8th grade math! (Picture attatched)

For the given infinite geometric sequence,

(a) the two expressions for r are [tex]r=\frac{6}{m-1}[/tex] and [tex]r= \frac{m+8}{6}[/tex]

(b) The two possible values of m are -8 and 5.

(c) The two possible values of r are [tex]-\frac{2}{3}[/tex] and [tex]\frac{3}{2}[/tex]

What are Geometric Series?

What is the Common Ratio of an Infinite Geometric Sequence?

In the question, the first three terms of an infinite geometric sequence are given as, m-1, 6, m+8

[tex]\implies a_{0} =m-1, a_{1}=6,[/tex] and [tex]a_{2}=m+8[/tex]

So, the common ratio is calculated as,

[tex]r=\frac{6}{m-1}=\frac{m+8}{6}[/tex]

Here, there are two expressions,

[tex]r=\frac{6}{m-1}[/tex]    ------(1), and  [tex]r= \frac{m+8}{6}[/tex]  ------(2)

Simplifying (1), we get

[tex]r(m-1)=6[/tex]  -----(3)

Simplifying (2), we get

[tex]6r=m+8\\\implies r= \frac{m}{6}+\frac{8}{6}[/tex]   ------(4)

Now, substituting the values of (4) in (3), we get

[tex](\frac{m}{6}+\frac{8}{6})(m-1)=6\\\implies \frac{m^{2} }{6}-\frac{m}{6} +\frac{2m}{3}-\frac{2}{3} =6\\\implies m^{2}+3m-4=36\\[/tex]

Solving further, we get

[tex]\implies m^{2}+3m-40=0\\\implies m=5, m=-8[/tex]

So, when [tex]m=5, r=\frac{6}{5-1}=\frac{3}{2}[/tex]

Also, when [tex]m=-8, r=\frac{6}{-8-1}=-\frac{2}{3}[/tex]

Therefore, (a) the two expressions for r are [tex]r=\frac{6}{m-1}[/tex] and [tex]r= \frac{m+8}{6}[/tex]

(b) The two possible values of m are -8 and 5.

(c) The two possible values of r are [tex]-\frac{2}{3}[/tex] and [tex]\frac{3}{2}[/tex]

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The given equation is a combined variation.

We have been given an equation is

[tex]\frac{xy}{z}[/tex]

This is an equation in which we have three variables and one constant.

This can not be the case of direct or inverse variation because in direct or inverse variation we have only two variables. Whereas, here we have three variables.

So, Direct and inverse options are discarded.

So, either It can be joint or combined

In joint variation both the variables are directly proportional

Hence, given equation is not joint because if we rewrite the equation we will get [tex]y = \frac{cz}{x}[/tex] both the variables are not directly proportional.

In combined, one variable is directly proportional and the other one is inversely proportional.

From [tex]y = \frac{cz}{x}[/tex]  we can see that z is directly proportional and x is inversely proportional.

Hence the answer is, the given equation is combined variation.

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The parabola has a minimum value of 5.75.

From the question, we have

f(x)=x^2-x+6

a=1, b=-1, c=6

Δ=b^2-4ac

=1-24

=-23<0

So f(x)=0 has no solution.

x=-b/2a=-(-1)/2=1/2

f(1/2)=1/4-1/2+6

=23/4  is the minimum

=5.75

The parabola has a minimum value of 5.75.

Multiplication:

Mathematicians use multiplication to calculate the product of two or more numbers. It is a fundamental operation in mathematics that is frequently utilized in everyday life. When we need to combine groups of similar sizes, we utilize multiplication. The fundamental concept of repeatedly adding the same number is represented by the process of multiplication. The results of multiplying two or more numbers are known as the product of those numbers, and the factors that are multiplied are referred to as the factors. Repeated addition of the same number is made easier by multiplying the numbers.

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Answer:

3k(7k + 4)

Step-by-step explanation:

1. Find the GCF of all terms in the polynomial

2. Express each term as a product of the GCF and another factor.

3. Use distributive property to factor out GCF

Answer:

[tex]f(x)=\frac{5}{6}(x+2)^2(x-1)(x-3)[/tex]

Step-by-step explanation:

The root -2 has a multiplicity of 2, and corresponds to a factor of [tex](x+2)^2[/tex].

The root 1 has a multiplicity of 1, and corresponds to a factor of [tex](x-1)[/tex].

The root 3 has a multiplicity of 1, and corresponds to a factor of [tex](x-3)[/tex].

So, [tex]f(x)=a(x+2)^2(x-1)(x-3)[/tex].

Since [tex]f(0)=10[/tex],

[tex]10=a(0+2)^2(0-1)(0-3) \\ \\ 10=12a \\ \\ a=\frac{5}{6} \\ \\ \therefore f(x)=\frac{5}{6}(x+2)^2(x-1)(x-3)[/tex]

The function for motorboat's distance from the shore is y = -4x + 50.

A function is a formula, rule, or law that specifies how one variable (the independent variable) and another variable are related. The linear equation is of the form Ax+By+C=0.

Where A and B are the coefficient and C is the constant.

After 9 minutes, boat is 14 km from the shore, so finding the rate of change,

Rate of Change [tex]=\frac{14-50}{9}\\[/tex]

[tex]=\frac{-36}{9}\\ =-4[/tex]

So, the rate of change of distance is 4 km / minutes.

Now, finding the function,

The function for the distance of motorboat from the shore will be,

y = -4x + 50

Where y will be the final distance and x is the minutes.

Therefore, the function for motorboat's distance from the shore is y = -4x + 50.

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The volume of the fully inflated balloon is 267.95 cubic inches, the volume of half-inflated balloon is 133.97 cubic inches and it's radius is 3.17in

The volume of sphere is the measure of space that can be occupied by a sphere. If we draw a circle on a sheet of paper, take a circular disc, paste a string along its diameter and rotate it along the string. This gives us the shape of a sphere.

The formula of a sphere is given as

v = 4/3πr³

But diameter = 2 * radius

radius = diameter / 2

radius = 8/2

radius = 4in

The volume of the fully inflated balloon will be

v = 4/3πr³

v = 4/3 * 3.14 * 4³

v = 267.95in³

The volume of the fully inflated balloon is 267.95in³

b) The volume of the half-inflated balloon will half the size of the current volume

v = 267.95in³ / 2 = 133.97in³

c) v = 4/3πr³

133.97 = 4/3 * 3.14 * r³

r = 3.17in

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Answer:

x ≈ 41.7 ° (a)

Step-by-step explanation:

b² = a² + c² - 2ac × cosβ

~~~~~~~~~~~

25² = 32² + 37² - 2(32)(37)cosx

625 = 2,393 - 2,368 cosx

cosx = [tex]\frac{-2393+625}{-2368}[/tex] ≈ 0.74662162

x ≈ 41.7 ° (a)

It takes them approximately 1.8 hours to paint the house together

Given,

Sam can paint a house in 5 hrs.

and, Gary can do it same work in 3 hrs.

To find the how long it will take the two working together?

Sam can paint a house in 5 hours. This means that Sam's unit rate of working is 1/5

Gary can paint the same house in 3 hours.

This means that Gary's unit rate of working is 1/3

If they work together, it would take them lesser time because they are working simultaneously. This means that their unit rates are additive. Combining their unit rates, it becomes

1/5 + 1/3 = 8/15

Assuming that if they work together, it will take them t hours. Therefore, Their unit rate of working will be 1/t.

Therefore,

8/15 = 1/t

8t = 15

t = 15/8  = 1.875

Hence, It takes them approximately 1.8 hours to paint the house together.

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Answer:

1.9

2.3

3.6

4.5

5.20

Step-by-step explanation:

hope it helps

Answer: x= -9

Step-by-step explanation:

Add like terms excluding x in: x+59+80+50=180

= x+189=180

[Transposing 189 to RHS]    x=180-189

So x= -9

Answer:

The average rate of change is 2

Step-by-step explanation:

First set divide your data into 2 sets of separate data:

Independent Data= Day 3        

Dependent Data   = 10 points

Independent Data= Day 6

Dependent Data   = 16 points

Then subtract your two independent data: 6-3=3

Then subtract your two dependent data:  16-10=6

Next divide: difference in dependent/ difference in independent

6/3=2

So, your rate of change is 2.

Rewriting the expression |3r-15| without using absolute value bars gives

3r < 15 for r < 5

Absolute value do not give negative values it gives only the positive value

The given expression is |3r-15|

|3r - 15| < 0

|3r| < |-15|

3r < 15

If r < 5 then  say 4

|3r - 15|

|3 * 4 - 15|

|12 - 15|

|-3| = 3

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The scale factor that was used in the dilation is 1.6.

A change called dilation lets you resize an object. Dilation is a method for enlarging or contracting objects. The result of this alteration is a picture with the exact same shape as the original. The form, however, has a size difference.

Now,

Scale factor = Dimension of the new shape/Dimension of the original shape

We have △R′S′T′ is the image of △RST under a dilation through point C. Also, given that RS=5 and R′S′=8.

So, the scale factor is,

Scale factor = Dimension of the new shape/Dimension of the original shape

Scale factor = R′S′ / RS

                    = 8/5

                     = 1.6

Hence, the scale factor that was used in the dilation is 1.6.

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In triangle ABC, the value of angle A  is 60°.

A triangle is a geometric figure with three edges, three angles and three vertices. It is a basic figure in geometry.

The sum of the angles of a triangle is always 180°

In the triangle ABC,

b=5cm, c=10cm

And by using Pythagoreans theorem,

a²=c²-b²

a²=10²-5²

a²=75

Use cosine rule to determine the angle A,

cosA=b²+c²-a²/2bc

cosA=25+100-75/2×5×10

cosA=50/100

cosA=1/2

cosA=cos60

⇒A=60°

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After simplifying the equation, the chemist used 18 liters of 20% solution, 12 liters of 35% solution and 24 liters of 80% solution.

From the given question,

Let that chemist used x, y, z liters of 20%, 35% and 80% solution respectively.

Then according to the given information,

x+y+z=54..........(1)

As we know that in x liters having 20% acid, in y liters having 35% acid, in z liters having 80% acid and in total mixture having 50% acid.

So the equation should be

x ×20%+y×35%+z×80%=54×50%

On simplifying we get

20x+35y+80z=2700

4x+7y+16z=540...............(2)

As given that the number of 80% solution used in 2 times the number of liters of 35% solution used.

So z=2y

Now putting the value of z in equation (1)

x+y+2y=54

x+3y=54

Subtract 3y on both side

x+3y−3y=54−3y

x=54−3y

Now putting the value of x and z in equation (2)

4(54−3y)+7y+16×2y=540

Simplifying

216−12y+7y+32y=540

216+27y=540

Subtract 216 on both side

216+27y−216=540−216

27y=324

Divide by 27 on both side

27/27 y=324/27

y=12

Now putting the value of y in x=54−3y to find the value of x

x=54−3×12

x=54−36

x=18

Now putting the value of y in z=2y to find the value of z

z=2×12

z=24

Hence, the chemist used 18 liters of 20% solution, 12 liters of 35% solution and 24 liters of 80% solution.

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The value of w in Angle  (w + 1)°  is 88

Given,

Angle (w + 1)°

Angle 91°

We have to find the value of w.

Here,

Linear pair of angles;

When two lines cross at one point, a linear pair of angles is created. If the angles follow the intersection of the two lines in a straight line, they are said to be linear. A linear pair's total angles are always equal to 180°.

Then,

Here the given angles are linear pair.

So,

(w + 1)° + 91 = 180

w + 1 + 91 = 180

w + 92 = 180

w = 180 - 92

w = 88

Then,

Angle (w + 1)° = 88 + 1 = 89°

Therefore,

The value of w in Angle  (w + 1)°  is 88

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a) After 1 day, the revenue for each item is the same.

b) Graph is shown in the attachment.

What is an equation?

An equation is a mathematical statement that proves two mathematical expressions are equal in algebra, and this is how it is most commonly used. In the equation 3x + 5 = 14, for instance, the two expressions 3x + 5 and 14 are separated.

M(d) = 2x² + 8x - 4

R(d) = 2x + 4

a) Find x by taking y=M(d)=R(d). ( same revenue=y)

     y= 2x² + 8x - 4

     y= 2x + 4

This shows that right-side expressions are equal.

 2x² + 8x - 4 = 2x + 4

 2x² + 8x - 4 - 2x - 4=0

2x² +6x-8=0

Find the factors

2(x-1)(x+4)=0

x-1=0,  x+4=0

x= 1, x=-4

consider positive value for days

So, after 1 day, the revenue for each item is the same

b)  The equations:

 y= 2x² + 8x - 4

     y= 2x + 4

By taking some random values for x and finding the corresponding y values, we get the ordered pairs in the form of (x,y).

By plotting and joining those points, we get the graph as shown in the attachment.

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Answer:

The x-intercept is the point where the graph crosses the x-axis; it can be found by setting y = 0 and solving for x.

The y-intercept is the point where the graph crosses the y-axis; it can be found by setting x = 0 and solving for y.

1) g(x) = (x + 5)⁴

2) f(x) = 7 - x⁴

3) h(x) = ¼(x - 3)⁴

The graphs are labeled below.

Answer:

slope: 4

Y intercept: -7

Step-by-step explanation:

slope intercept form is y=Mx+b

m- slope

b- y intercept

Looking at this equation where m is 4 is so the slope is 4

And where b is -7 is so the y intercept is -7

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The time taken for 14g of the radioactive isotope to remain is 25.70 years.

We know that the half life is the time that it would take for us to have only half of the number of the original radioactive isotopes in the sample still remaining in the sample.

We have the following information;

Half life of the sample = 11 years

Amount of the sample initially present = 70 grams

Amount of the sample after time t = 14 grams

Given that;

N/No = (1/2)t/[tex]t_{\frac{1}{2} }[/tex]

N = amount of radioatve isotpe at time t

No = amount of radioactive isotope initially present

t = time taken

[tex]t_{\frac{1}{2} }[/tex] = half life

14/70 = (1/2)^t/11

0.2 = (1/2)^t/11

ln 0.2 = t/11 ln0.5

t/11  = ln 0.2/ ln0.5

t = 11 *  ln 0.2/ ln0.5

t = 11 * (-1.61/-0.69)

t = 25.70 years

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Answer: 2.5

Step-by-step explanation:

got it right

The prime factorization of the number 68 is 2×2×17 and least prime number is 2 and greatest prime number is 17.

A number system is defined as a system of writing to express numbers.

We need to find the prime factorization of the number 68

Prime factorization is a process of writing all numbers as a product of primes.

The prime factorization of 68 is 2×2×17

The prime factors of 68 are 2 and 17.

The least prime number is 2 and greatest prime number is 17.

Hence the prime factorization of the number 68 is 2×2×17 and least prime number is 2 and greatest prime number is 17.

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Answer:

incomplete question..please complete the question to get the answer. there are some missing details

Answer:

y = x + 2

Step-by-step explanation:

Use any two points.

For example, let's use (-3, -1) and (1, 3).

First, we find the slope.

slope = m = (-1 - 3)/(-3 - 1) = -4/(-4) = 1

We have slope = m = 1.

Now we use the equation:

y = mx + b

We plug in the slope we found above.

y = 1x + b

y = x + b

Now we use one of the given points to find b.

We'll use point (1, 3).

3 = 1 + b

b = 2

Now that we know m and b, we write the equation:

y = x + 2

The Hiking club will be 0.6.

The completion for the row of the table will be:

Grade 7

Dance club: 0.25

Hiking club: 0.75

Total : 1

Grade 8:

Dance club: 0.40

Hiking club: 0.60

Total : 1

How to calculate the values?

The dance club for grade 7 will be calculated as:

= Number of dancers / Total number

= 5/20

= 0.25

The hiking club will be:

= 15/20

= 0.75

The dance club for grade 8 will be calculated as:

= Number of dancers / Total number

= 20/50

= 0.4

The hiking club will be:

= 30/50

= 0.6

Hence, The Hiking club will be 0.6.

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The expression we get from the given functions h for all real numbers x.

h(-3)= 8

Given that,

All real numbers x are defined by the functions h.

h(x)= -x+5

We have to find the expression

h(-3)

The core concept of mathematics' calculus is functions. The unique varieties of relations are the functions. In mathematics, a function is represented as a rule that produces a distinct result for each input x. In mathematics, a function is indicated by a mapping or transformation.

Take the functions,

h(x)= -x+5

Now, Take

h(-3) =-(-3)+5 = 3+5= 8

Therefore, The expression we get from the given functions h defined for all real numbers x.

h(-3)=8

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Answer:

h(2) = 0

Step-by-step explanation:

1) 2 * 2 = 4

2) 4 * 3 = 12

3) 5 * 2 = 10
4) 12 - 10 - 2 = 0

The probability of rolling a yahtzee on the first round is calculated to be  6/46656 or 1/7776

Given that,

A dice is rolled for one time .

To find : The probability that a yahtzee will be rolled on the first round.

Your first die will be one of the 6 possible results 6/6 times

That will match on the second die 1/6th of the time.

The third, fourth, and fifth dice all share this characteristic.

Multiplying the all probabilities that the first die will be a yahtzee which is

P( yahtzee will be rolled on the first round) = Product of all possible outcomes of each time a dice is rolled.

6/6*1/6*1/6*1/6*1/6*1/6=6/46656 or 1/7776

Therefore, probability that a yahtzee will be rolled on the first round is 1/7776.

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The volume of the jar is 1130.4 cubic inches and the number of marbles that can fit is 56.

In geometry, it is defined as a three-dimensional shape having two circular shapes at a distance called the height of the cylinder.

We know the volume of the cylinder is given by:

[tex]\rm V = \pi r^2 h[/tex]

It is given that:

A cylindrical jar has a radius of 6 inches and a height of 10inches.

The volume:

V = π(6)²(10)

V = (3.14)(6)²(10)

V = 1130.4 cubic inches

Number of marbles = 1130.4/20 = 56.52 = 56 marbles

Thus, the volume of the jar is 1130.4 cubic inches and the number of marbles that can fit is 56.

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