Please help!! this is rizzcalc

Please Help!! This Is Rizzcalc

Answers

Answer 1

The curve described by these parametric equations is the graph of the Cartesian equation: x = 2e^[(1 - y)/5]

What is the Cartesian equation?

To eliminate the parameter t, we need to find a way to express t in terms of x and y, and then substitute that expression into one of the equations to eliminate t.

x(t) = 2e^t

y(t) = 1 - 5t

Let's start by solving the second equation for t:

y = 1 - 5t

Adding 5t to both sides, we get:

5t = 1 - y

Dividing both sides by 5, we get:

t = (1 - y)/5

Now we can substitute this expression for t into the first equation:

x = 2e^t

x = 2e^[(1 - y)/5]

This is the Cartesian equation in terms of x and y. We have eliminated the parameter t and expressed the equation solely in terms of x and y.

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Related Questions

YALL I NEED HELPP PLEASE I need answers

Answers

Thus, the radius of circle KM for the given value of tangent is found as: MK = 6 mm.

Explain about the tangent:

A tangent is really the line that follows the curve's slope at a specific location. It is the line that contacts the curve at any specific point and follows the curve in that location.

At a particular point, tangents simulate the curve. By displaying the curve's angle of descent at that precise location, they provide the most accurate estimate.At one point, the tangent just touches the curve.A curve must really be differentiable at a point in order to have a tangent there.

Given:

Tangent NM = 8 mmRadius r = MKlength NK = 10 mm

MK ⊥ NM (property of tangent drawn to any circle)

Using the Pythagorean theorem in right triangle NMK

NK² = MK² + NM²

MK² = NK² -  NM²

MK² = 10² + 8²

MK² = 100 - 64

MK = 6 mm

Thus, the radius of the circle KM for the given value of tangent is found as: MK = 6 mm.

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what pattern would you not likely observe on a graph if you conduct a two-factor anova and there is a statistically significant interaction effect?

Answers

If there is a statistically significant interaction effect, we would not expect to see a pattern of parallel lines on a graph that plots the means of the dependent variable for each level of the two factors anova.

If there is a statistically significant interaction effect in a two-factor ANOVA, it means that the effect of one independent variable (factor) on the dependent variable depends on the level of the other independent variable. In other words, the effect of one factor on the dependent variable varies across the levels of the other factor.  

If there is no interaction effect, the lines would be parallel and have the same slope. But if there is an interaction effect, the lines would cross or have different slopes, indicating that the effect of one factor on the dependent variable depends on the level of the other factor.

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Solving
1. Nancy has 4 pencils. Byron has 3
times as many pencils as Nancy. Mac
has 5 fewer pencils than Byron. How
many pencils does Mac have?
A. 5
B. 7
C. 6 D.8

Answers

Answer:

7

Step-by-step explanation:

Since Nancy has 4 pencils, we know Byron has 3 times that amount

4 groups of 3 is equivalent to 12.

We know that is how many pencils Byron has. Mac has 5 less.

So, 12-5 = 7.

Adele wears a pedometer . On Monday she walked 2 3/5 miles . on Tuesday she walked one and 1 3/4 miles and on Wednesday she walked to 2 7/10 miles on Wednesday. How many miles has she walked so far this week

Answers

Adele has walked 7 1/20 miles so far this week.

To find out how many miles Adele has walked so far this week, we need to add the miles she walked on Monday, Tuesday, and Wednesday.
Step 1: Convert the mixed numbers into improper fractions.
- Monday: 2 3/5 = (2 × 5 + 3)/5 = 13/5
- Tuesday: 1 3/4 = (1 × 4 + 3)/4 = 7/4
- Wednesday: 2 7/10 = (2 × 10 + 7)/10 = 27/10
Step 2: Find a common denominator and add the fractions.
- Common denominator: 20 (least common multiple of 5, 4, and 10)
- Convert the fractions: 13/5 = 52/20,

7/4 = 35/20,

27/10 = 54/20
- Add the fractions:

52/20 + 35/20 + 54/20

= (52 + 35 + 54)/20

= 141/20
Step 3: Convert the improper fraction back to a mixed number.
- 141/20 = 7 + 1/20

= 7 1/20.

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I need help With this

Answers

Answer:

256ft²

Step-by-step explanation:

White squares=

2 (8x8) =

2 (64) = 256

Whole Rectangle=

8 + 8 = 16

A = bh = 16 x 32 = 512

512 - 256 = 256

which angles are corresponding angles ​

Answers

Answer:

A, D, and E

Hope this helps<3

PLEASE HELP I'LL GIVE 50 POINTS AND BRAINLIEST SUPER URGENT PLEASE DO THE PROOF OF ANY TYPE PLEASE PLEASE HELP ME THIS IS URGENT I WILL GIVE BRAINLIEST ​

Answers

The angle A and D will be equal by the help of side angle side theorm.

What is side angle side theorem?

The Side-Angle-Side (SAS) theorem is a concept in geometry that states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. In other words, if two triangles have the same length of two sides and the angle between those sides is also the same, then the two triangles are identical in shape and size. The SAS theorem is one of the several postulates that can be used to prove congruence between two triangles.

The angle ABC is equal to angle DBE by vertical opposite angle

AC  is equal to DE

and CB is equal to EB

therefore angle A and angle D

Congruent triangles are triangles that have the same shape and size. When two triangles are congruent, it means that they have the same angles and side lengths. In other words, they are identical in every way, and one can be superimposed onto the other by rotating, reflecting or translating.

There are several ways to prove that two triangles are congruent, including:

Side-Side-Side (SSS) theorem: if all three sides of one triangle are congruent to the corresponding sides of another triangle, then the two triangles are congruent.

Side-Angle-Side (SAS) theorem: if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Angle-Side-Angle (ASA) theorem: if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

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find the perimeter of triangle jkl round your answer to the nearest tenth if necessary. figures are not necessarily drawn to scale.

Answers

The perimeter of ΔJKL = KJ + JL + KL = 35 + 43.986 + 37.5 = 116.5

What is perimeter?

To find the perimeter of ΔJKL, we first need to find the length of JL. We can use the law of cosines to do this:

cos(J) = (KJ² + JL² - KL²) / (2KJJL)

cos(57) = ((35)² + x² - (37.5)²) / (235x)

x² - 23537.5*cos(57)*x + ((35)² - (37.5)²) = 0

Using the quadratic formula, we get:

x = [23537.5cos(57) ± √((23537.5cos(57))² - 4*((35)²-(37.5)²))]/2

x = [23537.5cos(57) ± √((435)²(37.5)²cos(57)² + 4*(37.5)²)]/2

x = 43.986 or x = 10.402

Since x represents the length of a side of a triangle, we know that the value of x we're looking for is the larger one, so x = 43.986.

Now we can find the perimeter of ΔJKL:

Perimeter = KJ + JL + KL = 35 + 43.986 + 37.5 = 116.486

Rounding to the nearest tenth, we get a final answer of 116.5.

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Complete question is: The perimeter of ΔJKL = KJ + JL + KL = 35 + 43.986 + 37.5 = 116.5

What is the factor form for 12a - b ?

Answers

Answer:

12a-b

Step-by-step explanation:

Given expression = 12a -b

The factored form for the given expression, cannot be determined because the given expression is a simplified expression its factor cannot be possible.

Answer: 1 (12a-b)

Step-by-step explanation: You would do distributive property to solve this, and 1 * 12 equals 12, so that works. 1 * b equals b, because multiplying it by 1 doesn't change its value. Hope this helps

find the circle and radius of the circle x2 + y2 +12x +6y +20=0

Answers

To find the circle and radius of equation x2 + y2 +12x +6y +20=0, we need to complete the square for both x and y terms.

First, we can simplify the equation by rearranging the constant term to the right-hand side:

x^2 + y^2 + 12x + 6y = -20

Next, we can complete the square for the x-terms by adding (12/2)^2 = 36 to both sides of the equation:

x^2 + 12x + 36 + y^2 + 6y = -20 + 36

Simplifying further:

(x + 6)^2 + (y + 3)^2 = 13

This is now in the standard form of a circle: (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.

So the center of the circle is (-6, -3) and the radius is the square root of 13. Therefore, the equation x^2 + y^2 + 12x + 6y + 20 = 0 represents a circle with center (-6, -3) and radius sqrt(13).

Pretest: Unit 3
Question 2 of 33
Which of the following best describes reflectional symmetry?
A. The quality a design has if its left and right halves are mirror
images of each other
B. The quality a design has if the top and bottom halves are mirror
images of each other
C. The quality a design has if it maintains some of its characteristics
when it is rotated about an axis lying in its plane
D. The quality a design has if it maintains all of its characteristics
when it is reflected over an axis lying in its plane
SUBMIT

Answers

Step-by-step explanation: The answer is D. The quality a design has if it maintains all of its characteristics when it is reflected over an axis lying in its plane. This is the definition of reflectional symmetry. A and B are examples of reflectional symmetry over a vertical or horizontal axis, but not the general case. C is the definition of rotational symmetry.

a quality control expert at life batteries wants to test their new batteries. the design engineer claims they have a variance of 8100 with a mean life of 1192 minutes. if the claim is true, in a sample of 72 batteries, what is the probability that the mean battery life would be greater than 1173.8 minutes? round your answer to four decimal places.

Answers

For a sample of 72 batteries, the probability that the average battery life of the sample exceeds 1173.8 minutes is 0.9918.

It states a battery life variance of 8100 minutes and a median battery life of 1192 minutes.

In a sample of 72 batteries, find the probability that the average battery life (x) of the sample exceeds 1173.8 minutes.

Due to the large sample size (n = 72), we can use the central limit theorem to assume that the sample mean follows a normal distribution with mean μ = 1192 and standard deviation σ = sqrt(8100/72) = 30.

So the z-score for x= 1173.8 is

z = (x - μ) / (σ / sqrt(n))

= (1173.8 - 1192) / (30 / square(72))

= -2.4

You can use a standard normal distribution table or calculator to find the probability that a  standard normal random variable is greater than -2.4. This is 0.9918 (rounded to four decimal places).

Therefore, for a sample of 72 batteries, the probability that the average battery life of the sample exceeds 1173.8 minutes is 0.9918 (rounded to four decimal places) if the claim is correct. 

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length = 16 m height = 10 m width = 11 m​

Answers

Answer:

892 m squared

Step-by-step explanation:

Answer:14.54

Step-by-step explanation: CALCULATOR

Using a scale of 1 centimeter = 5 meters, what are the ac-
tual dimensions of the swimming pool?
The length of the square in the grid is 1 cm long.
(Remember to count by 5 to find the length and width.)
u

Answers

The swimming pool's actual measurements are[tex]1500m^{2}[/tex], which solves the area issue that was presented.

What do you mean by area of a rectangle?

By calculating how much space would be needed to fully enclose its exterior, its overall size can be calculated. The surrounding region is considered when selecting a comparable product for the rectangular design.

We know, [tex]area = length *breadth[/tex] of a rectangle

[tex]length(l) = 10[/tex] squares and [tex]breadth (b) = 6[/tex] squares

Given [tex]1 cm = 5 m[/tex]

∴ length = [tex]5 *10 = 50 m[/tex], breadth = [tex]6*5 = 30 m[/tex]

    =[tex]50*30= 1500m^{2}[/tex]

Therefore area of the pool is [tex]1500m^{2}[/tex]

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Please explain and show mw how to work this please.

Answers

The answer of the given question based on the Similarity of triangles is the correct answer is (c) not similar.

What is Congruent triangle?

Congruent triangles are two or more triangles that have exactly the same size and shape. In other words, their corresponding sides and angles are equal in measure. When two triangles are congruent, we can superimpose one on top of the other so that they coincide exactly.

To determine whether two triangles APT and DPE are the similar, we need to check if their corresponding angles are the congruent and if their corresponding sides are the proportional.

Looking at the diagram, we can see that angle APT and angle DPE are both right angles, so they are congruent.

To check if the sides are proportional, we can compare the ratios of corresponding side lengths:

AP/DP = 39/26

PT/PE = 40/16

We can simplify these ratios:

AP/DP = 3/2

PT/PE = 5/2

The ratios are not equal, so the corresponding sides are not proportional.

Therefore, the triangles APT and DPE are not similar.

The correct answer is (c) not similar.

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a researcher wants to collect quantitative data about how much emergency department nurses use touch to comfort patients. the best way to collect these data would be for the researcher to use:

Answers

 Other methods that may be used to collect data on this topic include observation or self-report measures, but these may be less accurate than standardized measures.

When using terms in your answer, it is important to ensure that they are used correctly and in the appropriate context. This will help to provide clear and accurate information to the student asking the question.

The best way for a researcher to collect quantitative data about how much emergency department nurses use touch to comfort patients would be to use a standardized measure such as the Touch Questionnaire.

This measure assesses the frequency and nature of touch behaviors used by healthcare providers during interactions with patients, including comforting touches.

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7 divided by 70.85 please show how yu got the answer

Answers

When 7 is divided by 70.85, the quotient is 0.0988.

What is the quotient?

The quotient is the result of a division operation.

A division operation is one of the four basic mathematical operations, including addition, subtraction, and multiplication.

A division operation involves the dividend (in this situation, 7) divided by the divisor (70.85) to get 0.0988 (the quotient).

7 ÷ 70.85 = 0.0988

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ten weeks equal how many days

Answers

Answer:

70 days

Step-by-step explanation:

1 week = 7days

10 weeks = ? days

We Take

10 x 7 = 70 days

So, 10 weeks equal 70 days.

Solve.
10
=

4

+
3

2
10=−4x+3x
2
10, equals, minus, 4, x, plus, 3, x, squared
Choose 1 answer:
Choose 1 answer:
(Choice A)

=
1
,

1
2
x=1,−
2
1

x, equals, 1, comma, minus, start fraction, 1, divided by, 2, end fraction
A

=
1
,

1
2
x=1,−
2
1

x, equals, 1, comma, minus, start fraction, 1, divided by, 2, end fraction
(Choice B)

=

1
±
10
2
x=
2
−1±
10



x, equals, start fraction, minus, 1, plus minus, square root of, 10, end square root, divided by, 2, end fraction
B

=

1
±
10
2
x=
2
−1±
10



x, equals, start fraction, minus, 1, plus minus, square root of, 10, end square root, divided by, 2, end fraction
(Choice C)

=

2
±
34

3
x=
−3
−2±
34



x, equals, start fraction, minus, 2, plus minus, square root of, 34, end square root, divided by, minus, 3, end fraction
C

=

2
±
34

3
x=
−3
−2±
34



x, equals, start fraction, minus, 2, plus minus, square root of, 34, end square root, divided by, minus, 3, end fraction
(Choice D)

=

3
±
17
2
x=
2
−3±
17



x, equals, start fraction, minus, 3, plus minus, square root of, 17, end square root, divided by, 2, end fraction
D

=

3
±
17
2
x=
2
−3±
17


Answers

The correct option:D. x = -10. The value of 'x' for the given linear equation in one variable is found as -10.

Describe about the one variable linear equation:

If the degree n with in equation is equal to 1, then a linear equation only contains ONE variable. The highest exponent a single variable can have is referred to as the equation's degree.

Any arbitrary variable, such as x, y, or z, may be used as long as the linear equation is homogeneous.Another clue that an equation is linear is that it only contains one variable, and that variable's maximum degree is 1.

Given equation:

10 = −4x + 3x

Adding the variable'x' with sign, the coefficient will be added and the sign with the larger value will be taken.

10 = −1x

10 = -x

Divide both side by -1.

x = -10.

Thus,  the value of 'x' for the given linear equation in one variable is found as -10.

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Correct question:

Solve 10 = −4x + 3x

Option:

A. x = 2

B. x = 10

C. x = -6

D. x = -10

Micah deposited $2700 into a savings account that has an annual simple interest rate of 0.3%. Find the amount in the savings account after each number
of years.
A year
B. 3 years
C. 6 years
Please help I need it like right now

Answers

Answer:c

Step-by-step explanation:

The rate of change in graph one is a positive or negative? in graph B was it’s a positive or negative?

determine the rate of change , be sure to include your units for each answer . Explain work pls and work it out

Answers

Answer:

A) The slope is negative.  It is -30

B)The slope is positive.  It is 2.5

Step-by-step explanation:

Graph A:

I selected two points on the graph.

(1, 220) and (7, 40)

The slope is the change in y over the change in x.  The y values are 40 and 220.  The x values are 7 and 1.  You find the change by subtracting.

[tex]\frac{40 - 220}{7-1}[/tex] = [tex]\frac{-180}{6}[/tex] = -30

Graph B:

This graph is proportional because it is a straight line and it goes through the original.  This means that the slope can be found with any ordered pair in the form [tex]\frac{y}{x}[/tex].  I am going to use the point (2, 5).  [tex]\frac{5}{2}[/tex] = 2.5

Helping in the name of Jesus.

A shopkeeper allow 20% discount on the market price of an article and 10% Vat is levied on it. if a customer pay rupees 500 for Vat then find the discount amount.

Answers

In conclusion the discount amount is Rs. 1250.

Why it is?

Let's start by using algebra to represent the problem:

Let's assume the market price of the article is "M".

The shopkeeper offers a 20% discount, so the selling price (SP) would be:

SP = M - 0.20M

SP = 0.80M

Then, a 10% VAT is levied on the selling price, so the final price (FP) would be:

FP = SP + 0.10SP

FP = 1.10SP

FP = 1.10(0.80M)

FP = 0.88M

We know that the customer paid Rs. 500 for VAT, so:

0.10SP = 500

Substituting SP = 0.80M, we get:

0.10(0.80M) = 500

Solving for M, we get:

M = 6250

So, the market price of the article is Rs. 6250.

Now, to find the discount amount, we can subtract the selling price from the market price:

Discount amount = Market price - Selling price

Discount amount = Rs. 6250 - 0.80 x Rs. 6250

Discount amount = Rs. 1250

Therefore, the discount amount is Rs. 1250.

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luis says that the probability of getting a yahtzee in one roll of the dice is a 1 6b 5 . explain why luis is wrong.

Answers

In the given problem, we find that Luis is wrong because the probability of getting a Yahtzee in one roll of the dice is 1/6^5.

Luis is wrong because the probability of getting a Yahtzee in one roll of the dice is 1/6^5. This is approximately 0.00077 or 0.077%.

A Yahtzee is a combination of five dice with the same number on each dice. The probability of rolling any specific combination of five dice is 1/6^5 and this is because each dice has a 1/6 probability of showing the required number.

The probability of rolling a Yahtzee in one roll of the dice is 6 times the probability of rolling any specific combination, which is 6 x 1/6^5 or 1/6^4  since there are only six possible combinations of a Yahtzee (one for each number on the dice). Therefore, the correct probability of getting a Yahtzee in one roll of the dice is 1/6^5 and not 1/6b5 as Luis had stated.

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A boat covered 15km against the current and then 6km with the current. It spent the same time for the entire trip as if it would cover 22km on the lake. What is the speed of the boat in still water if the speed of the current is 2km/h

Answers

The length of the entire trip is equal to the time required to travel [tex]22 km[/tex], the boat's average speed on calm water is [tex]3.24 km/h[/tex]

What do you mean by equation?

In mathematics, an equation is a statement that all equations are the same.

A variable is a symbol that represents an unknowable value or a value that is subject to vary within a given range, and an equation may contain one or more of these symbols.

Mathematics can be used to identify hidden quantities in problems and to express interactions between variables.

As an example, the equations [tex]2x+5= 13[/tex] contain the response variable, which stands for an indeterminate value.

To determine the value of[tex]x[/tex] that makes the equation remain true, this equation can be solved.

Since [tex]2(4)+5=13[/tex], the solution in this case is [tex]x=4[/tex] Equations come in a variety of different forms, such as linear equations, quadratic equations, and systems of equations.

Given

We can create another equation since the duration of the full journey is equal to the duration needed to travel [tex]22 km[/tex] along the lake:

The sum of the times against and with the current is the total time.

By simplifying and substituting the formula

[tex]15/(b-2) +6/(b+2) = 22/b[/tex]

After multiplying both sides by [tex]b(b-2)(b+2)[/tex] we get

[tex]22(b-2)(b+2) = 15b(b+2) +6b(b-2)[/tex]

Adding and subtracting

[tex]15b^{2} +30b +6b^{2} - 12b =22(b^{2} -4)\\21b^{2} +42b -22b^{2} +88 = 0\\-b^{2} +2b+4[/tex]

The boat speed cannot be negative so we considered positive value

[tex]3.24 km/h for b =1+\sqrt{5}[/tex]

Therefore the length of the entire trip is equal to the time required to travel [tex]22 km[/tex], the boat's average speed on calm water is [tex]3.24 km/h[/tex]

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a group of seven people are attending the movies together. (a) two of the seven insist on sitting side-by-side. in how many ways can the seven be seated together in a row?

Answers

The total number of ways in which the group of seven people can be seated together in a row are 1440.

The group of seven people are attending the movie together and two of them are insisting that they will sit side by side to each other.

Now if all the people were to be seated in any random order then the number of positions would have been seven but now as two people are sitting side by side so we will consider only six positions in which we have to make the combinations.

And as we know that the seven people will be unique and there will be no repetition total number of ways in which the seven can be fitted together in a row is,

= 6 x 5 x 4 x 3 x 2 x 1 x 2

= 1440

It is to be noted that we have multiplied by the number of positions with to because the two people who are sitting together can be seated in two possible ways.

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the lengths of a professor's classes has a continuous uniform distribution between 50.0 min and 52.0 min. if one such class is randomly selected, find the probability that the class length is more than 51.7 min.

Answers

The probability that the class length is more than 51.7 min is 0.3 or 30%.

To solve the problem, we first need to find the total possible outcomes, which is the range of class lengths between 50.0 min and 52.0 min, which is 2.0 min. Since the distribution is uniform, each possible class length within this range is equally likely.

Next, we need to find the favorable outcomes, which is the range of class lengths that are more than 51.7 min. This range is 52.0 min minus 51.7 min, which equals 0.3 min.

To find the probability, we divide the number of favorable outcomes by the total number of possible outcomes:

Probability = (favorable outcomes) / (total outcomes) = 0.3 min / 2.0 min = 0.15 or 15%.

Therefore, the probability of a randomly selected class length being more than 51.7 min is 0.3 or 30%.

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Find the missing dimension. Use the scale factor 1 : 12.

Answers

According to the information, the actual width of the item is 21.6 yards.

How to find the actual width of the item?

We can use the scale factor to find the actual width of the item from its model width. The scale factor is 1:12, which means that every 1 unit on the model represents 12 units in actual size. The model width is given in feet, so we need to convert it to yards before we can use the scale factor.

1 yard = 3 feetModel width in yards = 5.4 ft ÷ 3 = 1.8 yd

Now we can use the scale factor to find the actual width:

1 unit on the model = 12 units in actual size

So,

1.8 yd on the model = x yd in actual size1 : 12 = 1.8 : x

To solve for x, we can cross-multiply:

1.8 × 12 = xx = 21.6

Therefore, the actual width of the item is 21.6 yards.

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Help please!! 40pts!!

Answers

Answer: 3,64

Step-by-step explanation:

Help with these 2 algebra questions!!

1. A number is four more than twice another number. If the difference between the numbers is 27, determine the numbers.

2. Suppose that 2 sides of a triangle are equal and the third side is 10cm greater than each of the other two. the perimeter of the triangle is 100cm. find the length of each side.

Answers

1. Let's call the first number "x" and the second number "y". We know that:

x = 4 + 2y (equation 1)

x - y = 27 (equation 2)

We can use equation 1 to substitute x in equation 2:

(4 + 2y) - y = 27

Simplifying the equation:

3y = 23

So the second number is:

y = 23/3

Using equation 1, we can find the first number:

x = 4 + 2(23/3) = 50/3

Therefore, the numbers are 50/3 and 23/3.

2. Let's call the length of the equal sides of the triangle "x" and the length of the third side "y". We know that:

y = x + 10 (equation 1)

2x + y = 100 (equation 2)

We can use equation 1 to substitute y in equation 2:

2x + (x + 10) = 100

Simplifying the equation:

3x = 90

So the length of the equal sides is:

x = 30

Using equation 1, we can find the length of the third side:

y = 30 + 10 = 40

Therefore, the lengths of the sides are 30, 30, and 40.

MNOP is a trapezium. If MN//OP and A is the midpoint of NO prove : Area of triangle MAP = 1/2area of trapezium MNOP​

Answers

Answer: Given that MNOP is a trapezium with MN parallel to OP and A is the midpoint of NO.

To prove that the area of triangle MAP is half the area of trapezium MNOP, we need to use the following theorem:

Theorem: If a line segment joins the midpoint of one side of a triangle to a vertex opposite to that side, then the segment divides the triangle into two equal areas.

Proof:

Since A is the midpoint of NO, we can draw a line segment AP from vertex P to point A on NO as shown in the figure below:

      M _______N

       |\      |

       |  \    |

       |    \  |

       |      \|

      O--------P

            A

We can see that triangle MAP is formed by the line segment AP and side MP of the trapezium.

Also, we can see that triangle MAN is congruent to triangle NOP because they are both right triangles with corresponding sides parallel. Therefore, they have the same area.

Since MNOP is a trapezium, the area of trapezium is given by:

Area of trapezium MNOP = (1/2)(MN+OP) × height

Since MN is parallel to OP, the height of the trapezium is the same as the height of triangle MAN and NOP.

Therefore, the area of trapezium MNOP can be written as:

Area of trapezium MNOP = (1/2)(MN+OP) × height

= (1/2)(MA+AP+OP) × height

= (1/2)(MA+MP) × height (Since AP = NO/2 = height)

So, we have proved that:

Area of trapezium MNOP = (1/2)(MA+MP) × height

Using the theorem, we know that AP divides triangle MAP into two equal areas. Therefore,

Area of triangle MAP = (1/2) × Area of triangle MAP

Substituting the above in the expression for the area of trapezium, we get:

Area of trapezium MNOP = Area of triangle MAP + (1/2) × Area of triangle MAP

= (3/2) × Area of triangle MAP

Therefore, we have:

Area of triangle MAP = (1/2) × Area of trapezium MNOP

Thus, we have proved that the area of triangle MAP is half the area of trapezium MNOP.

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