Really need help with this GeoMETry qUIz
The following is an incomplete paragraph proving that ∠WRS ≅ ∠VQT, given the information in the figure where segment UV is parallel to segment WZ. :
(picture attached)
According to the given information, segment UV is parallel to segment WZ while angles SQU and VQT are vertical angles. Angle VQT is congruent to angle SQU by the Vertical Angles Theorem. Because angles SQU and WRS are _____________ angles, they are congruent according to the ____________ Angles Postulate. Finally, angle VQT is congruent to angle WRS by the Transitive Property of Equality.

Which term accurately completes the proof?

1Same-side interior
2Corresponding
3Alternate interior
4Vertical

Answer:

2.264 (3 d.p.)

Step-by-step explanation:

Given integral:

[tex]\displaystyle \int^0_{-1} \left(4x^6+2x\right)^3\left(12x^5+1\right)\;\text{d}x[/tex]

First, evaluate the indefinite integral using the method of substitution.

[tex]\textsf{Let} \;\;u = 4x^6+2x[/tex]

Find du/dx and rewrite it so that dx is on its own:

[tex]\dfrac{\text{d}u}{\text{d}x}=24x^5+2 \implies \text{d}x=\dfrac{1}{24x^5+2}\; \text{d}u[/tex]

Rewrite the original integral in terms of u and du, and evaluate:

[tex]\begin{aligned}\displaystyle \int\left(4x^6+2x\right)^3\left(12x^5+1\right)\;\text{d}x&=\int \left(u\right)^3\left(12x^5+1\right)\cdot \dfrac{1}{24x^5+2}\; \text{d}u\\\\&=\int \left(u\right)^3\left(12x^5+1\right)\cdot \dfrac{1}{2(12x^5+1)}\; \text{d}u\\\\&=\int \dfrac{u^3\left(12x^5+1\right)}{2(12x^5+1)}\; \text{d}u\\\\&=\displaystyle \int \dfrac{u^3}{2}\; \text{d}u\\\\&=\dfrac{u^{3+1}}{2(3+1)}+C\\\\&=\dfrac{u^4}{8}+C\end{aligned}[/tex]

Substitute back u = 4x⁶ + 2x:

                                              [tex]=\dfrac{(4x^6+2x)^4}{8}+C[/tex]

Therefore:

[tex]\displaystyle \int \left(4x^6+2x\right)^3\left(12x^5+1\right)\;\text{d}x=\dfrac{(4x^6+2x)^4}{8}+C[/tex]

To evaluate the definite integral, we must first determine any intervals within the given interval -1 ≤ x ≤ 0 where the curve lies below the x-axis. This is because when we integrate a function that lies below the x-axis, it will give a negative area value.

Find the x-intercepts by setting the function to zero and solving for x.

[tex]\left(4x^6+2x\right)^3\left(12x^5+1\right)=0[/tex]

Therefore:

[tex]\begin{aligned}\left(4x^6+2x\right)^3&=0\\4x^6+2x&=0\\x(4x^5+2)&=0\end{aligned}[/tex]

[tex]x=0[/tex]

[tex]\begin{aligned}4x^5+2&=0\\4x^5&=-2\\x^5&=-\frac{1}{2}\\x&=\sqrt[5]{-\dfrac{1}{2}}\end{aligned}[/tex]

[tex]\begin{aligned}12x^5+1&=0\\12x^5&=-1\\x^5&=-\dfrac{1}{12}\\x&=\sqrt[5]{-\dfrac{1}{12}}\end{aligned}[/tex]

Therefore, the curve of the function is:

So to calculate the total area, we need to calculate the positive and negative areas separately and then add them together, remembering that if you integrate a function to find an area that lies below the x-axis, it will give a negative value.

Integrate the function between -1 and ⁵√(-1/2).

As the area is below the x-axis, we need to negate the integral so that the resulting area is positive:

[tex]\begin{aligned}A_1&=-\displaystyle \int_{-1}^{\sqrt[5]{-\frac{1}{2}}} \left(4x^6+2x\right)^3\left(12x^5+1\right)\;\text{d}x\\\\&=-\left[\dfrac{(4x^6+2x)^4}{8}\right]_{-1}^{\sqrt[5]{-\frac{1}{2}}}\\\\&=-\left[\left(\dfrac{\left(4\left(\sqrt[5]{-\frac{1}{2}}\right)^6+2\left(\sqrt[5]{-\frac{1}{2}}\right)\right)^4}{8}\right)-\left(\dfrac{(4(-1)^6+2(-1))^4}{8}\right)\right]\\\\&=-[0-2]\\\\&=2\end{aligned}[/tex]

Integrate the function between ⁵√(-1/2) and ⁵√(-1/12).

[tex]\begin{aligned}A_2&=\displaystyle \int_{\sqrt[5]{-\frac{1}{2}}} ^{\sqrt[5]{-\frac{1}{12}}} \left(4x^6+2x\right)^3\left(12x^5+1\right)\;\text{d}x\\\\&=\left[\dfrac{(4x^6+2x)^4}{8}\right]_{\sqrt[5]{-\frac{1}{2}}}^{\sqrt[5]{-\frac{1}{12}}}\\\\&=\left(\dfrac{\left(4\left(\sqrt[5]{-\frac{1}{12}}\right)^6+2\left(\sqrt[5]{-\frac{1}{12}}\right)\right)^4}{8}\right)-\left(\dfrac{\left(4\left(\sqrt[5]{-\frac{1}{2}}\right)^6+2\left(\sqrt[5]{-\frac{1}{2}}\right)\right)^4}{8}\right)\\\\\end{aligned}[/tex]

     [tex]\begin{aligned}&=\dfrac{625}{648\sqrt[5]{12^4}}-0\\\\&=0.132117398...\end{aligned}[/tex]

Integrate the function between ⁵√(-1/12) and 0.

As the area is below the x-axis, we need to negate the integral so that the resulting area is positive:

[tex]\begin{aligned}A_3&=-\displaystyle \int_{\sqrt[5]{-\frac{1}{12}}}^0 \left(4x^6+2x\right)^3\left(12x^5+1\right)\;\text{d}x\\\\&=-\left[\dfrac{(4x^6+2x)^4}{8}\right]_{\sqrt[5]{-\frac{1}{12}}}^0\\\\&=-\left[\left(\dfrac{(4(0)^6+2(0))^4}{8}\right)-\left(\dfrac{\left(4\left(\sqrt[5]{-\frac{1}{12}}\right)^6+2\left(\sqrt[5]{-\frac{1}{12}}\right)\right)^4}{8}\right)\right]\\\\&=-\left[0-\dfrac{625}{648\sqrt[5]{12^4}}\right]\\\\&=\dfrac{625}{648\sqrt[5]{12^4}}\\\\&=0.132117398...\\\\\end{aligned}[/tex]

To evaluate the definite integral, sum A₁, A₂ and A₃:

[tex]\begin{aligned}\displaystyle \int^0_{-1} \left(4x^6+2x\right)^3\left(12x^5+1\right)\;\text{d}x&=2+2\left( \dfrac{625}{648\sqrt[5]{12^4}}\right)\\\\&=2+ \dfrac{625}{324\sqrt[5]{12^4}}\right}\\\\&=2.264\; \sf (3\;d.p.)\end{aligned}[/tex]

The result of the operation 11/12 + (-7/12) simplifies to 1/3, because the sum of the numerators is 4, and we keep the same denominator of 12, giving us 4/12. In simplest form, this fraction is 1/3.

The question is asking us to perform an addition operation on two fractions. Specifically, we are asked to add 11/12 and -7/12. When adding or subtracting fractions, the most important thing is that they have the same denominator, which our fractions do. Here's how you perform the operation:

Add the numerators of the fractions: 11 + (-7) = 4.

You keep the same denominator: 12.

So, 11/12 + (-7/12) = 4/12.

To simplify this fraction, we find the greatest common divisor of 4 and 12, which is 4, and divide both the numerator and denominator by it. So, 4/12 simplifies to 1/3.

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Answer:

Feet is the appropriate unit.

Step-by-step explanation:

Feet will be an appropriate measure to describe the depth of a lake.

Miles is a very large unit usually used to describe distance between two places.

Cubic centimeters is a unit of volume.

Millimeters is a very small unit used to describe small objects like the diameter of a penny etc.

Therefore, feet is the most appropriate unit to measure the depth of the lake.

A matrix with a determinant of zero is called a singular matrix and doesn't have an inverse because it indicates the system of equations it represents doesn't have a unique solution, and the transformation it performs is not reversible.

When a matrix has a determinant of zero (det A = 0), it is classified as a singular matrix and is not invertible. This characteristic implies that the matrix cannot be used to find unique solutions for a set of linear equations because such a matrix corresponds to a system of equations that has either no solution or an infinite number of solutions.

The ability to have an inverse matrix is crucial as it allows us to solve matrix equations and essentially 'undo' the transformations applied by the original matrix.

The reason why a zero determinant indicates the absence of an inverse lies in the mathematics of linear transformations.

A determinant of zero suggests that the transformation associated with the matrix collapses the dimensionality of the space, which means some information about the original vectors is lost, and hence, an inverse operation to recover the original vectors cannot exist.

To further illustrate this, consider the matrix equation AB = I, where A is our original matrix and B is its supposed inverse yielding the identity matrix I. It follows from the properties of determinants that det(AB) = det(A) x det(B).

If det(A) is zero, then the product det(A) x det(B) will also be zero, not equal to 1, which is the determinant of the identity matrix. Therefore, B cannot serve as the inverse of A.

In other words, having a non-zero determinant is a prerequisite for a matrix to have an inverse, as it ensures that the system of equations it represents is solvable and that the matrix transformation is reversible.

3,−6,12,−24,...

You can rewrite this as:

(−1)n3⋅(1,2,4,8,...)

If we focus on 1,2,4,8...

an+1=2an
with a0=1.

This can be written out as a nice sum:
N∑n=02n=20+21+22+23+...
=1+2+4+8+...

Thus, now we can recombine everything to get:

3N∑n=0(−1)n2n

The options A, B, and D have a proportional relationship.

It is the relationship between two variables where their ratios are equivalent.

Consider the first equation, x = 2y.

It can be written like the ratio of x and y.

i.e              [tex]\frac{x}{y} -\frac{2}{1}[/tex]

So, option A is proportional at x = 2 and y = 1.

For option B, It can rewrite as

[tex]\frac{a}{b} =\frac{1}{3}[/tex]

So, option B is proportional when a = 1 and b = 3.

Option C is not proportional since it can't be written as a ratio.

Option D can be written as

[tex]\frac{7}{3} =\frac{n}{x}[/tex]

It is proportional when n = 7 and x = 3.

Option E is not proportional since it is not equal.

Therefore the proportional options are A,B, and D.

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The value of x is 4.

To solve for x, we'll use the property of parallelograms that opposite angles are supplementary.

Given that one angle is [tex]\(140^\circ\)[/tex], we can set up the equation:

140 + 10x = 180

Now let's solve for x:

10x = 180 - 140

10x = 40

[tex]x = \frac{40}{10}[/tex]

x = 4

Therefore, the value of x is 4. Adjacent angles in a parallelogram are indeed supplementary.

Answer:

Option c

Step-by-step explanation:

A parent function is given by f(x) = x²

If the graph of the parent quadratic function is shifted a units to the left then new function will become

f'(x) = (x+a)²

Now the new function f'(n) is shifted b units down then the shifted function will be

f" (x) = (x + a)² - b

Therefore, Option c. is the answer.

Answer:

B is your answer

Step-by-step explanation:

g(x)=(1/4x)^2

Answer:

The car travels 53 miles per hour.

Step-by-step explanation:

time (hours)           4       6      8       10

distance (miles)   212   318   424   530

The rate of change can be given as:

[tex]\frac{318-212}{6-4}[/tex] = [tex]\frac{106}{2}[/tex] = 53 mph

[tex]\frac{424-318}{8-6}[/tex] = [tex]\frac{106}{2}[/tex] = 53 mph

Hence, the rate of change is 53 mph or we can say the car travels 53 miles per hour.

Answer:

25 red flowers

Step-by-step explanation:

yeah

The sum of four consecutive even integers that equal 36 is broken down to find the smallest integer. Once the smallest integer is found (6), we can determine that the third largest integer in the sequence is 8.

To solve for the third largest integer in a set of four consecutive even integers that sum to 36, we can define the smallest integer as x. The next integers would be x + 2, x + 4, and x + 6. Setting up the equation:

x + (x + 2) + (x + 4) + (x + 6) = 36

Combining like terms, we get:

4x + 12 = 36

Subtracting 12 from both sides:

4x = 24

Dividing by 4:

x = 6

Now we have our integers: 6, 8, 10, 12. The third largest integer, which is the second smallest, is 8.