Use mathematical induction to prove the statement is true for all positive integers n. 8 + 16 + 24 + ... + 8n = 4n(n + 1)

8 1/3 you have to do keep, change, flip

[tex]\frac{25}{3}[/tex]

change division to multiplication and turn the fraction upside down

5 ÷ [tex]\frac{3}{5}[/tex]

= 5 × [tex]\frac{5}{3}[/tex] = [tex]\frac{25}{3}[/tex] in simplest form

Either A or B on Spinner 1 can be paired with any of 1, 2, 3, 4 on Spinner 2. The appropriate choice is ...

... C.) {A1, A2, A3, A4, B1, B2, B3, B4}

The function is linear, so its rate of change is the same everywhere. That rate of change is the coefficient of the variable x, so is $2.90. Since the units of x are feet, and the units of P(x) are dollars, the rate of change $2.90 must be ...

... A. $2.90 per square foot

The average rate of change for the given cost function over the interval [2,000, 2,500] is calculated using the slope formula, resulting in an average rate of $2.90 per square foot, corresponding to option A.

The question asks to find the average rate of change of the cost function P(x) = $2.90x + $103.00, where x represents the square footage of the carpet, over the interval of [2,000, 2,500 square feet]. To calculate the average rate of change, we use the formula of the slope between two points on the function:

(P(x_2) - P(x_1)) / (x_2 - x_1).

So for the interval [2,000, 2,500], we have:

(P(2500) - P(2000)) / (2500 - 2000) = (($2.90 \times 2500 + $103.00) - ($2.90 \times 2000 + $103.00)) / (2500 - 2000).

This simplifies to:

($7250 + $103 - $5800 - $103) / 500 = $1450 / 500 = $2.90 per square foot.

Therefore, the average rate of change over the interval [2,000, 2,500] is $2.90 per square foot, which corresponds to option A.

Answer:

  3 8/33

Step-by-step explanation:

You want the mixed-number equivalent of the repeating decimal 3.2424....

A repeating decimal fraction that starts at the decimal point can be converted to a ratio of integers by expressing the repeating digits over the same number of 9s.

Here, the two repeating digits "24" mean the fraction equivalent is ...

  0.2424... = 24/99 = 8/33

The mixed number equivalent of 3.2424... is 3 8/33.

__

Additional comment

If the repeating portion does not start at the decimal point, you can use the following technique to do the conversion.

Effectively, this multiplies and divides the number by ((10^n) -1)/((10^n) -1). In the case here, that would be multiplication by 1 in the form 99/99.

  3.242424... × 99/99 = 321/99 = 3 8/33

The "321" will be manifested on a calculator as 320.99999.... If you multiply by (100 -1) with pencil and paper, you can see that you get ...

  324.2424... - 3.2424... = 321 . . . . . . repeating digits cancel

If the repeat starts somewhere else, you can still use the "fraction with equal number of 9s" technique, but with a multiplier:

  3.6242424... = 3.6 + 0.0242424 ... = 3.6 + (1/10)(24/99)

  = 36/10 + 8/330 = 3 103/165

<95141404393>

3.242424... as a mixed number is [tex]\(3 \frac{8}{33}\)[/tex].

To express the repeating decimal 3.242424... as a mixed number, we can first write it as an infinite geometric series.

Let x = 3.242424.... Then, multiplying x by 100 gives 100x = 324.242424...

Now, subtract x from 100x:

100x - x = 324.242424... - 3.242424...

99x = 321

Now, solve for x:

[tex]\[ x = \frac{321}{99} \][/tex]

Now, express the fraction [tex]\(\frac{321}{99}\)[/tex]as a mixed number.

[tex]\[ \frac{321}{99} = 3 \frac{24}{99} \]\\Now, simplify the fraction:\[ \frac{24}{99} = \frac{8}{33} \][/tex]

So, 3.242424... as a mixed number is [tex]\(3 \frac{8}{33}\)[/tex].

800 - 900 = -100. = 100/800=.125 x 100 = 12.5% increase.

C

given x² + 16x = 41

to complete the square on this equation

since the coefficient of the x² term is 1

add (half the coefficient of the x-term )² to both sides

x² + 2(8)x + 64 = 41+ 64

(x + 8 )² = 105 → C

Answer:

Option C is correct

Step-by-step explanation:

Given the equation

[tex]x^2+16x=41[/tex]

we have to find the equation results from completing the square.

For completing the square method, since the coefficient of [tex]x^2[/tex] is 1 therefore we have to divide the coefficient of x by 2 and then squaring of that value adding on both sides of equation, we get

Here coefficient of x is 16 therefore square of half the number adding both sides

[tex]x^2+16x+64=41+64[/tex]

[tex]x^2+8x+8x+64=105[/tex]

[tex]x(x+8)+8(x+8)=105[/tex]

[tex](x+8)(x+8)=105[/tex]

[tex](x+8)^2=105[/tex]

Option C is correct

Answer:

  about 35

Step-by-step explanation:

This question is asking for an estimate, which means you round the number(s) to something convenient to perform the calculation.

Here, 47% is conveniently rounded to 50% = 50/100 = 1/2. Then 1/2 of 70 is ...

  1/2·70 = 70/2 = 35

A reasonable estimate of 47% of 70 is 35.

___

As you get more sophisticated in your estimating, you can also estimate the error. This estimate is about 3% (of 70) high, so is high by 3/100·70 = 2.1. That means the real answer is 35 - 2.1 = 32.9.

You could also estimate the error as 3% of 100 = 3/100·100 = 3, so you could say the actual value is between 35-3 = 32 and 35. Since 100 is larger than 70, you know that 3% of 100 is larger than 3% of 70, so this is an over-estimate of the error.

To find the number of 30-day periods until the value of the gift cards will be equal, we set up and solve an equation. The value of each card will be $10 when they have equal value.

To solve this problem, we can set up an equation to represent the value of each gift card over time and find when they will be equal.

Let x be the number of 30-day periods that have passed. After x periods, the value of the first gift card will be $50 - $2x, and the value of the second gift card will be $40 - $1.50x.

Setting the two expressions equal, we get:

$50 - $2x = $40 - $1.50x

Simplifying the equation, we have:

$10 = $0.5x

Dividing both sides by $0.5, we get:

x = 20

So, after 20 periods (or 20 months), the value of each gift card will be equal.

To find the value of each card at that point, we can substitute x = 20 into one of the expressions. Let's use the first gift card:

Value = $50 - $2(20) = $50 - $40 = $10

Therefore, when they have equal value, each gift card will be worth $10.

Answer:

Option A is the correct answer.

Explanation:

 Any vector can be resolved in to two components. Horizontal component and vertical component.

Consider a vector F which is at angle θ⁰ to the horizontal, we can resolve this vector in to two.

Horizontal component = F cos θ

Vertical component = F sin θ

Here we have Force , F = 50 pounds

Angle with horizontal = 80°

Horizontal component = F cos θ = 50* cos 80 = 8.68 pounds

                                      ≅ 9 lb.

Option A is the correct answer.

Graph [tex]f(x) = x^2[/tex]

quadratic function is of the form [tex]f(x) = ax^2 + bx +c[/tex]

In our f(x) there is no x term and constant so we put 0

[tex]f(x) = 1x^2+0x+0[/tex]

The value of a=1 , b=0  and c=0

To find vertex , use formula [tex]x= \frac{-b}{2a}[/tex]

Plug in the values

[tex]x= \frac{-0}{2(1)}[/tex] =0

Now plug in x=0 in f(x) equation

[tex]f(x) = x^2[/tex]

[tex]f(0) = 0^2[/tex]=0

So vertex is (0,0)

Now we pick some number for x below and above 0

Make a table

x            y=x^2

-2           4

0             0

2            4

Now plot all the points (-2,4) (0,0) and (2,4)

The graph is attached below

Answer:  [tex]7:8:9 = 10.5:12:13.5[/tex]

Step-by-step explanation:

Suppose,  [tex]7:8:9= x:12:y[/tex]

Now according to the ratio, the equations will be........

[tex]\frac{7}{8}=\frac{x}{12}\\ \\ 8x=7*12=84\\ \\ x=\frac{84}{8}=10.5[/tex]

and

[tex]\frac{8}{9}=\frac{12}{y}\\ \\ 8y=9*12=108\\ \\ y=\frac{108}{8}=13.5[/tex]

So, the ratio will be  [tex]7:8:9 = 10.5:12:13.5[/tex]

The ratio is [tex]\boxed{7:8:9 = 10.5:12:13.5}[/tex].

Further Explanation:

Given:

The ratio is [tex]7:8:9[/tex] and is equal to [tex]{\text{\_\_\_}}:{\text{12}}:{\text{\_\_\_}}[/tex].

Calculation:

Consider the number in the first blank as [tex]x[/tex].

Consider the number in the second blank as [tex]y[/tex].

Therefore, the ratio is [tex]7:8:9[/tex] and is equal to [tex]{{x}}:{\text{12}}:{{y}}[/tex]

From the given equation the ratio of [tex]7:8[/tex] is equal to the ratio of [tex]x:12[/tex].

Now equate [tex]\dfrac{7}{8}[/tex] and [tex]\dfrac{x}{12}[/tex] to obtain the value of [tex]x[/tex].

[tex]\begin{aligned}\frac{7}{8} &= \frac{x}{{12}} \\ \frac{7}{8} \times 12 &= x \\ \frac{{21}}{2} &= x \\ 10.5 &= x \\ \end{aligned}[/tex]

Therefore, the value of [tex]x[/tex] is [tex]10.5[/tex].

From the given equation the ratio of [tex]8:9[/tex] is equal to the ratio of [tex]12:y[/tex].

Now equate [tex]\dfrac{8}{9}[/tex] and [tex]\dfrac{12}{{y}}[/tex] to obtain the value of [tex]y[/tex].

[tex]\begin{aligned}\frac{8}{9} &= \frac{{12}}{y} \\ \frac{9}{8} &= \frac{y}{{12}} \\ \frac{9}{8} \times 12 &= y \\ \frac{{27}}{2} &= y \\ 13.5&= y \\\end{aligned}[/tex]

Therefore, the value of [tex]y[/tex] is [tex]13.5[/tex].

Hence, the ratio is [tex]\boxed{7:8:9 = 10.5:12:13.5}[/tex].

Learn more:

1. Learn more about product of binomial and trinomial https://brainly.com/question/1394854

2. Learn more about line segments https://brainly.com/question/909890

Answer details:

Grade: Middle School

Subject: Mathematics

Chapter: Ratio and Proportion

Keywords: number, reciprocal, fraction, ratio, equation, proportion, 7:8:9, fill blank.

Answer: Please, see the attached file.

Thanks.

Solution:

We can graph using the intercepts:

(1) y-intercept, when x=0

y=4x-3→y=4(0)-3→y=0-3→y=-3

Point=(x,y)=(0,-3)

(2) x-intercept, when y=0

y=4x-3→0=4x-3

Solving for x: Adding 3 both sides of the equation:

0+3=4x-3+3

3=4x

Dividing both sides of the equation by 4:

3/4=4x/4

3/4=x

x=3/4=0.75

Point=(x,y)=(3/4,0)=(0.75,0)

With these pair of points (0, -3) and (0.75,0) we can draw the right line

Final answer:

The inequality models a situation where you have at least 45 dollars and you are adding $10 for each additional unit.

Explanation:

The inequality 5 + 10w ≥ 45 can be modeled by a situation where you have at least 45 dollars and you are adding $10 for each additional unit. Let's solve it step by step:

1. Subtract 5 from both sides of the inequality: 10w ≥ 40.

2. Divide both sides by 10: w ≥ 4.

The solution to the inequality is w ≥ 4. This means that the situation being modeled is where you have at least 4 units of something.

The empirical rule tells you 68% of the population is within 1 standard deviation of the mean. Your limits are 1 standard deviation from the mean, so your percentage is ...

... 68%

_____

98.20 - 0.62 = 97.58 . . . 1 standard deviation below the mean

98.20 + 0.62 = 98.82 . . . 1 standard deviation above the mean

Approximately 68% of healthy adults have body temperatures between 97.58°F and 98.82°F, as per the empirical rule which states that about 68% of data falls within one standard deviation of the mean in a bell-shaped distribution.

Using the empirical rule (also known as the 68-95-99.7 rule), we can determine the approximate percentage of healthy adults with body temperatures between 97.58°F and 98.82°F. The empirical rule states that for a bell-shaped distribution:

Approximately 68% of the data falls within one standard deviation of the mean.

Approximately 95% falls within two standard deviations.

Approximately 99.7% falls within three standard deviations.

Given that the mean body temperature is 98.20°F and the standard deviation is 0.62°F:

One standard deviation from the mean includes temperatures from 98.20°F - 0.62°F to 98.20°F + 0.62°F, which is 97.58°F to 98.82°F.

Therefore, approximately 68% of healthy adults have body temperatures between 97.58°F and 98.82°F according to the empirical rule.

The range is the output value

f(-2) = 2(-2)+7 =3

f(3)= 2(3)+7 = 13

f(8) = 2(8)+7 = 23

The range is 3,13,23

To complete the square, you can add (and subtract) the square of half the x coefficient.

... y = x² -10x + 30

... y = (x² -10x +25) + (30 -25)

... y = (x -5)² +5

D. 5x = 45 because you are trying to find x, the number of tickets that must be sold to earn $45.

Answer:

5x=45

Step-by-step explanation:

We are given a pattern:

-9 , -4 , 1 , 6 , ...

We have to describe the pattern and find the next two terms

-9+5= -4

-4+5= 1

1+5= 6

Hence, the numbers are found by adding five to the previous term

The next two numbers are found as:

6+5 =11

11+5 = 16

Hence, next two numbers are:

11 and 16

I'd start with setting up the equations. Calling x the number of students in most popular country 1, and y in country 2.

x + y = 44740

x - y = 5672

now solve for x1 and x2. You can subtract second from first:

0 + 2y = 39068

y = 19534

so

x = 5672 + 19534 = 25206

Add a sentence to answer it in words.

total students= 44740 divided by 2 countries= 22370 for each country. if were trying to get a difference of 5672 students, then were gonna divide that 5672 by 2 which is gonna = 2836 students. country 1= 22370 students + 2836 of country 2's students. country 2 is 22370 students - 2836 students that we gave to country 1. so now, country 1 ends up with 25206 students, country 2 end up with 19534 students. a difference of 5672.

the last digits of the numbers being multiplied are 6 and 3

resulting in a product of 18

thus the last digit in her answer would have to be an 8 not a 3

4 years=48 months

A new car costs 25,000, but she will get 13,000 for her used car as long as she takes good car of it. If she takes good care of her used car and gets 13,000 for it, she will only need to save 12,000 total. We can divide 12,000 by 48 to find out how much Courtney should save minimum each month to afford her new car. 12,000/48=250

You can check to see if this is correct by multiplying 48*250=12,000. Paying 250 for each month of the 48 months will equal the $12,000 that Courtney needs.

Courtney needs to save at least $250 per month to afford her new car in 4 years.

I hope this helps :)

Courtney should save at least $250 per month for the next 4 years to be able to pay cash for the new car after considering the amount she'll receive from selling her used car.

Courtney needs to save money to buy a new car in 4 years. The estimated cost of the new car is $25,000, and she expects to get $13,000 for her used car. To find out the least amount she should save each month, we will subtract the expected amount from the sale of her used car from the cost of the new car, and then divide by the number of months in 4 years (48 months).

The calculation is as follows:

Total amount needed = Cost of New Car - Sale of Used Car
= $25,000 - $13,000
= $12,000

Monthly savings required = Total amount needed \/ Number of months
= $12,000 \/ 48
= $250

Therefore, Courtney should save at least $250 per month to pay cash for her new car in 4 years.

Expanded, the function is ...

... p(x) = x² -x +1

Compared to

... ax² +bx +1

_____

The negative discriminant means there are no real roots, hence no x-intercepts.

Answer: The value of a is 1. The value of b is -1. The value of c is 1. The value of the discriminant is -3. The quadratic function will not intersects the  x-axis.

Explanation:

The standard form of a quadratic equation is,

[tex]ax^{2} +bx +c[/tex]

It can be written as,

[tex]x^{2} -x+1[/tex]

For this case we have the following equation:

[tex](2x + 3) ^ 2 + 8 (2x + 3) + 11 = 0[/tex]

Let [tex]u = 2x + 3[/tex]

We have:

[tex]u ^ 2 + 8u + 11 = 0[/tex]

By definition, given an equation of the form [tex]ax ^ 2 + bx + c = 0[/tex]

The quadratic formula, to find the solution can be written as:

[tex]x = \frac{-b+/-\sqrt{b ^ 2-4 (a) (c)} }{2(a)}[/tex]

In this case we have:

[tex]a = 1\\b = 8\\c = 11[/tex]

Substituting in the quadratic formula we have:

See attached image

Answer:

Option B

x = (- 7 ± √5) / 2

use the substitution u = 2x + 3

equation can now be written as

u² + 8u + 11 = 0

solve for u using the quadratic formula with a = 1, b = 8 and c = 11

u = ( - 8 ± √( 64 - 44 ) )/2

  = ( - 8 ± √20 )/2 = ( - 8 ± 2√5 ) / 2 = - 4 ± √5

change the variable back into terms of  x

2x + 3 = - 4 ± √5 ( subtract 3 from both sides )

2x = - 7 ± √5 ( divide both sides by 2 )

x = ( - 7 ± √5 ) / 2

The length of the three sides of the triangle is 17cm, 25cm, and 12cm.

To solve this problem, let's assign variables to the three sides of the triangle. Let x represent the shortest side, (x + 8) represent the longest side, and (x - 5) represent the third side.

Using the perimeter given, we can set up an equation:

x + (x + 8) + (x - 5) = 56

Simplifying the equation, we have 3x + 3 = 56, then 3x = 53, and finally x = 17.

Therefore, the three sides of the triangle are:
Shortest side: 17cm
Longest side: 17 + 8 = 25cm
Third side: 17 - 5 = 12cm

https://brainly.com/question/32440372

#SPJ3

Answer:

-20

Step-by-step explanation:

We are given

[tex]f(n+1)=-10f(n)[/tex]

we are given

[tex]f(1)=1[/tex]

At n=1:

we can plug n=1 into formula

[tex]f(1+1)=-10f(1)[/tex]

[tex]f(2)=-10f(1)[/tex]

[tex]f(2)=-10*1[/tex]

[tex]f(2)=-10[/tex]

At n=2:

we can plug n=2 into formula

[tex]f(2+1)=-10f(2)[/tex]

[tex]f(3)=-10f(2)[/tex]

[tex]f(3)=-10*-10[/tex]

[tex]f(3)=100[/tex]................Answer

A function assigns the value of each element of one set to the other specific element of another set. The value of f(3) is 100.

A function assigns the value of each element of one set to the other specific element of another set.

As the recursive function is given to us, f(n + 1) = –10f(n), also the value of f(1)=1 , therefore, the value of f(2) can be written as,

[tex]f(n + 1) = -10f(n)\\\\f(1 + 1) = -10f(1)\\\\f(2)= -10 \times 1\\\\f(2)=-10[/tex]

Now, the value of f(3) can be written as,

[tex]f(n + 1) = -10f(n)\\\\f(2 + 1) = -10f(2)\\\\f(3)= -10 \times -10\\\\f(3)= 100[/tex]

Hence, the value of f(3) is 100.

Learn more about Function:

https://brainly.com/question/5245372

The point-slope form of the equation of a line with slope m through point (h, k) can be written as

... y = m(x -h) +k

For your problem, where m = -4 and (h, k) = (2, 2), this becomes

... y = -4(x -2) +2

... y = -4x +8 +2 . . . . eliminate parentheses

... 4x +y = 10 . . . . . . .add 4x to put into standard form

f(1) = 16

to evaluate f(1) substitute x = 1 into f(x)

f(1) = 3 - 2 - 3 + 18 = 16

When 3x³ - 2x² - 3x + 18 is divided by (x - 1 ) the remainder is 16