Problem 6. Determine if the pair of planes is parallel or intersecting. If they are parallel, find the distance between them. If they intersect, find a parametrization for the line of intersection. 1. 3.x + 2y +z = 8 and x + 3y – z = 5. 2. 2.c – y +z = 4 and 4.0 – 2y + 2z = 3.

The surface area of the ramp is 3672 in².

Option A is the correct answer.

A rectangle is a two-dimensional shape where the length and width are different.

The area of a rectangle is given as:

Area = Length x width

We have,

The ramp consists of three rectangular surfaces and two triangles.

Now,

Triangle.

Base = 48 in

Height = 14 in

Area of a triangle.

= 1/2 x base x height

= 1/2 x 48 x 14

= 48 x 7

= 336 in²

Rectangle.

Since the back and the bottom rectangle are not included.

Front rectangle.

Length = 50 in

Width = 60 in

Area = 50 x 60

Area = 3000 in²

Now,

The surface area of the ramp.

= 300 + 2 (336)

= 3000 + 672

= 3672 in²

Thus,

The surface area of the ramp is 3672 in².

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The function [tex]\lim_{x \to 0} \frac{1 - \sqrt[3]{x^2 + 1}}{x^2}[/tex] has a limit

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

[tex]\lim_{x \to 0} \frac{1 - \sqrt[3]{x^2 + 1}}{x^2}[/tex]

Substitute 0 for x in the above expression

So, we have the following representation

[tex]\lim_{x \to 0} \frac{1 - \sqrt[3]{x^2 + 1}}{x^2} = \frac{1 - \sqrt[3]{0^2 + 1}}{0^2}[/tex]

Evaluate the expression

[tex]\lim_{x \to 0} \frac{1 - \sqrt[3]{x^2 + 1}}{x^2}=\infty[/tex]

This means that the limit of the expression is to infinity

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Answer: Let's call the length of the cake "l" and the width of the cake "w". The total area of the cake would be l * w.

Half of the cake is frosted with vanilla icing, so the area of the vanilla half would be (l * w) / 2.

And 2/3 of the vanilla half is to be covered with sprinkles, so the area covered with sprinkles would be (l * w) / 2 * 2/3 = (l * w) / 3.

The other half of the cake is frosted with chocolate icing, so the area of the chocolate half would be (l * w) / 2.

And 1/4 of the chocolate half is to be covered with sprinkles, so the area covered with sprinkles would be (l * w) / 2 * 1/4 = (l * w) / 4.

So, the total area of the cake covered with sprinkles would be (l * w) / 3 + (l * w) / 4.

Step-by-step explanation:

The variable y does not vary directly with x is wrong since when x increases by 1 unit y increases by 1/2 unit for all real x.

In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line. The slope of a line is a measure of its steepness. Mathematically, slope is calculated as "rise over run" (change in y divided by change in x).

here, we have,

Given that line passes through (0,0) and (2,1)

(Note that any two points are sufficient to determine the equation of the line)

The slope = change in y/change in x

                 = (1-0)/(2-0)

                 = 1/2

Hence answers are:

Variable y varies directly with x. (if x increases y also increases and vice versa)

The constant of variation = slope of line = 1/2

The constant of variation is 2 is wrong since slope = 1/2

The variable y does not vary directly with x is wrong since when x increases by 1 unit y increases by 1/2 unit for all real x.

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Answer:

first 2 pairs

Step-by-step explanation:

[tex]\frac{6}{100}[/tex] = 6 ÷ 100 = 0.06 ← equivalent

[tex]\frac{7}{10}[/tex] = 7 ÷ 10 = 0.7 ← equivalent

[tex]\frac{5}{10}[/tex] = 5 ÷ 10 = 0.5 ≠ 0.05

[tex]\frac{32}{100}[/tex] = 32 ÷ 100 = 0.32 ≠ 3.2

The statement that corresponds to a probability of an event occurring is the probability mass of observing 50 successes in 50 binomial trials.

Probability represents comprehensive analysis of the reasonable likelihood that a particular outcome shall typically occur. This occurrence is dependent on the ratio of favorable instances to probable instances. The integral of the chi-squared distribution's density among complex chi-squared values of zero and five and probability density of the chi-squared distribution at chi-squared value four are functionally related to the overall chi-squared distribution. However, neither of them directly reflects the likelihood that an event will actually take place.

The strength of evidence against null hypothesis in a chi-squared test is exhibited by p-value of a chi-squared value of numeral four, but it does not directly translate to likelihood that an event will take place. Therefore, the statement that the probability mass of observing 50 successes in 50 binomial trials corresponds with the given question.

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The formula that can be used to describe the sequence will be aₙ = –81 · (–4/3)ⁿ⁻¹.

A series of non-zero integers where every term after the first is obtained by increasing the one before it by a constant, non-zero value known as the scale factor.

Let a₁ be the first term and r be the common ratio. Then the nth term of the geometric sequence is given as,

aₙ = a₁ · (r)ⁿ⁻¹

The sequence is given below.

–81, 108, –144, 192, ...

The first term of the sequence is –81. And the common ratio is given as,

r = 108 / (–81)

r = – 4/3

The formula can be used to describe the sequence is given as,

aₙ = –81 · (–4/3)ⁿ⁻¹

The formula that can be used to describe the sequence will be aₙ = –81 · (–4/3)ⁿ⁻¹.

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The center and radius of the circle with equation (x – 14)2 + (y – 2)2 = 9 is  (14, 2), r = 3, the correct option A.

A circle can be characterized by its center's location and its radius's length.

Let the center of the considered circle be at (h,k) coordinate.

Let the radius of the circle be 'r' units.

Then, the equation of that circle would be:

(x-h)^2 + (y-k)^2 = r^2

We are given that;

The equation of circle=  (x – 14)2 + (y – 2)2 = 9

Now,

We know the equation of a circle with center (h, k) and radius r is:

(x - h)^2 + (y - k)^2 = r^2

Comparing the given equation to this form, we can see that the center of the circle is (14, 2) and the radius is √9 = 3.

Therefore, by the given equation of circle answer will be (14, 2), r = 3.

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1. The equation that correctly relates the remaining debt (D) to the number of months (1) I've been making payments is:

a. 170t + D = 8500.

2. If you continue paying at the rate above (assuming there is no additional interest), it will take 38 months for you to pay off the credit card balance.

An equation is a mathematical statement showing the equality of two or more mathematical expressions using the equation symbol (=).

The total debt on the credit card = $8,500

The promotional interest rate for 24 months = 0%

Monthly repayment = $170

Total repayment after 12 months = $2,040 ($170 x 12)

Balance on the credit card after 12 months of repayment = $6,460 ($8,500 - $2,040)

The number of months to repay the balance without interest = 38 months ($6,460/$170).

Thus, D, the remaining debt can be determined using this equation, a. 170t + D = 8500 and it will take 38 months to repay without interest.

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Answer:

y=67

z=113
x=67

Step-by-step explanation:

opposite angles are parallel in a rhombus and they all equal 360

Answer:

-1

Step-by-step explanation:

1. Plug the numbers into the equation

      6-12/6

2. Solve.

  6-12=-6

  -6/6=-1

Answer:

4

Step-by-step explanation:

A= 6

C= 12

6-12÷6

With BIDMAS the order becomes:

6-(12÷6) —> 6-2 = 4

Answer:-17

Step-by-step explanation:

b=7,then

b^2=49

square root of 49-4*3*2=-17

The mode of the data 20,26,16,22,20,30,22,20,26

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

20,26,16,22,20,30,22,20,26

The mode of a dataset is the data element that has the highest frequency

Using the above as a guide, we have the following:

20 has the highest frequency of 3

Hence, the mode is 3

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The total amount spend is 391.88

The noun discount refers to an amount or percentage deducted from the normal selling price of something.

If 14 bag of mix cement are bought and and every third bag is discounted

no of bag on discount = 14/3 = 4 remainder 2

therefore 4 bags will be on discount

discount on a bag = 3.88× 75/100

= 291/100 = 2.91

therefore each bag on discount will cost = 3.88 - 2.91

= 0.97

for 4 bags = 0.97 ×4 = 3.88

for remaining 10 bags = 3.88 × 10

= 388

therefore the total amount = 388+3.88

= $391.88

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Answer: 2a^4b^3

Step-by-step explanation: Divide the base like normal. Subtract the exponents.

a6 - a2 = a4

b4 - b = b3

A single variable will always equal to 1 so b will be its own exponent.

After divide the solution is,

⇒ 2a⁴b³

Division method is used to distributing a group of things into equal parts. Division is just opposite of multiplications.

For example, dividing 20 by 2 means splitting 20 into 2 equal groups of 10.

We have to given that;

Divide 10a⁶b⁴ by 5a²b.

Hence, We get;

⇒ 10a⁶b⁴ / 5a²b.

⇒ 2a⁶⁻² b⁴⁻¹

⇒ 2a⁴b³

Thus, After divide the solution is,

⇒ 2a⁴b³

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Answer & Step-by-step explanation:

The domain of the function g(x) = ⟨÷⟩6x + 6 is the set of all real numbers, since dividing by zero is not defined.

And it is a linear function, which means that it is a straight line in a 2-dimensional graph. The slope of the line is 6, and the y-intercept is 6, meaning that when x=0, the value of g(x) is 6.

The volume of the unused roll is approximately 10,316.97 cubic centimeters.

what is radius ?

A radius is a line segment that connects the center of a circle or a sphere to any point on its circumference or surface, respectively. It is half the length of the diameter, which is a line segment that passes through the center of the circle or sphere and has endpoints on its circumference or surface, respectively. The radius is an important measurement in geometry and is used to calculate other important properties of circles and spheres, such as their area, circumference, and volume.

Given by the question:
To find the area of the base of the cylinder, we need to know the radius. Since the paper towel roll has an inner radius of 4.2 cm and an outer radius of 12 cm, we can calculate the area of the base as follows:

[tex]Area of outer circle - Area of inner circle = πr_outer^2 - πr_inner^2[/tex]

[tex]= \pi (12 cm)^2 - \pi (4.2 cm)^2[/tex]

= [tex]452.16\pi cm^2 - 55.3896\pi cm^2[/tex]

= [tex]396.7704\pi cm^2[/tex]

Therefore, the volume of the unused roll can be calculated as:

V = Bh = [tex](396.7704\pi cm^2)(26 cm)[/tex]

[tex]= 10,316.97 cm^3[/tex]

Therefore, the volume of the unused roll is approximately 10,316.97 cubic centimeters.

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The answer that makes the comparison statements true, The sum is more than 17. So Option B is correct.

Addition is a basic mathematical operation that combines two or more numbers to give a total or sum. It is one of the four elementary arithmetic operations, along with subtraction, multiplication, and division.

The equation 14+17=n represents the total number of cans Danica collected at the end of the first week, where n is the number of cans Danica collected.

We can solve for n by adding 14 and 17, which gives us:

n = 14 + 17 = 31

Therefore, Danica collected 31 cans at the end of the first week.

To make the comparison statements true:

The sum is more than 17:

This statement is true, since the sum of 14 and 17 is 31, which is greater than 17.

The sum will be 14:

This statement is false, since the sum of 14 and 17 is 31, not 14.

The sum will be 17:

This statement is false, since the sum of 14 and 17 is 31, not 17.

So, the correct answer is: The sum is greater than 17.

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For complete question image is attached:

1. The ratio of the shaded area to the area of the whole figure is 3/8 and the ratio of the shaded area to the unshaded area is 3/5.

2. The value of AC/CD. CD/BD, and BD/BC is 8/5, 5/2 and 10/7 respectively.

A ratio is a comparison of two quantities, and it can be expressed in different ways, such as a fraction, a decimal, or a percentage. In this case, we will explore how to find ratios of areas in geometric figures.

Firstly, let's consider the rectangle on the right. To find the ratio of the shaded area to the area of the whole figure, we need to calculate both of these values. The shaded area is a rectangle with dimensions 6 by 2, which means its area is 6 x 2 = 12 square units.

The whole figure is a rectangle with dimensions 8 by 4, so its area is

=> 8 x 4 = 32 square units.

Therefore, the ratio of the shaded area to the area of the whole figure is 12/32, which simplifies to 3/8.

To find the ratio of the shaded area to the unshaded area, we need to subtract the shaded area from the whole area. The unshaded area is a rectangle with dimensions 8 by 2, so its area is

=> 8 x 2 = 16 square units.

Subtracting the shaded area from the whole area gives us

=> 32 - 12 = 20 square units for the unshaded area.

Thus, the ratio of the shaded area to the unshaded area is 12/20, which simplifies to 3/5.

Moving on to the second question, we need to use the figure below to find three ratios: AC/CD, CD/BD, and BD/BC.

Let's start with the first ratio, AC/CD. We know that AC is 8 units long and CD is 5 units long, so

=> AC/CD is 8/5.

Next, CD/BD. We know that BD is 10 units long, so to find CD/BD, we need to subtract the length of AC from the length of BD. AC is 8 units long, so

BD - AC = 10 - 8 = 2 units.

Therefore, CD/BD is 5/2.

Finally, BD/BC. We know that BC is 12 units long, so to find BD/BC, we need to subtract the length of CD from the length of BC. CD is 5 units long, so

=> BC - CD = 12 - 5 = 7 units.

Therefore, BD/BC is 10/7.

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The total weight in the bag is given as follows:

8.2 pounds.

The total weight is obtained applying the proportions in the context of the problem.

From the table, the weight of each equipment is given as follows:

Hence the total weight of a bag with 4 balls, 2 gloves, and 2 bats is given as follows:

4 x 0.35 + 2 x 0.77 + 2 x 2.63 = 8.2 pounds.

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Answer:

2:5

Step-by-step explanation:

We know

There are 9 girls and 6 boys taking golf lessons.

So, there are a total of 15 taking a golf lesson.

Write the ratio that compares the number of girls taking golf lessons to the total number of students taking golf lessons.

The ratio is

6:15 = 2:5

So, the ratio is  2:5

The Average Minute to Clean up a Car is 0.7794.

The slope of the line is the ratio of the rise to the run, or rise divided by the run. It describes the steepness of line in the coordinate plane.

The slope intercept form of a line is y=mx+b, where m is slope and b is the y intercept.

The slope of line passing through two points (x₁, y₁) and (x₂, y₂) is

m=y₂-y₁/x₂-x₁

The equation of the graph is y=0.7794x-1435.5.

When we compare the equation with the standard form of equation y=mx+b.

The m will be 0.7794 which means slope is 0.7794.

Hence,  0.7794 is the Average Minute to Clean up a Car.

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x int = 15

y int = -10

y intercept can be found bt setting X to 0

x intercept can be found by setting Y to 0

2(0) - 3y = 30 --> -3y = 30 --> y = -10

2x - 3(0) = 30 --> 2x = 30 --> x = 15

alternatively, y int is "b" in eq. y = mx + b

-3y = 30 -2×

y = -10 + (2/3)x

y = (2/3)x - 10

b = -10

Answer: The rate of change for Miquel's position in the race can be calculated by his speed, which is 18 kilometers per hour. So, his rate of change is 18 kilometers per hour. This means that for every hour that passes, Miquel will have covered 18 kilometers in the race.

Step-by-step explanation:

Answer:

See below for answers and explanations

Step-by-step explanation:

Part A

[tex]x^2+4x+4\\\rightarrow x^2+2x+2x+4\\\rightarrow x(x+2)+2(x+2)\\\rightarrow (x+2)(x+2)[/tex]

Part B

[tex]x^2-12x+27\\\rightarrow x^2-9x-3x+27\\\rightarrow x(x-9)-3(x-9)\\\rightarrow (x-3)(x-9)[/tex]

Part C

[tex]x^2-x-42\\\rightarrow x^2+6x-7x-42\\\rightarrow x(x+6)-7(x+6)\\\rightarrow (x-7)(x+6)[/tex]

Part D

[tex]x^2+8x-9\\\rightarrow x^2+9x-x-9\\\rightarrow x(x+9)-1(x+9)\\\rightarrow (x-1)(x+9)[/tex]

Answer:

[tex]x=\dfrac{1}{2},\;\;x=1,\;\;x=-\dfrac{4}{3}[/tex]

Step-by-step explanation:

Given polynomial function:

[tex]f(x) = 6x^3-x^2-9x+4[/tex]

If (2x - 1) is a factor of the given function, then 2x - 1 = 0.

Therefore:

[tex]\implies 2x-1=0[/tex]

[tex]\implies x=\dfrac{1}{2}[/tex]

To divide the given function by the factor (2x - 1), we can perform synthetic division.

As the leading coefficient of the divisor is not 1, divide the dividend and divisor by the leading coefficient of the divisor (2):

[tex]\textsf{Dividend}: \quad 3x^3-\dfrac{1}{2}x^2-\dfrac{9}{2}x+2[/tex]

[tex]\textsf{Divisor}: \quad \left(x-\dfrac{1}{2}\right)[/tex]

Perform Synthetic Division

Place ¹/₂ in the division box.

Write the coefficients of the dividend in descending order.

(Note: As no terms are missing, we do not need to use any zeros to fill in missing terms).

[tex]\begin{array}{c|crrr}\vphantom{\dfrac12}\frac{1}{2} & 3 & -\frac{1}{2}& -\frac{9}{2} & 2\\\cline{1-1}\end{array}[/tex]

Bring the leading coefficient straight down:

[tex]\begin{array}{c|crrr}\vphantom{\dfrac12}\frac{1}{2} & 3 & -\frac{1}{2}& -\frac{9}{2} & 2\\\cline{1-1}&\downarrow&&&\\\cline{2-5}&3\end{array}[/tex]

Multiply the number you brought down with the number in the division box and put the result in the next column (under the -¹/₂ ):

[tex]\begin{array}{c|crrr}\vphantom{\dfrac12}\frac{1}{2} & 3 & -\frac{1}{2}& -\frac{9}{2} & 2\\\cline{1-1}\vphantom{\dfrac12}&\downarrow&\frac{3}{2}&&\\\cline{2-5}&3\end{array}[/tex]

Add the two numbers together and put the result in the bottom row:

[tex]\begin{array}{c|crrr}\vphantom{\dfrac12}\frac{1}{2} & 3 & -\frac{1}{2}& -\frac{9}{2} & 2\\\cline{1-1}\vphantom{\dfrac12}&\downarrow&-\frac{3}{2}&&\\\cline{2-5}&3&1\end{array}[/tex]

Repeat:

[tex]\begin{array}{c|crrr}\vphantom{\dfrac12}\frac{1}{2} & 3 & -\frac{1}{2}& -\frac{9}{2} & 2\\\cline{1-1}\vphantom{\dfrac12}&\downarrow&-\frac{3}{2}&\frac{1}{2}&\\\cline{2-5}&3&1&-4\end{array}[/tex]

[tex]\begin{array}{c|crrr}\vphantom{\dfrac12}\frac{1}{2} & 3 & -\frac{1}{2}& -\frac{9}{2} & 2\\\cline{1-1}\vphantom{\dfrac12}&\downarrow&-\frac{3}{2}&\frac{1}{2}&-2\\\cline{2-5}&3&1&-4&0\end{array}[/tex]

The bottom row (except the last number) gives the coefficients of the quotient. The degree of the quotient is one less than that of the dividend.

The last number in the bottom row is the remainder.

Therefore, the factored function after synthetic division is:

[tex]f(x)=(2x-1)(3x^2+x-4)[/tex]

The quadratic factor can be factored further:

[tex]\implies 3x^2+4x-3x-4[/tex]

[tex]\implies x(3x+4)-1(3x+4)[/tex]

[tex]\implies (x-1)(3x+4)[/tex]

Therefore, the fully factored function is:

[tex]f(x)=(2x-1)(x-1)(3x+4)[/tex]

To find the roots of the function, set each factor to zero and solve for x:

[tex]2x-1=0 \implies x=\dfrac{1}{2}[/tex]

[tex]x-1=0 \implies x=1[/tex]

[tex]3x+4=0 \implies x=-\dfrac{4}{3}[/tex]

Therefore, the roots of the given function are:

[tex]x=\dfrac{1}{2},\;\;x=1,\;\;x=-\dfrac{4}{3}[/tex]

Answer:

What is X?

Step-by-step explanation:

4 multiplied by X + 8 =Ps

Answer:

[tex]x=-3,x=-3,x=7[/tex]

Step-by-step explanation:

[tex]We\ know\ that\ the\ roots\ of\ an\ equation\ are\ its\ intercepts\ cut\ on\ the\ x-axis.\\We\ can\ observe\ that\ the\ graph\ of\ y=f(x)\ cuts\ the\ x\ axis\ at\ the\ point (7,0)\\ and\ touches\ the\ x-axis\ at\ (-3,0).\\[/tex]

[tex]Hence,\\The\ curve\ represents\ a\ cubic\ polynomial\ with\ double\ roots\ (-3,0),(-3,0)\\ as\ well\ as\ a\ distinct\ root:\ (7,0)[/tex]

Answer: c[tex]\leq[/tex]-17/4

Step-by-step explanation:

-8c+13[tex]\geq[/tex]47

-8c[tex]\geq[/tex]47-13

-8c[tex]\geq[/tex]34

c[tex]\leq[/tex] -17/4 or -4 1/4

The inequality changes if both sides of the expressions are switched or multiplied by a negative sign

Inequalities are expressions not separated by an equal sign.

From the given inequalities

For -10.5 < 3e

3e > -10.5

e > -10.5/3

This shows that the sign changes

For b/6≤ 7

b ≤ 6(7)

b≤ 42

The inequality sign does not change.

For the inequality -4.5c > 9

4.5c < -9
c < -9/4.5

c < -2

The inequality sign changes.

For the inequality -21/5 ≤  -5/2 d

21/5 ≥ 5/2d

5/2d≤ 21/5

d ≤42/25

The inequality sign does not change

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