a-b+2c=7
b+c=1
a-2b-3c=0

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]

we have that

[tex]PR = 4x - 2\\RS = 3x - 5[/tex]

we know that

[tex]PS=PR+RS[/tex]

substitute the values of PR and RS in the formula above

[tex]PS=(4x-2)+(3x-5)[/tex]

Combine like terms

[tex]PS=(7x-7)[/tex]

therefore

the answer is

[tex]PS=(7x-7)[/tex]

The expression which represents PS is; 7x -7

According to the question;

From observation;

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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.

Answer:

Step-by-step explanation:

The length of her lawn=64.15 feet.

Width of her lawn=22.70 meters.

we are given that:

1 foot=0.3048 meters.

Hence,

64.15 feet=21.86232 meters.

Hence, length of lawn=21.86232 meters.

Hence, width of the lawn is greater than the length of the lawn.

( since 22.70 meters> 21.862332 meters)

Also 1 meter=1.0936 yards.

so, width of lawn in yards is:

22.70×1.0936=24.82472 yards.

length of lawn is: 23.9086462752 yards.

Hence, width of lawn is 0.9160737 yards more than the length of lawn.

The student's problem involves the use of the trapezoid rule to approximate total distance traveled by a race car, finding the speed at a particular time point (Sv(6)), and calculating the car's average speed to see if it meets the qualifying speed.

To solve this problem, we need to conduct several mathematical operations. Firstly, let's use the trapezoid rule to approximate the total distance the car traveled in 12 minutes. Then we'll find an approximation for Sv(6) and explain its meaning. Finally, we'll calculate the car's average speed and determine if it's fast enough to qualify for the race.

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they receive 25 per entry but 5 goes towards the cost of the race

 so 25-5 = 20 per entry is for the charity

 so 20x + 5000 = 25000

subtract 5000 from each side

20x= 20000

 divide both sides by 20

 x = 20000/ 20  

x = 1000

 they need at least 1000 entries to make 25000

 so to make more than 25000, they need 1001 entries

we know that

Vertical angles are a pair of opposite and congruent angles formed by intersecting lines

In this problem

[tex](7x-8)=(6x+11)[/tex] --------> by vertical angles

Solve for x

Combine like terms

[tex](7x-6x)=(11+8)[/tex]

[tex]x=19\ degrees[/tex]

therefore

the answer is

the value of x is [tex]19\ degrees[/tex]

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.

The surface area of the triangular prism is 468 (square cm).

The surface area A of a triangular prism is given by the formula:

[tex]\[ A = 2A_{\text{base}} + P_{\text{base}} \times h \][/tex]

where:

- [tex]\( A_{\text{base}} \)[/tex] is the area of one of the triangular bases,

- [tex]\( P_{\text{base}} \)[/tex] is the perimeter of one of the triangular bases,

- h is the distance between the bases.

The area of a right triangle is given by:

[tex]\[ A_{\text{base}} = \frac{1}{2} \times \text{base} \times \text{height} \][/tex]

And the perimeter of a right triangle is the sum of the lengths of its three sides.

Given that the legs of the right triangles forming the bases are 9 cm and 12 cm, and the distance between the bases is 10 cm, we can calculate the surface area.

1. Area of the triangular base [tex](\(A_{\text{base}})\)[/tex]:

[tex]\[ A_{\text{base}} = \frac{1}{2} \times 9 \times 12 \][/tex]

2. Perimeter of the triangular base [tex](\(P_{\text{base}})\)[/tex]:

[tex]\[ P_{\text{base}} = 9 + 12 + \sqrt{9^2 + 12^2} \][/tex]

3. Surface area of the triangular prism A:

[tex]\[ A = 2 \times A_{\text{base}} + P_{\text{base}} \times 10 \][/tex]

Calculate each part and find the total surface area.

[tex]\[ A_{\text{base}} = \frac{1}{2} \times 9 \times 12 = 54 \, \text{cm}^2 \][/tex]

[tex]\[ P_{\text{base}} = 9 + 12 + \sqrt{9^2 + 12^2} = 9 + 12 + 15 = 36 \, \text{cm} \][/tex]

[tex]\[ A = 2 \times 54 + 36 \times 10 = 108 + 360 = 468 \, \text{cm}^2 \][/tex]

So, the surface area of the triangular prism is [tex]\(468 \, \text{cm}^2\)[/tex].

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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Final answer:

Elga's age is determined by setting up an equation E + 3E = 32 and solving for E. After simplifying, we find that Elga is 8 years old.

Explanation:

To solve this problem, we can use algebra to set up two equations based on the information given that Alvin's age is three times Elga's age and the sum of their ages is 32.

Let's let E represent Elga's age. According to the problem, Alvin's age will be 3E because it is three times Elga's age. We can write the following equation to represent the sum of their ages:

E + 3E = 32

Combining like terms, we have:

4E = 32

Dividing both sides by 4 to solve for E, we get:

E = 32 / 4

Therefore, Elga's age is:

E = 8

Elga is 8 years old.