It takes Amelia 45 minutes to rake her yard every Saturday during the fall. It takes Andy 1 hour to rake the same yard. If Amelia and Andy work together, how long will it take them to rake the yard? Round your answer to the nearest minute.

a. The next term in the sequence is 303.75.

b. The sequence is a geometric sequence.

c. The recursive function for the sequence is: aₙ = aₙ₋₁ (1.5) for n≥2.

Given the sequence is : 60,90,135,202.5,....

a. the next term in the sequence is:

Find the common ratio by dividing any term in the sequence by the term that comes before it:

a₂/a₁ = 90/60 = 1.5

a₃/a₂ = 135/90 = 1.5

a₄/a₃ = 202.5/135 = 1.5

The common ratio (r) of the sequence is constant and equals the quotient of two consecutive terms.

r = 1.5

find the sum:

To find the sum of the series, plug the first term: a = 60, the common ratio(r): 1.5 and the number of elements n = 4 into the geometric series sum formula:

Sₙ = a(1₋rⁿ/1₋r)

Sₙ = 60(1 ₋ 1.5⁴/1₋1.5)

Sₙ = 60(1 ₋ 5.0625/1₋1.5)

S₄ = 60(8.125)

S₄ = 478.5

To find the general form of the series, plug the first term:

aₙ = 60×1.5ⁿ⁻¹

use the general form to find the next term:

a₁ = 60

a₂ = a₁rⁿ⁻¹ = 60(1.5)¹ = 90

a₃ = a₁rⁿ⁻¹ = 60(1.5)² = 135

a₄ = a₁rⁿ⁻¹ = 60(1.5)³ = 202.5

a₄ = a₁rⁿ⁻¹ = 60(1.5)⁴ = 303.75

hence the required next term is 303.75

(b) The above sequence is geometric sequence.

(c) recursive function for the sequence : aₙ = aₙ₋₁ (1.5) for n≥2.

Hence we get the required answers.

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Answer:

105.

Step-by-step explanation:

Total score for 10 students = 50 * 10 = 500.

When 1 student joins:

mean score = (500 + x) / 11 = 55    where x is the test score of the new student.

500 + x = 55 * 11

x  = 55*11 - 500

  =  605 - 500

   = 105.

In the given case , we can conclude The test score of the student who joined the class is 105 marks.

Let's solve this problem step by step:

The mean test score of the first 10 students is 50 marks. Therefore, the total marks scored by these 10 students is 10 x 50 = 500 marks.

When the new student joins, the mean score of all 11 students becomes 55 marks.

Therefore, the total marks scored by all 11 students is 11 x 55 = 605 marks.

To find the test score of the new student, we subtract the total marks of the first 10 students from the total marks of all 11 students: 605 - 500 = 105 marks.

Therefore, the test score of the student who joined the class is 105 marks.

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See below for the proof that the expression x^2 +2x+2 can only have positive values

The expression is given as

x^2 + 2x + 2

The above expression is a quadratic expression

A quadratic expression is represented as

ax^2 + bx + c

Where the discriminant (d) is

d = b^2 - 4ac

So, we have

d = 2^2 - 4 * 1 * 2

Evaluate

d = -4

Because d is negative, it means that the expression has only complex roots

Quadratic expressions that have only complex roots can only have positive values

Hence, the expression x^2 +2x+2 can only have positive values

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The initial amount and rate of change given a table for a linear function are $1160 and -$30, respectively

The table of linear function represents the given parameter

On the table, we have the following points

(x, y) = (6, 980) and (12, 800)

The rate of change is calculated as

Rate = (y2 - y1)/(x2 - x1)

So, we have

Rate = (980 - 800)/(6 - 12)

Evaluate

Rate = -30

This means that the rate of change is -$22.5

The initial amount is calculated as

y = m(x - x1) + y1

Where

m = -30

So, we have

y = -30(x - x1) + y1

Using one of the points, we have

y = -30(x - 6) + 980

Set x = 0

So, we have

y = -30(0 - 6) + 980

Evaluate

y = 1160

Hence, the initial amount and rate of change given a table for a linear function are $1160 and -$30, respectively

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Answer: 8m

Step-by-step explanation:

The dimension that give the maximum volume will be the sides.

The volume will be 324cm³.

It should be noted that the maximum value is when the rectangle is a square.

Let s be the side of the square.

The perimeter will be 4s.

Solving for s:

= 36/4

= 9cm

It should be noted that area = s²

where s = side

= 9² = 81cm

Therefore, the volume will be

= Area × Side

= 81 × 4

= 324cm³

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1. 9miles

2. 6miles

3. 3miles

==================================================

Explanation:

We need to divide both sides by -2 to fully isolate x.

Dividing both sides by a negative number will flip the inequality sign

We go from -6 < -2x to 3 > x

Then we flip 3 > x to x < 3 which is choice C

-----------------

A slightly longer approach:

-6 < -2x

-6+2x < 0 .... add 2x to both sides

2x < 6 .... add 6 to both sides

2x/2 < 6/2 .... divide both sides by 2

x < 3

We get the same answer as the previous section. It helps show why the sign flip happened when dividing both sides by a negative number.

Answer:

the answer is 5.

Step-by-step explanation:

used math-way to solve

The value of term a11 is 21k - 43.

The difference between any two consecutive integers in an arithmetic progression (AP) sequence of numbers is always the same value.

The formula of Arithmetic progression is

a(n) = a1 + (n - 1)d

given that the term of arithmetic sequence and common difference,

a1 = k+7 and d = 2k-5.

now find the value of term a11.

a(n) = a1 + (n - 1)d

now, substitute a(n) = a11, a1 = k+7, d = 2k-5.

a11 = k+7 + (11 - 1)(2k-5)

a11 = k + 7 + 10(2k-5)

a11 = k +7+ 20k - 50

a11 = 21k - 43

Hence,The value of term a11 is 21k - 43.

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SSS postulate of congruence, to prove ∠ L N M ≅ ∠ L N P.

What do you mean by congruent?

In triangles MLN and PLN,

MN ≅ PN,

LM ≅ LP

To prove :

Δ MLN ≅ Δ PLN

  Statement                                              Reason

1. LN ≅ LN                                           1. Common segment

2. MN ≅ PN;                                        2. Given

LM ≅ LP

3. Δ MLN ≅ Δ PLN                              3. SSS  postulate of congruence,

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Answer:

5200 people

Step-by-step explanation:

1. Divide 100 by 5 (How many sets of 5 are there in 100)

100/5 = 20

2. Multiply 260 by 20 (To find the number of people, you need to find 100%)

260*20 = 5200

Answer:

5200

Step-by-step explanation:

Answer: −

3

x

+

5

y

Step-by-step explanation:

Answer:

2 3/4 divided by 5/6 = 3 3/10

1 1/3 divided by 1/4 = 5 1/3

2/5 divided by 1 2/5 = 2/7

5/7 divided by 3 1/5 = 25/112

2 2/7 divided by 2 2/7 = 1

Answer:

the volume of the sphere is 1436.76

Answer:

Step-by-step explanation:

a² + b² = c²

18² + b² = 30²

324 + b² = 900

b² = 576

b = 24

Answer: 24

Step-by-step explanation: you use the formula: a^2 + b^2 = c^2. If you drew a picture, it would form a triangle with 30 as the length of the hypotonuse, and 18 as one of the legs. This given, we would have a^2 + 18^2 = 30^2 (hypotonuse = c) Then, solve.

The amount that you earned last month would be the total of $17.25

We have the following data

online pay bill per month = 50

The monthly online deposit = 50

The points that are made per car payment = 100

The point that is made per credit card = 1.5 points

The monthly credit card charges of $1,400

The monthly point would be 1400 * 1.5 = $2100

We have to solve for the total points

= 50 + 50 + 100 + 2100

= 2300

We have  $75 per 10,000

For 2300 points, this would be 75/10000 * 2300

= $17.25                                          

Hence the amount that is made in dollars from the points in the last month is $17.25

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[tex]~~~~~~ \textit{Simple Interest Earned} \\\\ I = Prt\qquad \begin{cases} I=\textit{interest earned}\\ P=\textit{original amount deposited}\dotfill & \$5000\\ r=rate\to 4.5\%\to \frac{4.5}{100}\dotfill &0.045\\ t=years\dotfill &6 \end{cases} \\\\\\ I = (5000)(0.045)(6) \implies I = 1350[/tex]

Answer:

Step-by-step explanation:

68 / 4 = 17 = length or width (because it is a square both are the same)

19 + 2 = 19

4(19) = 76 (perimeter of outer fence)

68(1.31) + 76(1.31) = $188.64

The equation which represents the situation is 2[(b + 3) + b] = 26 and the equation has one solution.

The correct answer option from above is option C

The perimeter of a rectangle is the addition of all the sides of the rectangle, that is, the addition of the two length and two width.

Perimeter of the rectangle = 2(length + width)

26 = 2{(b + 3) + b}

26 = 2(b + 3 + b)

26 = 2(2b + 3)

26 = 4b + 6

26 - 6 = 4b

20 = 4b

b = 20/4

b = 5

Width of the rectangle = b

= 5 meters

Length of the rectangle = (b + 3)

= (5 + 3)

= 8 meters

So therefore, length and width of the rectangle given the perimeter is 8 meters and 5 meters respectively.

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(a) A = 1/2 (bh)

(b) b = P/R

(c) 8?

Given that K is the midpoint on segment JL, the numerical value of segment JL is 54.

A midpoint is simply a point that divides a segment into two equal halves.

Given the data in the question;

Given that K is the midpoint on segment JL, this means segment JK is divided into two equal halves, Segment JK is equal to Segment KL.

Segment JK = Segment KL

8x + 11 = 14x - 1

Collect like terms

8x - 14x = -1 - 11

-6x = -12

x = -12/-6

x = 2

Hence, the numerical values of Segment JK and Segment KL will be;

Segment JK = 8x + 11 = 8(2) + 11 = 16 + 11 = 27

Segment KL = 14x - 1 = 14(2) - 1 = 28 - 1 = 27

Since Segment JK and Segment KL makes up segment JL, the numerical  value of Segment JL is the sum of Segment JK and Segment KL.

Segment JL = 27 + 27 = 54

Given that K is the midpoint on segment JL, the numerical value of segment JL is 54.

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Using a system of equations, the prices are given as follows:

A system of equations is when two or more variables are related, and equations are built to find the values of each variable.

For this problem, the variables are given as follows:

3 reference and 2 exercise cost RM63, hence:

3x + 2y = 63.

5 reference and 3 exercise cost RM102, hence:

5x + 3y = 102.

Multiplying the first equation by 3 and the second by -2, we have that:

Adding them:

-x = -15 -> x = 15.

Hence:

3(15) + 2y = 63

2y = 18.

y = 9.

Hence the costs are:

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According to the question, the total amount of rope needed to extend the two stakes supporting the tree in trigonometry can be determined by using the functions sine and cosine.

Explain trigonometric function with example.

If the rope can be tied with the tree trunk then the staked rope makes an angle with respect to the ground. And it can be calculated with the help of the trigonometric functions. And it states that the ratio of the length of the opposite sides and the hypotenuse of the right angle triangle. And the result of the ratio is sine function.

Therefore, the total amount of two ropes needed so that it extends to the two stakes of the supporting trees.

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Given polygons a right triangle and an isosceles triangle sometimes be similar.

As given ,

Given polygons : A right triangle and isosceles triangle.

Polygon a right triangle and isosceles triangle never be similar.

Reason:

Therefore, given polygons a right triangle and an isosceles triangle sometimes be similar.

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The coordinates of the midpoint of a segment with the given endpoints will be (-19, 5).

Let AB be the line segment and C be the mid-point of line segment AB. Let the coordinate of point A (x₁, y₁), the coordinate of point B (x₂, y₂), and the coordinate of the mid-point (x, y).

Then the coordinate of the mid-point of the line segment is given as,

(x, y) = [(x₁ + x₂) / 2, (y₁ + y₂) / 2]

The endpoints are A(-28, 8) and C(-10, 2).

Then the coordinates of the midpoint of a segment with the given endpoints will be

(x, y) = [(-28 - 10) / 2, (8 + 2) / 2]

(x, y) = (-38/2, 10/2)

(x, y) = (-19, 5)

The coordinates of the midpoint of a segment with the given endpoints will be (-19, 5).

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The temperature at 6pm based on the information will be 16.6°F.

From the information, at 9 am, the temperature was -1.7°F, between 9am and noon, the.temperature increased by 18.1°F, between noon and 3pm, the temperature increased by 14.8°F and between 3pm and 6pm, the temperature dropped 14.6°F.

Therefore, the temperature at 6pm will be:

= -1.7°F + 18.1°F + 14.8°F - 14.6°F

= 16.6°F.

Therefore, the temperature at 6pm based on the information will be 16.6°F.

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The result of the expression 4a+7b+3a-2b+2a for the variables a= -5 and b= 3 is -30

Given,

The algebraic expression = 4a+7b+3a-2b+2a

Rearrange the terms

=4a+3a+2a+7b-2b

Combine the like terms

=9a+5b

We know

a= -5

b= 3

Substitute the values in the expression

9(-5)+5(3)= -45+15

=-30

Hence, the result of the expression 4a+7b+3a-2b+2a for the variables a= -5 and b= 3 is -30

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