Arithmetic and Geometric Sequences Grade 11

A toy car is rolling down an inclined track and picking up speed as it goes. The car travels 4 cm in the first second, 8cm in the second second, 12cm in the next second, and so on. Determine the total distance travelled by the car in 30 seconds.

Step-by-step explanation:

To have an equal number of buns and sausages, Arthur needs to find the least common multiple (LCM) of 6 and 10.

The LCM of 6 and 10 is 30.

Therefore, he should buy at least 30 packs of buns and 30 packs of sausages to get an equal number of buns and sausages.

That way, he would have 30/6 = 5 packs of buns and 30/10 = 3 packs of sausages.

The tin with 250ml is better value for money. The solution has been obtained using the unitary method.

What is a unitary method?

The unitary method is used to calculate the value of a single unit from a given multiple, to put it simply. With the unitary method, you first determine the value of a single unit before determining the value of the necessary quantity of units.

We are given two tins.

First : 345ml tin of paint costing £4.80

Second: 250ml tin of paint costing £3.35

Now, using the unitary method, we get

First tin

For 350ml, cost is £4.80

So, for 1ml, cost will be

£4.80/350 = £0.014

Second tin

For 250ml, cost is £3.35

So, for 1ml, cost will be

£3.35/250 = £0.013

Hence, the tin with 250ml is better value for money.

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the inequality model. 5x-6y <0 is attached below.

What is inequality?

Inequalities are mathematical expressions where neither side is equal. In inequality, as opposed to equations, we compare two values. Less than (or less than or equal to), greater than (or greater than or equal to), or not equal to signs are used in place of the equal sign in between. Sometimes it can be about a "not equal to" relationship, where one thing is more than the other or less. In mathematics, an inequality is a relationship that results in a non-equal comparison between two numbers or other mathematical expressions.

2x + 3y > 7x-3y ​

5x-6y <0

So we have to plot this inequality.

Hence, the inequality model. 5x-6y <0 is attached below.

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Answer: To find the percent of the decrease in the number of laps jogged from Saturday to Sunday, we can use the following formula:

percent decrease = (change in value / original value) x 100

In this case, the original value is the number of laps jogged on Saturday, which is 21 laps. The change in value is the difference in the number of laps jogged from Saturday to Sunday, which is 21 - 16.5 = 4.5 laps.

So, substituting these values into the formula:

percent decrease = (4.5 / 21) x 100

percent decrease = 0.214 x 100

percent decrease = 21.4%

Rounded to the nearest tenth, the percent of decrease in the number of laps jogged from Saturday to Sunday is 21.4%.

Step-by-step explanation:

The required algebraic expression is x-3

An algebraic expression in mathematics is an expression which is made up of variables and constants, along with algebraic operations (addition, subtraction, etc.). Expressions are made up of terms.

Given here: The statement "A number subtracted from 3 "

Thus translating the given text we get the following:

Let the number be x

And we have have to subtract 3 from x

Thus the algebraic expression is given by x-3

Hence, The required algebraic expression is x-3

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The area of the given image design is; 15 in²

The formula to find the area of a triangle is;

Area = ¹/₂(base * height)

Let us divide the image into two triangles and as such;

Area of biggest triangle = ¹/₂(14 * 15)

= 85 in²

We will now deduct the rectangle that was removed by finding its' area to get;

Area of rectangle = length * width

= 10 * 7 = 70 in²

Thus;

Area of image = 85 - 70

Area of image = 15 in²

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It take for the value of the car is 4.7 years.

Now, According to the question:

A new car is purchased for $23,000

and, over time its value depreciates by one half every 3 years.

It take for the value of the car to be $7,800.

According to the statement :

The equation will be :

y = 23,000 × [tex](\frac{1}{2} )^n[/tex]

7,800 = 23,000  × [tex](\frac{1}{2} )^n[/tex]

By simplification, we get:

n = 1.56

Over time its value depreciates by one half every 3 years.

So, 1.56 × 3 = 4.68 ≈ 4.7 years.

Hence,  It take for the value of the car is 4.7 years.

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The inverse equation of the function is y = 5x - 12

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

f(x) = 1/5x + 12

Express as an equation

So, we have

y = 1/5x + 12

Swap x and y

This gives

x = 1/5y + 12

Next, we make y the subject

1/5y = x - 12

This gives

y = 5x - 12

Hence, the inverse is y = 5y - 12

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From the data points given, the linear equation is obtained as y = 6x - 3.

What is a linear equation?

A linear equation is one that has a degree of 1 as its maximum value. No variable in a linear equation, thus, has an exponent greater than 1. A linear equation's graph will always be a straight line.

The data points given are -

X = 1, 2, 3, 4, 5

f(x) = 3, 9, 15, 21, 27

The slope of the line is given as m.

The slope can be deduced using the formula -

(y2 - y1) / (x2 - x1)

m = (9 - 3) / (2 - 1)

m = 6/1

m = 6

The slope-intercept form of a equation is y = mx + b

Substitute the values of x, y and m in the equation -

3 = 6(1) + b

3 = 6 + b

b = 3 - 6

b = -3

So, the equation becomes -

y = 6x - 3

This equation forms a straight line in the graph.

Therefore, y = 6x - 3 is a linear equation.

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The segment lengths in the diagram are given as follows:

Using the Pythagorean Theorem, the length of segment GE is given as follows:

(GE)² = 5² + 12²

(GE)² = 169

GE = 13.

Applying the geometric mean theorem, the length AM is obtained as follows:

(GM)² = ME x AM

25 = 12 AM

AM = 2.083.

Then, applying the Pythagorean Theorem, the length AG is found as follows:

(AG)² = 2.083² + 5²

AG = 5.42.

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The quotient is n^2+5n-5 and the remainder is 9. The solution has been obtained by using the synthetic division.

What is synthetic division?

Synthetic division is typically used to identify the zeroes of polynomials and is described as "a simplified method of dividing a polynomial with another polynomial equation of degree 1."

We are given the expression as (n^3+3n^2-15n+19)

The expression is to be divided by (n-2)

So, by using the synthetic division, we get

2 |  1     3     -15      19    

     -     2      10     -10        

     1     5      -5       9    

Quotient =  n^2+5n-5

Remainder = 9

Hence, the quotient is n^2+5n-5 and the remainder is 9.

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Answer: 1400

Step-by-step explanation: If it is saving 40% of the original price than it is 60% of the original price.  840=60/100*x, and so we divide both sides by 60/100. 840*100/60=1400. Therefore, 1400 is the original price.

Answer:

62.8 kg or 128.43 lbs

Step-by-step explanation:

if 18 percent of all the chocolate bars is is cashews at 11.3 kgs, the chocolate makes up the other 82 percent so theres a total of 62.78 kgs of candy total and you can use the eqaution for 1kg equal 2.205 lbs to find a total of 138.43 lbs of total chocolate bars

We will apply the percentage method to find that 48 students said red is their favourite colour.

Probability is a constituent part of computation that deals with the occurrence of a random event. The value is defined from zero to one. Probability is familiarised to predict how likely events are to happen. Probability is the degree to which something is possible to happen.

Total number of pupils in class 9 = 80

We need to calculate the number of students choosing red as their favourite colour.

We are given that 80 pupils are there in year 9.

Out of these, 3/5 of the pupils say that their favourite colour is red.

Let n be the number of pupils who say red is their favourite colour.

∴ n = (3/5) × 80

∴ n = 48

--The given question is incorrect; the correct question is

"3/5 of the pupils in a year 9 class say their favourite colour is red. There are 80 pupils in Year 9. How many students said red is their favourite colour?"--

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Answer:

I need more information

Step-by-step explanation:

The solution to the system of equations cannot be determined from the information provided. The tables provide a set of x and y values for each equation, but they do not provide enough information to determine the equations themselves. In order to find the solution, we would need to know the form of the equations and be able to solve them simultaneously. The information provided is just a set of points that are plotted on a coordinate plane that is connected by using a straight line, but it is not enough to find the solution of the system of equations.

The probability of getting blue and an odd number will be 1/10.

Probability simply means the chance that a particular thing or event will happen. It is the occurence of likely events. It is simply the area of mathematics that deals with the numerical estimates of the chance that an event will occur or that a particular statement is true.

The probability of getting blue is 2/10 and the probability of getting odd number is 5/10.

Therefore, the probability of getting blue and an odd number will be:

= 2/10 × 5/10

= 1/10.

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Step-by-step explanation:

this distance YZ is the height of the triangle WXY.

it splits the main triangle into 2 right-angled triangles WYZ and XYZ.

we can use Pythagoras

c² = a² + b²

with any of the 2 smaller right-angled triangles to get YZ.

remember, "c" is the Hypotenuse (the side opposite to the 90° angle). a and b are the legs.

so, when picking the larger one :

26² = 24² + YZ²

676 = 576 + YZ²

100 = YZ²

YZ = sqrt(100) = 10 = distance of Y to WX.

The domain of the function will be All Real Numbers except x = -9. The correct option is B.

The range of values that we are permitted to enter into our function is known as the domain of a function. The x values for a function like f make up this set (x). A function's range is the set of values it can take as input. After we enter an x value, the function outputs this set of values.

Given that the function is y=−9x−26/x+3. As the domain is defined as all the input values going into the function. The domain will be all the real numbers except -9. Because the function is undefined on -9.

The graph of the function is attached with the answer below.

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Answer:

5

Step-by-step explanation:

diameter = 4 so radius =2

open up the cylinder so its top & bottom circle's circumference = 2πr =4π

top & bottom circle's area = πr² = 4π each, 8π for both

total surface = 28π

so lateral surface = 28π - 8π = 20π

so 20π = 4π* height

so height is 5

So, the last three digits of m*n is 001.

Let p be the probability that Alfred wins given that he goes first. Then, the probability that Bonnie wins given that she goes first is 1-p. Therefore, the probability that Alfred wins the second game given that Bonnie went first in the first game is 1-p. Similarly, the probability that Alfred wins the third game given that he went first in the second game is p, and so on.

Therefore, the probability that Alfred wins the sixth game given that he went first in the first game is:

p(1-p)(1-p)(p)(p)(p) = p^4 (1-p)^2

Since m and n are relatively prime, the last three digits of m*n are the last three digits of p^4 * (1-p)^2, which is the last three digits of p^4 and the last three digits of (1-p)^2. Since p is the probability of winning given that you go first, it is a number between 0 and 1. Therefore, the last three digits of p^4 and (1-p)^2 are 001, resulting in the last three digits of the final answer being 001.

Therefore, the last three digits of m*n is 001.

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The ratios 3/11 and 17/6 is not proportional.

Proportions are defined as the concept where two or more ratios are set to be equal to each other.

Suppose we have two ratio p : q and r : s. If both these ratios are proportional, then we can write it as p: q : : r : s.

This is same as p : q = r : s or p/q = r/s

The given ratios are 3/11 and 17/6.

For two ratios a/b and c/d to be proportional, a/b must be equal to c/d.

If 3/11 and 17/6 are proportional,

3/11 must be equal to 17/6.

Or doing the cross multiplication,

3 × 6 must be equal to 11 × 17.

18 must be equal to 187.

But 18 ≠ 187

So the ratios are not proportional.

Hence 3/11 and 17/6 are not proportional.

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If the vertices of the triangle BCD with vertices B(-8,3), C(-4, 6), and D(2, 0) , and if they are reflected across x axis and dilated by a scale factor of 0.5 then the coordinates of B"C"D" are B''(-4 , -1.5) , C"(-2 , 3) and D"(1 , 0)    .

If the point (x,y) is reflected across the x axis , then the reflected points are (x,-y) .

the vertices of the triangle BCD are : B(-8,3), C(-4, 6), and D(2, 0) ;

So , the reflected vertices are : B'(-8,-3) , C'(-4,-6) and D'(2,0) ;

if the point (x,y) is dilated by a scale factor of "k" , then the dilated vertices are written as : (kx , ky) .

the dilation factors is = 0.5 ;

So , the vertices B'(-8,-3) , C'(-4,-6) and D'(2,0) after dilation will be written as :

⇒ B''(-8×0.5 , -3×0.5) , C"(-4×0.5 , 6×0.5) and D"(2×0.5 , 0) ;

After simplifying further ,

we have ;

⇒ B''(-4 , -1.5) , C"(-2 , 3) and D"(1 , 0) ;

Therefore , the coordinates of the triangle B"C"D" are B''(-4,-1.5) , C"(-2,3) and D"(1,0)  .

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The scale factor is equivalent to {k} = 4/3.

        {K} = L{final}/L{initial}

Given is that figure 2 is a scale image of figure 1.

We can write the scale factor as -

K = {y}/{x} = 8/6 = 4/3

Therefore, the scale factor that is applied to figure {1} to get figure {2} is equivalent to {k} = 4/3.

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The length of MN is 14 units.

A type of triangle known as an isosceles triangle has any two sides that are both the same length. The theorem that describes the isosceles triangle states that "if the two sides of a triangle are congruent, then the angle opposite to them are also congruent." This means that the two angles of an isosceles triangle opposite to equal sides are equal in measure. For example, in the triangle ΔABC, if sides AB and AC are equal, then ABC is an isosceles triangle where ∠B = ∠C.

Given;

ΔMNO,

∠M = ∠N

Also, NO=12 and OM=14

The angles are equal then triangle is isosceles and for given triangles the two sides are equal;

Sides opposite to angles are;

NO and MO

MO = NO = 14 units

Therefore, the length of third side of triangle MNO is 14 units.

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The cheaper plan is given as follows:

Shop ZYX.

The cost of a single plum at Shop ZYX is given as follows:

$2.4.

To obtain the cheaper plan, we must obtain the cost of a single plum for each shop.

The cost of a single plum for each shop is obtained using a proportion, with the division of the total cost by the number of plums.

Hence the cost per plum in each shop is given as follows:

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The lines best approximate the directrices of the ellipse y = 10.6 or y = -6.6

The term called directrices of the ellipse is defined as a line parallel to the latus rectum of ellipse and is perpendicular to the major axis of the ellipse.

Here we know that first of all here the known form is an ellipse with a vertical major axis so it would have to be

=> d = a²/c      

Here d refers the distance between the center and the directrix.

When we take the value of a= 8, b=3, then we get the value of C as,

=> c² = a²- b²

Apply the value on it, then we get

=> c ² = 64-9

Then the value of is about 7.4

Then the value of d is calculated as,

=> d = 64/7.4 = 8.6

So the value of y is defined as,

=> 2 + 8.6 = 10.6

Or if it falls the negative direction, then we get

=> y = 2 - 8.6 = -6.6

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A convex polyhedron have 241 space diagonals.

As we know that the every pair of vertices of the convex polyhedron determines either an edge, a face diagonal or a space diagonal.

Here, we have 26 vertices.

so, the total line segments determined by the vertices:

²⁶C₂

= 26! / 2! × (26 - 2)!

= 325

Out of 325  line segments, 60 are edges (Given).

We know that each triangular face has zero face diagonals and each quadrilateral face has two face diagonals.

So there are 2 × 12 = 24 face diagonals.

This means, 325 - 60 - 24 = 241 line segments segments must be the space diagonals.

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Answer: The PMT function in Sheets is used to calculate the monthly payment for a loan. The syntax of the function is PMT(rate, nper, pv, [fv], [type]).

Rate: the interest rate per period

Nper: the total number of payments

Pv: the present value (the total amount of the loan)

Fv: the future value of the loan (optional)

Type: the number 0 or 1 indicating when payments are due (0 = end of period, 1 = beginning of period)

Given the information in the problem, Ben's mortgage is for 10 years, so the total number of payments is 10*12 = 120. The present value of the loan is $196000 and the down payment is 25% of the home's cost so 25%*196000 = $49000. Therefore, the present value of the loan is $196000 - $49000 = $147000

Based on this information, the following commands would be used to determine the monthly payment:

PMT(0.076/12,120,-$147000)

PMT(0.076/12,120,-196000+$49000)

The first command uses the annual interest rate divided by 12 to convert it to a monthly rate, and the total number of payments is 120 (10 years * 12 months/year). The second command also uses the same monthly rate and total payments and the present value which is $147000

The other commands

Step-by-step explanation:

Answer:

Step-by-step explanation:

- Hope this helps!

The quadratic equation 4 · x² - 7 · x = 3 · x + 24 has the following roots: x₁ = 19 / 4 and x₂ = - 9 / 4.

In this question we find the case of a quadratic equation, whose form is defined below:

y = a · x² + b · x + c

Where:

The procedure find the solutions to the quadratic equations:

Now we proceed to determine the solutions to the quadratic equation:

4 · x² - 7 · x = 3 · x + 24

4 · x² - 10 · x - 24 = 0

4 · [x² - (5 / 2) · x - 6] = 0

4 · [x² - (5 / 2) · x + 25 / 4] = 4 · 6 + 4 · (25 / 4)

4 · (x - 5 / 4)² = 49

(x - 5 / 4)² = 49 / 4

x - 5 / 4 = ± 7 / 2

x = 5 / 4 ± 7 / 2

The solutions to the quadratic equation 4 · x² - 7 · x = 3 · x + 24 are x₁ = 19 / 4 and x₂ = - 9 / 4.

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