2^-x - 1

This means that

g(x) = 2^-x - 1

Hence, the equation of the function g(x) is g(x) = 2^-x - 1

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The equation represents a linear function, as it is a proportional relationship, and thus the rate of change of the output variable relative to the input variable is constant.

A proportional relationship is defined according to the equation presented as follows:

y = kx.

In which k is the constant of proportionality, representing the increase in the output variable y when the constant variable x is increased by one.

A proportional relationship is a special case of a linear function, as it has an intercept of zero.

The equation for this problem is given as follows:

t = 10.8w.

Which is a proportional relationship with a constant of 10.8.

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The pairs of equivalent expressions matched together are;

(-14+3/2b)+(1+8/2b) = -15-5/2b

(5+2b)+(2b+3/2) = 4b+13/2

(7/2b-3)-(8+6b) = -5/2b-11

(-10+b)+(7b-5) = 8b-15

Following PEMDAS

P = parenthesis

E = Exponents

M = Multiplication

D = Division

A = Addition

S = subtraction

(-14+3/2b)-(1+8/2b)

open parenthesis

= -14 + 3/2b - 1 - 8/2b

combine like terms

= -15 + 3/2b - 4b

= -15 + (3b-8b)/2

= -15 + 5/2b

(5+2b)+(2b+3/2)

open parenthesis

= 5 + 2b + 2b + 3/2

= 4b + (10+3)/2

= 4b + 13/2

(7/2b-3)-(8+6b)

= 7/2b - 3 - 8 - 6b

= (7b-12b)/2 -11

= -5/2b - 11

(-10+b)+(7b-5)

= -10 + b + 7b - 5

= -15 + 8b

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The angle must be measured with an accuracy of 2.291º for the percent error in estimating the height of the tree to be less than 4%.

To find the required accuracy of the angle measurement for the percent error in estimating the height of the tree to be less than 4%, follow these steps:

1. Calculate the height of the tree using the given angle of elevation (71.4º) and distance from the base (50 feet) with the tangent function: height = 50 * tan(71.4º).
2. Calculate 4% of the tree's height.
3. Use the small angle approximation to find the change in angle (∆θ) that would produce a 4% change in height: ∆θ = (4% of height) / (50 * tan(71.4º)).
4. Convert the change in angle (∆θ) to degrees and round your answer to three decimal places.

Following these steps:

1. height = 50 * tan(71.4º) ≈ 139.296 feet
2. 4% of height = 0.04 * 139.296 ≈ 5.572 feet
3. ∆θ = 5.572 / (50 * tan(71.4º)) ≈ 0.039988
4. ∆θ ≈ 0.039988 radians = 2.291 degrees

So, the angle must be measured with an accuracy of 2.291º for the percent error in estimating the height of the tree to be less than 4%.

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A variable is any property, number, or quantity that can be measured or counted. Variables can also be referenced as data items.

Inference: Inference uses selected samples from a population to estimate the properties of unknown people.

A record (or record) is a collection of data. For tabular data, a record corresponds to one or more database tables, each table column represents a specific variable, and each row corresponds to a particular record in that data set.

Therefore:

All the data collected in particular is referred to as Data Set.

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Arranging the given numbers in ascending order:

11, 36, 40, 54, 68, 88, 90, 90

There are eight numbers in the set, and since it's an even number, we need to find the average of the two middle values: 54 and 68.

Median = (54 + 68) / 2 = 122 / 2 = 61

Therefore, the correct answer is that the median of the given set of numbers is 61.

Answer:

40, 90, 36, 68,90, 11,88,54.

Step-by-step explanation:

Just took it

Use the 3-point curved line drawing tool to draw the average cost curve for quantities q = 1, q = 2, and q = 3. Connect these points smoothly to form the average cost curve.

To calculate the formulas for average fixed cost (AFC), marginal cost (MC), average variable cost (AVC), and average cost (AC) based on the cost function C = 6 + 8q, we can use the following equations: Average Fixed Cost (AFC): AFC = Total Fixed Cost (TFC) / Quantity (q). Since the cost function C = 6 + 8q does not have any fixed cost component, AFC would be zero. Marginal Cost (MC): MC = Change in Total Cost (ΔTC) / Change in Quantity (Δq). The cost function C = 6 + 8q has a constant marginal cost of 8. Average Variable Cost (AVC): AVC = Total Variable Cost (TVC) / Quantity (q). Since the cost function C = 6 + 8q does not have any variable cost component, AVC would be the same as MC, which is 8.

Average Cost (AC): AC = Total Cost (TC) / Quantity (q); AC = (Total Fixed Cost + Total Variable Cost) / Quantity; AC = (6 + 8q) / q; AC = 6/q + 8. Now, for the graphical representation: Use the line drawing tool to draw the marginal cost curve, which is a straight line with a slope of 8. Label this line 'MC'. Use the 3-point curved line drawing tool to draw the average cost curve for quantities q = 1, q = 2, and q = 3. Connect these points smoothly to form the average cost curve. Label this curve 'AC'. Please note that the shape and position of the curves will depend on the specific quantities chosen, but the general trend will be a downward-sloping MC curve intersecting the U-shaped AC curve.

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Answer:

Mr. A initially paid approximately N8000 for the land.

Step-by-step explanation:

Step 1: Let's assume Mr. A initially purchased the land for a certain amount, which we'll call "x" in currency units.

Step 2: Mr. A sold the land to Mr. B at a profit of 10%. This means Mr. A sold the land for 110% of the amount he paid (1 + 10/100 = 1.10). Therefore, Mr. A received 1.10x currency units from Mr. B.

Step 3: Mr. B sold the land to Mr. C at a gain of 5%. This means Mr. B sold the land for 105% of the amount he paid (1 + 5/100 = 1.05). Therefore, Mr. B received 1.05 * (1.10x) currency units from Mr. C.

Step 4: According to the given information, Mr. C paid N1240 more for the land than Mr. A paid. This means the difference between what Mr. C paid and what Mr. A paid is N1240. So we have the equation: 1.05 * (1.10x) - x = N1240

Step 5: Simplifying the equation: 1.155x - x = N1240

Step 6: Solving for x: 0.155x = N1240

x = N1240 / 0.155

x ≈ N8000

Therefore, in conclusion, Mr. A initially paid approximately N8000 for the land.

We have to determine whether the given series converges or diverges. The given series is as follows: `(n+4)! / 4!(n!)` Let's use the ratio test to find out if this series converges or diverges.

The Ratio Test: It is one of the tests that can be used to determine whether a series is convergent or divergent. It compares each term in the series to the term before it. We can use the ratio test to determine the convergence or divergence of series that have positive terms only. Here, a series `Σan` is convergent if and only if the limit of the ratio test is less than one, and it is divergent if and only if the limit of the ratio test is greater than one or infinity. The ratio test is inconclusive if the limit is equal to one. The limit of the ratio test is `lim n→∞ |(an+1)/(an)|` Let's apply the Ratio test to the given series.

`lim n→∞ [(n+5)! / 4!(n+1)!] * [n!(n+1)] / (n+4)!` `lim n→∞ [(n+5)/4] * [1/(n+1)]` `lim n→∞ [(n^2 + 9n + 20) / 4(n^2 + 5n + 4)]` `lim n→∞ (n^2 + 9n + 20) / (4n^2 + 20n + 16)`

As we can see, the limit exists and is equal to 1/4. We can say that the given series converges. The series converges. To determine the convergence of the given series, we use the ratio test. The ratio test is a convergence test for infinite series. It works by computing the limit of the ratio of consecutive terms of a series. A series converges if the limit of this ratio is less than one, and it diverges if the limit is greater than one or does not exist. In the given series `(n+4)! / 4!(n!)`, the ratio test can be applied. Using the ratio test, we get: `

lim n→∞ |(an+1)/(an)| = lim n→∞ [(n+5)! / 4!(n+1)!] * [n!(n+1)] / (n+4)!` `= lim n→∞ [(n+5)/4] * [1/(n+1)]` `= lim n→∞ [(n^2 + 9n + 20) / 4(n^2 + 5n + 4)]` `= 1/4`

Since the limit of the ratio test is less than one, the given series converges.

The series converges to some finite value, which means that it has a sum that can be calculated. Therefore, the answer is a).

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Step-by-step explanation:

6(3h - 4) = 18h + (-24) = 18h -24

standard form for a linear equation means

• all coefficients must be integers, no fractions

• only the constant on the right-hand-side

• all variables on the left-hand-side, sorted

• "x" must not have a negative coefficient

\((\stackrel{x_1}{4}~,~\stackrel{y_1}{1})\qquad (\stackrel{x_2}{2}~,~\stackrel{y_2}{-2}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{-2}-\stackrel{y1}{1}}}{\underset{run} {\underset{x_2}{2}-\underset{x_1}{4}}}\implies \cfrac{-3}{-2}\implies \cfrac{3}{2} \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{1}=\stackrel{m}{\cfrac{3}{2}}(x-\stackrel{x_1}{4})\)

\(\stackrel{\textit{multiplying both sides by }\stackrel{LCD}{2}}{2(y-1)=2\left( \cfrac{3}{2}(x-4) \right)}\implies 2y-2 = 3(x-4)\implies 2y-2=3x-12 \\\\\\ -3x+2y-2=-12\implies -3x+2y=-10\implies \stackrel{\times -1\textit{ to both sides}}{3x-2y=10}\)

Answer:

x = 66/7 & area = 132/7

Step-by-step explanation:

7x-9=0

x=9/7

substitute value of x in 5x-3;

5(9/7)+3=66/7

Then, a*a,

thus, 2(66/7) = 132/7

Answer:

1/4 ÷ 6 would be 1/24

Step-by-step explanation:

Reduce the expression, if possible, by cancelling the common factors.

Exact Form: 1/24

Decimal Form: 0.041¯6

Answer:

The answer is 1/24.

Step-by-step explanation:

1/4 divided by 6

Here , 4 is the denominator in 1/4 and 6 is the numerator in 6/1

So, denominator and numerator will be multiplied .

=1/(4 x6)

=1/24

The remainder when f(x) is divided by x-1 can be calculated using the division theorem.

The remainder when f(x) is divided by x-1 is 6.

remainder is 6 and divisor is x-1.

According to the division theorem, when a polynomial function is divided by another polynomial function, the remainder is equal to the function evaluated at the point where the divisor of the division equation is equal to zero.

In this case, the divisor is x-1, and when x-1 is equal to zero, then x = 1.

Therefore, the remainder when f(x) is divided by x-1 can be calculated by evaluating the function f(x) at x = 1:

\(f(1) = 2(1)^3 + (1)^2 + 3\)

= 2 + 1 + 3

= 6

f(1) = 2(1)³ + (1)² + 3 = 6

Therefore, the remainder when f(x) is divided by x-1 is 6.

remainder is 6 and divisor is x-1.

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