Find the first three terms of the arithmetic series described.

a_1=1,a_n=19,S_n=100

[tex](a^2+3a)^2+(2a+5)-(a^4-3a+2)=\\a^4+6a^3+9a^2+2a+5-a^4+3a-2=\\6a^3+9a^2+5a+3[/tex]

Answer:

The border will be 18.6 inches wide

Step-by-step explanation:

Let the width be w

The length = 43 inches

Total wood = 800 square inches

Formula for area of rectangle = length x width

800 = 43 x w

w = 800/43

w = 18.6 inches

Therefore, the border will be 18.6 inches wide.

!!

Answer:

5 inches

Step-by-step explanation:

Let x be the width ( in inches ) of the border,

Given,

The length of the picture = 43 inches,

And, width of the picture = 27 inches,

So, the length of whole frame ( picture with border ) = 43 + 2x,

Its width = 27 + 2x

Thus, the area of the whole frame,

[tex]A=(43+2x)(27+2x)-----(1)[/tex]

Now, the area of the picture,

[tex]A_1=43\times 27=1161\text{ square inches}----(2)[/tex]

We have, the area of the border,

[tex]A_2=800\text{ square inches}----(3)[/tex]

Thus,

[tex]A=A_1+A_2[/tex]

Equation (1) (2) and (3),

[tex](43+2x)(27+2x) = 1161+800[/tex]

[tex]1161 + 86x+54x+4x^2=1161+800[/tex]

[tex]4x^2+140x-800=0[/tex]

[tex]x^2+35x-200=0[/tex]

[tex]x^2+40x-5x-200=0[/tex]

[tex]x(x+40)-5(x+40)=0[/tex]

[tex](x-5)(x+40)=0[/tex]

[tex]\implies x=5\text{ or }x=-40[/tex]

∵ Width can not be negative,

Hence, the width of the border is 5 inches.

Answer:

700 cm^2

Step-by-step explanation:

The area of the picture + frame = L * W

Area = 50 * 30

Area = 1500

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

Area of inside area

Area = L * W

Area = 40 * 20

Area = 800

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

Now the frame is going to take up the difference.

Area Frame = Entirie area - inside area

Area Frame = 1500 - 800

Area Frame = 700

Answer:

11.2 miles

Step-by-step explanation:

What has been describe here is a right triangle.

He went 10 miles east, and he turned left and went 5 miles.  He started at home ended at work.  We want to make a road from his home directly to his workplace (no turns; straight there).

So we have by Pythagorean Theorem:

[tex]5^2+10^2=c^2[/tex]

[tex]25+100=c^2[/tex]

[tex]125=c^2[/tex]

Square root both sides:

[tex]\sqrt{125}=c[/tex]

[tex]11.2 \approx c[/tex]

11.2 miles would be the length of that direct path.

Answer:

According to the graph of f(x), the function has three real solutions to x= -2, x= -1 and x=1.

Then, given that the function is a 5 degree polynomial, we can conclude that the function has 2 complex zeros.

Answer:

Step-by-step explanation:

[tex]3-\sqrt{x}=0\\\\\text{Domain:}\ x\geq0\\\\3-\sqrt{x}=0\qquad\text{add}\ \sqrt{x}\ \text{to both sides}\\\\3=\sqrt{x}\to\sqrt{x}=3\\\\\text{Use the de}\text{finition of a square root:}\ \sqrt{a}=b\iff b^2=a\\\\\sqrt{x}=3\iff x=3^2\\\\x=9[/tex]

Answer:

11 cartons

Step-by-step explanation:

Number of large boxes that can be contained in a carton = 8

Number of small boxes that can be contained in a carton = 10

Total number of boxes sent in one shipment = 96

This shipment contains both large and small boxes. Let there be x cartons of large boxes and y cartons of small boxes in one shipment. So, we can say,

Total number of large boxes in one shipment = Number of boxes in one carton * Total number of cartons = 8x

Similarly,

Total number of small boxes in one shipment = 10y

Since, total number of boxes in one shipment is 96, we can set up the equation as:

8x + 10y = 96

This equation can have following possible solutions:

We are given in the statement that are more large boxes than the small boxes, the valid solutions are:

Assuming that he sent both large and small boxes, the only valid solution will be:

This means, the merchant sent 7 cartons of large boxes i.e. 56 large boxes and 4 cartons of small boxes i.e. 40 small boxes. So in total he sent 11 cartons.

Answer:

11 boxes total   4 small    7 large  

Step-by-step explanation:

1 small box *10 = 10             96-10 = 86 (NO MULTIPLE OF 8)

2 small boxes*10=20          96-29=76 ( no multiple of 8)

3 small boxes  *10=30        96-30= 66(no multiple of 8)

4 small boxes*10 = 40        96-40 = 56 (YES multiple of 8)      56/8 = 7   (7 large boxes)

4 large boxes and 7 large boxes

Answer:

Step-by-step explanation:

a)

y=−1/12x^2+6x+3

y=−1/12(0)^2+6(0)+3

y = 3

b)

y=−1/12x^2+6x+3

y = -1/12 (x^2-72x) + 3

y = =-1/12 (x^2-72x+1296-1296) +3

y = -1/12(x^2 -72x +1296) + 108 + 3

y = -1/12 (x - 36)^2 +111

maximum height of the ball is 111 feets

c)

y = -1/12 (x - 36)^2 +111

0 = -1/12 (x - 36)^2 +111

-111 = -1/12 (x - 36)^2

1332 = (x - 36)^2

36.497 = x - 36

x = 72.497

How far from the child does the ball strike the ground = 72.497 feets

The ball is 3 feet high when it leaves the child's hand. The maximum height of the ball is 6 feet. The ball strikes the ground 2 feet from the child.

(a) To find the height of the ball when it leaves the child's hand, we need to substitute x=0 into the equation for y. Therefore, y=(-1/12)(0)^2+6(0)+3=3. So, the ball is 3 feet high when it leaves the child's hand.

(b) The maximum height of the ball can be found by determining the vertex of the parabolic equation. The equation is in the form of y=ax^2+bx+c, where the vertex is located at x=-b/2a. In this case, a=-1/12 and b=6. Therefore, x=-6/(2(-1/12))=-6/(-1/6)=-6(-6)=6 feet. So, the maximum height of the ball is 6 feet.

(c) To determine how far from the child the ball strikes the ground, we need to find when the ball's height is 0. Set y to 0 and solve for x. 0=(-1/12)x^2+6x+3. This equation can be solved using factoring or the quadratic formula. The solutions are x=-36 and x=2. Therefore, the ball strikes the ground 2 feet from the child.

Answer:

9y +9x - 10a-10c -10b = 105

Step-by-step explanation:

Given

a+b+c = -6 and x+y = 5

We have to find 9y +9x - 10a-10c -10b

Rearranging the terms

9x+9y-10a-10b-10c

Simplifying will give us:

= 9 (x+y) - 10(a+b+c)

Putting the values of x+y and a+b+c

= 9(5) -10(-6)

=45+60

=105

Therefore,

9y +9x - 10a-10c -10b = 105 ..

Answer:

9y + 9x - 10a - 10c - 10b  = 105

Step-by-step explanation:

It is given that,  a+ b + c = - 6 and x + y = 5

To find the value of given expression

Let the expression be 9y +9x - 10a-10c -10b

9y +9x - 10a -10c -10b  = 9(y + x) - 10(a + c + b )

 = 9(x + y) - 10(a + b + c)

 = 9 * (5) - 10 * (-6)

 = 45 + 60

 = 105

Therefore 9y + 9x - 10a - 10c - 10b  = 105

In finance, the buyer pays more when choosing to finance the boat rather than paying cash upfront due to additional fees included in the payments.

The subject area of the question is Mathematics, specifically Financial Mathematics. The question is asking about the total cost Ken will pay for the boat if he chooses to finance it. The down payment is $3000 and he will make 48 monthly payments of $300, totaling $14,400. Therefore, Ken will pay a total of $17,400 when financing the boat, which is $2,400 more than the cash purchase price.

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To find the total amount paid for the boat, add the down payment to the product of the monthly payments and the number of payments.

To find the total amount paid for the boat, we need to calculate the total future amount. The down payment is $3,000 and there are 48 monthly payments of $300 each.

The total future amount can be calculated using the formula:

Total Future Amount = Down Payment + Monthly Payments * Number of Payments

Plugging in the values, we get:

Total Future Amount = $3,000 + $300 * 48 = $17,400

Therefore, Ken will pay a total of $17,400 for the boat.

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Answer:

The distance formula d = rt, 'd' stands for distance, 'r' stands for rate and 't' stands for time.

It's a well known formula used in physics to solve problems that involves questions like "how fast", "how far", or "for how long".

In the formula d = rt, the variable t represents time. It's a consistent representation across situations whenever you are dealing with equations for motion, speed, or other physics contexts.

The variable t in the formula d = rt stands for time. This formula is used to calculate distance based on the rate of speed (r) and the time (t) traveled. To solve for time, you would rearrange the formula to t = d/r, which means time equals distance divided by rate.

In various physics and mathematical contexts, as shown in the reference information, the letter t consistently represents time. It could be the time an object takes to cover a certain distance at a given rate, or more complex scenarios involving non-Euclidean worlds or quantum mechanics. Nevertheless, whenever you see t in such equations, it is typically referring to time, unless otherwise specified.

Answer:it would take him five weeks to lose ten pounds

Step-by-step explanation:

Answer: To determine Rocco´s weight jake must use this expression

Step-by-step explanation:

Hi to solve this problem we have to analyze the information given:

So, we have to write an equation where p is Rocco's weight after w weeks.

Each week Rocco will lose 0.5 pounds so the expression is 0.5w, where w is the number of weeks.

Mathematically speaking:

By replacing w by the number of weeks, Jake will obtain Rocco´s weight.

For example, if the number of weeks is 20:

p= 55-0.5x 10

p= 55 - 20

p= 35

After 20 weeks Rocco will lose 10 pounds.

Answer:

105 frames

Step-by-step explanation:

Given that 25 frames are played per second

25 frames= 1 sec

The video already played 3 and 2/5 seconds before the player started to count 0

Write 3 and 2/5 seconds as an improper fraction

=(5*3)+2 / 5 = 17/5 seconds

Multiply by 25 frames

17/5 *25 =85 frames

So according to the video counter,after 17/5 seconds,it should count 85 frames.However,at 0 seconds,it indicated a count of 190 frames.Thus,to get the number of frames that were already in count you subtract 85 frames from the 190 frames.

190-85=105 frames.

The video had played 275 frames when the timer showed [tex]-3\frac{2}{5}[/tex]    seconds.

To determine how many frames had been played when the timer showed [tex]-3\frac{2}{5}[/tex] seconds, we first need to convert [tex]-3\frac{2}{5}[/tex] seconds to an improper fraction or decimal. We have [tex]-3\frac{2}{5}[/tex] = [tex]-\frac{17}{5}[/tex] seconds.

Since the video plays at a rate of 25 frames per second, we can calculate the number of frames played during [tex]-\frac{17}{5}[/tex] seconds as follows:

Total frames = 25 frames/second × [tex]-\frac{17}{5}[/tex] seconds = -85 frames.

This means that 85 frames had been played before the timer started counting (since negative time indicates frames played before the recorded start time).

Since at 0 seconds, the video had already played 190 frames, we add the frames played during the negative time interval:

Total frames = 190 frames (initial) + 85 frames = 275 frames.

Answer:

to find the vertical asymptote, you have to put the rational function in the simplest form, which means to cancel any common factor between the numerator and denominator. here we don't have anything to cancel. then take the denominator and equal it to 0. x-3=0 ,x=3

to find the horizontal asymptote, in this situation, the degree of the numerator and denominator are the same which is 1. therefore, y=the coefficient of the numerator ÷ the coefficient of the denominator. y=6÷1 ,y=6

Answer with explanation:

The given function is:

   [tex]f(x)=\frac{6x-2}{x-3}[/tex]

Horizontal Asymptote

   [tex]y= \lim_{x \to \infty} \frac{6x-2}{x-3}\\\\\text{Dividing Numerator and Denominator by ,x}\\\\\Rightarrow y=\lim_{x \to \infty} \frac{6-\frac{2}{x}}{1-\frac{3}{x}}\\\\\Rightarrow y=\frac{6-0}{1-0}\\\\\Rightarrow y=6[/tex]

Vertical Asymptote

→Substitute Denominator of f(x) =0

⇒x-3=0

⇒x=3

⇒x=3 is Vertical Asymptote , and y=6 is horizontal Asymptote.

Matching with all the options

 Option D

Answer:

2

Step-by-step explanation:

There are four 11-year olds and x amount of 14-years old.

We are given the average of this group of 11 and 14 year olds is 12.

This gives us the following equation to solve for x:

[tex]\frac{4(11)+x(14)}{4+x}=12[/tex]

Doing some simplifying on the left and writing 12 as 12/1 on the right because I'm going to cross-multiply in the step to follow:

[tex]\frac{44+14x}{4+x}=\frac{12}{1}[/tex]

Cross multiplying:

[tex]1(44+14x)=(4+x)(12)[/tex]

Distribute:

[tex]44+14x=48+12x[/tex]

Subtract 12x on both sides:

[tex]44+2x=48[/tex]

Subtract 44 on both sides:

[tex]2x=4[/tex]

Divide both sides by 2:

[tex]x=2[/tex]

There are two 14-year olds.

Let's confirm.

You have four 11 year olds and two 14 years old.

So you have:

11,11,11,11,14,14

We want to find the average of this set of numbers.

You add the numbers and divide by the number of numbers.

[tex]\frac{11+11+11+11+14+14}{6}=\frac{72}{6}=12[/tex]

This is the average the problem wanted us to obtain so the work we did above is correct.

[tex]\huge{\boxed{10}}[/tex]

We can rewrite this using an exponent. [tex]\sqrt{10}^2[/tex]

When you square a square root, you basically just cancel out the square root, leaving whatever was inside the square root symbol. In this case, you end up with a final answer of [tex]\boxed{10}[/tex].

Step-by-step explanation:

First, you must find the square root of 10, which is 3.16.

Now, multiply 3.16x3.16, or find 3.16 squared.

This is 9.98

For this case we have the following functions:

[tex]f (x) = 4x ^ 2 + 1\\g (x) = x ^ 2-5[/tex]

We must find[tex](f-g) (x):[/tex]

By definition of operations with functions we have:

[tex](f-g) (x) = f (x) -g (x)[/tex]

So:

[tex](f-g) (x) = f (x) -g (x) = 4x ^ 2 + 1- (x ^ 2-5) = 4x ^ 2 + 1-x ^ 2 + 5 = 3x ^ 2 + 6[/tex]

Answer:

Option A

[tex]3x ^ 2 + 6[/tex]

Answer:

(x + 4)(x + 2)

Step-by-step explanation:

Consider the factors of the constant term ( + 8) which sum to give the coefficient of the x- term ( + 6)

The factors are + 4 and + 2, since

+ 4 × + 2 = + 8 and + 4 + 2 = + 6, hence

x² + 6x + 8 = (x + 4)(x + 2) ← in factored form

The two binomials that are factors of this trinomial x^2+6x+8 would be (x + 4)(x + 2).

Factorization is expressing a mathematical quantity in terms of multiples of smaller units of similar quantities.

For an integer, factorization is the decomposition of that integer in terms of smaller integers which when multiplied with each other give back that considered number.

Those composing integers are called factors of that considered integers.

Consider the factors of the constant term ( + 8) which sum to give the coefficient of the x- term ( + 6)

x² + 6x + 8

The factors are - 4 and + 2, since

+ 4 × + 2 = + 8 and + 4 + 2 = + 6,

hence , in the factored form

x² + 6x + 8 = (x + 4)(x + 2

Learn more about factorization here:

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Answer:

Step-by-step explanation:

He uses (2/3)(12 eggs) = 8 eggs.

He has (12-8), or 4, eggs left.

Answer:

Step-by-step explanation:

Number of eggs = 1 dozen = 12 eggs

Number of used eggs = 2/3 * 12 (Total eggs will be multiplied by used eggs)

2/3*12

2*4

=8 eggs

Number of used eggs= 8 eggs

To find the number of left eggs. Simply subtract the used eggs from total eggs

12-8 = 4eggs

4 eggs were left....

Answer:

90°

Step-by-step explanation:

we know that

The side TD is perpendicular to the plane formed by the square base ABCD of the cube

therefore

The measure of angle ∠TDB is 90 degrees

The correct answer is d) 90

Step-by-step explanation:

[tex]_nP_k=\dfrac{n!}{(n-k)!}\\\\\\_5P_5=\dfrac{5!}{0!}=120[/tex]

Answer:

A.  120.

Step-by-step explanation:

5P5  =  5! / (5-5)!

= 5! / 0!

= 5!/1

= 5!

= 120.

Answer:

Step-by-step explanation:

Answer:

-6/7

Step-by-step explanation:

To find the slope of a line given two points on that line, you can use the slope formula: [tex]\frac{y_2-y_1}{x_2-x_2}[/tex] where you have points [tex](x_1,y_1) \text{ and } (x_2,y_2)[/tex].  

When using that formula, do not put the subscripts when plugging in the answers.

Something that is equivalent to that formula and I think is easier to remember is:

Lining up the points and subtracting vertically and then putting 2nd difference over first difference.

Like this:

 (-2 ,  3)

- ( 5 , -3)

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

-7         6

So the slope is 6/-7 or -6/7.

It doesn't matter what order you do the points. You have done the (5,-3) on top instead.

Like this:

  ( 5, -3)

-  (-2,  3)

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

  7     -6

The slope is still -6/7.

If you don't like that and you do just directly like using that formula, then I will do it that way too.

So applying the formula either gives you: [tex]\frac{3-(-3)}{-2-5)}[/tex] or [tex]\frac{-3-3}{5-(-2)}[/tex].  Both will give us the same number.

[tex]\frac{3-(-3)}{-2-5)}=\frac{6}{-7}=\frac{-6}{7}[/tex]

[tex]\frac{-3-3}{5-(-2)}=\frac{-6}{7}[/tex]

Answer:

6

Step-by-step explanation:

In this question it appears that for each burger Tom then has to eat 1 portion of fries. This combination would take 15 + 5 = 20 minutes.

Since 60/20 = 3, he would be able to eat 3 burgers and 3 portions of fries in 60 minutes. Making up a total of 3+3 = 6

Answer:

Tom eats 3 burgers and 3 portions of fries in 1 hour.

Step-by-step explanation:

Tom eats a burger in 15 minutes and a portion of fries in 5 minutes.

Total time he spent = 15 + 5  = 20 minutes in eating both

We have to calculate the number of burgers and fries he is able to eat in one hour.

∵ In 20 minutes Tom eats burger and fries = 1

∴ In 1 minutes he would be able to eat = [tex]\frac{1}{20}[/tex]

∴ In 60 minutes he would be able to eat = [tex]\frac{60}{20}=3[/tex] sets of burgers and fries.

Therefore, Tom eats 3 burgers and 3 portions of fries in 1 hour.

Answer:

14/18 and 49/63

Step-by-step explanation:

14/18 and 49/63 are equivalent to 7/9.

7 x 2 = 14

9 x 2 = 18

Therefore, 14/18

7 x 7 = 49

9 x 7 = 63

Therefore, 49/63

Answer:

21/27 and 42/54

Step-by-step explanation:

therefore, 21/27 and 42/54 are equivalent to 7/9

7 x 3=21

9 x 3= 27

Therefore, 21/27

7 x 6= 42

9 x6 = 54

Therefore, 42/54

Answer:

The answer is 7.433

Step-by-step explanation:

Bc:

  25.643  the 5 turned to 15, and the 2 is a 1.

-   18.210

    7.433

Hope my answer has helped you, if not i'm sorry.

Answer:

7.433

Step-by-step explanation:

To solve this problem, you'd need to subtract the numbers from left to right.

25.643-18.21=07.433=7.433

So, the correct answer is 7.433.

I hope this helps!

Thank you for submitting your question!

Answer:

The final step in solving inequality is x>3.

Step-by-step explanation:

To solve this problem, first thing to do is switch sides of -2x>-6, and divde -2 from both sides.

-2x>-6

-2x/-2>-6/-2

Divide numbers from left to right.

-6/-2=3

The final step in solving the inequality is x>3, which is our answer.

I hope this helps!

Thank you! for submitting question on Brainly.

Answer:

5.65 cm

Step-by-step explanation:

We are given that the length of each leg of an isosceles right triangle is 4 cm and we are to find the length of the hypotenuse.

For this, we will use the Pythagoras Theorem:

[tex] a ^ 2 = b ^ 2 + c ^ 2 [/tex]

where [tex]a[/tex] is the hypotenuse.

[tex] a ^ 2 = 4 ^ 2 + 4 ^ 2 [/tex]

[tex] \sqrt { a ^ 2} = \sqrt { 3 2 } [/tex]

[tex] a = 5 . 6 5 [/tex]

Therefore, the length of the hypotenuse is 5.65 cm.

Answer:

hypotenuse = 4√2

Step-by-step explanation:

It is given that, the  length of each leg of an isosceles right triangle is 4 cm

To find the hypotenuse

From the given information we get the given triangle is a right triangle with angles 45°, 45° and 90°

Therefore the side are in the ration 1 : 1 : √2

Here two legs are give

leg1 : leg2 : hypotenuse = 4 : 4 : 4√2

Therefore hypotenuse = 4√2