A 6 ounce package of fruit snacks contains 45 pieces how many pieceswould would you expect in a10 ounce package

(a) Wild guess:

[tex]f(x)=\dfrac1{(6+x)^2[/tex]

Recall the power series

[tex]\displaystyle\frac1{1-x}=\sum_{n=0}^\infty x^n[/tex]

With some manipulation, we can write

[tex]\displaystyle\frac1{6+x}=\frac16\frac1{1-\left(-\frac x6\right)}=\frac16\sum_{n=0}^\infty\left(-\frac x6\right)^n=\sum_{n=0}^\infty\frac{(-x)^n}{6^{n+1}}[/tex]

Take the derivative and we get

[tex]\displaystyle-\frac1{(6+x)^2}=-\sum_{n=0}^\infty\frac{n(-x)^{n-1}}{6^{n+1}}[/tex]

[tex]\displaystyle=-\sum_{n=1}^\infty\frac{n(-x)^{n-1}}{6^{n+1}}[/tex]

[tex]\displaystyle=-\sum_{n=0}^\infty\frac{(n+1)(-x)^n}{6^{n+2}}[/tex]

so we have

[tex]\displaystyle\frac1{(6+x)^2}=\sum_{n=0}^\infty\frac{(n+1)(-x)^n}{6^{n+2}}[/tex]

By the ratio test, this series converges if

[tex]\displaystyle\lim_{n\to\infty}\left|\frac{\frac{(n+2)(-x)^{n+1}}{6^{n+3}}}{\frac{(n+1)(-x)^n}{6^{n+2}}}\right|=\left|\frac x6\right|\lim_{n\to\infty}\frac{n+2}{n+1}=\left|\frac x6\right|<1[/tex]

or [tex]|x|<6[/tex], so that the radius of convergence is [tex]R=6[/tex].

(b). If we take the second derivative, we get

[tex]\displaystyle\frac2{(6+x)^3}=\sum_{n=0}^\infty\frac{n(n+1)(-x)^{n-1}}{6^{n+2}}[/tex]

[tex]\displaystyle=\sum_{n=1}^\infty\frac{n(n+1)(-x)^{n-1}}{6^{n+2}}[/tex]

[tex]\displaystyle=\sum_{n=0}^\infty\frac{(n+1)(n+2)(-x)^n}{6^{n+3}}[/tex]

[tex]\displaystyle\frac1{(6+x)^3}=\frac12\sum_{n=0}^\infty\frac{(n+1)(n+2)(-x)^n}{6^{n+3}}[/tex]

Apply the ratio test again and we get [tex]R=6[/tex].

(c) Multiply the previous series by [tex]x^2[/tex] and we get

[tex]\displaystyle\frac{x^2}{(6+x)^3}=\frac12\sum_{n=0}^\infty\frac{(n+1)(n+2)(-x)^nx^2}{6^{n+3}}[/tex]

[tex]\displaystyle=\frac12\sum_{n=0}^\infty\frac{(n+1)(n+2)(-1)^nx^{n+2}}{6^{n+3}}[/tex]

The ratio test yet again tells us [tex]R=6[/tex].

are you trying to combine them?

if so, then when you multiply parenthesis stuffs, you should FOIL.

foil stands for first, outside, inside, and last.

when we combine (x -2)(3x-4)

1. we multiply the first parts together x & 3x to get 3x^2.

2. then we multiply the "outsides," x & -4 to get -4x.

3. insides would be -2 & 3x, so -6x.

4. finally last is -2 & -4 to make 8.

putting it all together we get 3x^2 - 4x - 6x + 8.

or simply 3x^2 - 10x + 8 by combing like terms.

for (x + 6)^2 keep in mind that squaring is multiplying something by itself. so (x+6)^2 is the same as (x+6)(x+6).

applying the same foil technique:

1. x * x = x^2

2. x * 6 = 6x

3. 6 * x = 6x

4. 6 * 6 = 36

so we get x^2 + 6x + 6x + 36.

simplified it's x^2 + 12x + 36

keep in mind also that FOILing is only really good for binomials. (something like (x+6)(6x^2+7x+6) wouldnt work).

also also remember that all were really doing is just distributing

Answer:

Option C  [tex]10^{6}[/tex]

Step-by-step explanation:

we have

[tex]10^{2}*10^{4}[/tex]

we know that

When multiplying two powers that have the same base, you can add the exponents (product rule exponent)

so

[tex]10^{2}*10^{4}=10^{2+4}=10^{6}[/tex]

Answer:

10 to the power of 6 :)

Step-by-step explanation:

Answer:

[tex](x',y')-->(x-3,y-2)[/tex]

Step-by-step explanation:

Notice that we can get from the x-coordinate of A, 1, to the x-coordinte of A', -2, by subtracting 3 from the x-coordinate of A. More formaly:

[tex]1+a=-2[/tex]

[tex]a=-2-1[/tex]

[tex]a=-3[/tex]

Similarly, we can get from the y-coordinate of A, 5, to the y-coordinate of A', 3, by subtracting 2 from the y-coordinte of A. More formaly:

[tex]5+b=3[/tex]

[tex]b=3-5[/tex]

[tex]b=-2[/tex]

Now we now that to get to A' from A, we need to subtract 3 to the x-coordinate of A and subtract 2 to the y-coordinate. Knowing this, we can create the expression to translate any point of the polygon ABCD to create the polygon A'B'C'D':

[tex](x',y')-->(x-3,y-2)[/tex]

Answer:18.375

Step-by-step explanation:8x/7+9=30

We simplify the equation to the form, which is simple to understand  

8x/7+9=30

Simplifying:

+1.14285714286x+9=30

We move all terms containing x to the left and all other terms to the right.  

+1.14285714286x=+30-9

We simplify left and right side of the equation.  

+1.14285714286x=+21

We divide both sides of the equation by 1.14285714286 to get x.  

x=18.375

Brainliest please :)

Answer: x=18 3/8

Step-by-step explanation:

I got it right on odyssey ware

Answer:

64.9 inches

Step-by-step explanation:

3 * 7.8 = 23.4

5 * 9.2 = 46

23.4 + 46 = 64.9 inches

69.4 inches of string does susie have altogether.

What is Addition?

Addition (usually denoted by a plus sign +) is one of the four basic arithmetic operations, the other three being subtraction, multiplication, and division. Adding two integers gives the sum or sum of their values. The example in the adjacent image shows a combination of three apples and two apples, making a total of five apples. This observation is equivalent to the mathematical expression "3 + 2 = 5" (ie "3 plus 2 equals 5").

In addition to counting items, addition can also be defined and performed without reference to specific objects, using instead abstractions called numbers, such as integers, real numbers, and complex numbers. Addition belongs to arithmetic, a branch of mathematics. In algebra, another branch of mathematics, addition can also be performed on abstract objects such as vectors, matrices, subspaces, and subgroups.

Addition has several important properties. It is commutative, which means that the order of the operands does not matter, and it is associative, which means that if you add more than two numbers, the order of addition does not matter (see Addition). Repeated addition of 1 is the same as counting (see successor function). Adding 0 does not change the number. Addition also follows predictable rules for related operations like subtraction and multiplication.

Here given,Susie has 8 pieces of string and 3 pieces are 7.8 inches long ,and 5 pieces are 9.2 inches long.

So,total length of 3 pieces of string [tex] = (3 \times 7.8) = 23.4 \: inches[/tex]

and total length of 5 pieces of string [tex] = (5 \times 9.2) = 46 \: inches[/tex]

So,Susie have altogether (23.4+46) = 69.4 inches string.

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Answer:

She can do 8.5 anwsers a minute.

Step-by-step explanation:

Divide 205 by 25. You get 8.5

To check, you can multiply 8.5 by 25, with your result being 205. Hope this helps love! :)

Answer:

K

Step-by-step explanation:

The factor that will change in the expression will be K.

n = number of walls

K = units of $/ft²

L = length of each wall in feet

h = height of each wall in feet

If the customer will request a more expensive brand of paint, the only factor that will change is K, because K holds the price of the paint over how many square feet.

By using more expensive brand of paint, the cost "K" would change.

Given:

painter's fee = nKLh

length= L

h= Height

n = number of walls.

The constant "K" represents the cost per square foot of the paint.

By using a more expensive brand of paint, the cost per square foot would increase.

Therefore, the value of "K" in the expression would change to reflect the new cost associated with the more expensive paint.

The other factors in the expression, "n" (number of walls), "L" (length of each wall), and "h" (height of each wall) remains same.

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Answer:

C = π(6)

C = 6π

C = 18.84

Step-by-step explanation:

The circumference is the distance around the circle. It relates the number of times the diameter will encircle the circumference as 3.14 or π. As a result, the formulas for the circumference of a circle are C = 2πr or C = πd. The information given is the diameter so use C = πd by substituting d = 6.

C = π(6)

C = 6π

C = 18.84

Answer:

A 3.14x6

Step-by-step explanation:

Answer:

Random Sample

Step-by-step explanation:

Answer:

Part 1) The dimensions are

Length is [tex]20\ ft[/tex], Width is [tex]12\ ft[/tex]

Part 2) The perimeter is [tex]64\ ft[/tex]

Step-by-step explanation:

Part 1) Determine both dimensions

Let

x-----> the length of the pool

y----> the wide of the pool

we know that

The area of the rectangle (pool) is equal to

[tex]A=xy[/tex]

we have

[tex]A=240\ ft^{2}[/tex]

so

[tex]240=xy[/tex] -----> equation A

[tex]x=y+8[/tex] ----> equation B

substitute equation B in equation A and solve for y

[tex]240=(y+8)y[/tex]

[tex]240=y^{2}+8y[/tex]

[tex]y^{2}+8y-240=0[/tex]

using a graphing tool to solve the quadratic equation

the solution is

[tex]y=12\ ft[/tex]

see the attached figure

Find the value of x

[tex]x=y+8[/tex] ------> [tex]x=12+8=20\ ft[/tex]

Part 2) Find the perimeter

The perimeter of rectangle (pool) is equal to

[tex]P=2(x+y)[/tex]

we have

[tex]x=20\ ft, y=12\ ft[/tex]

substitute

[tex]P=2(20+12)=64\ ft[/tex]

The dimensions of the pool is 12ft by 20ft and the perimeter of the pool is 64feet

The perimeter of a rectangle is calculated as;

P= 2(l+w)

where

l is the length

w is the width

If the pool is to be 8ft longer than it is wide and cover 240 square feet then the equation becomes;

240 = w(w+8)

240 = w² + 8w

w² + 8w - 240 = 0

w² +20w -12w -240 = 0

w(w+20) -12(w+20) = 0

w = 12feet

Determine the length of the rectangle. Recall that:

l = 8 + w

l = 8+12

l = 20feet

Hence the dimensions of the pool is 12ft by 20ft

Perimeter = 2(12+20)

P = 2(32)

P = 64ft

Hence the perimeter of the pool is 64feet

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Answer:

M=11.438 n=23.186

Step-by-step explanation:

Answer:

Step-by-step explanation:

Answer:

The two numbers would be -51 and 51

Step-by-step explanation:

To find these, first set the equation for the first number as x. You can then set the second number as x + 102. Now, find their product.

x(x + 102) = x^2 + 102x

Now, to find the minimum, find the value of x in the vertex of this equation.

-b/2a = -102/2(1) = -102/2 = -51

So we know -51 is the first number. Now we find the second using the prewritten equation.

x + 102 = -51 + 102 = 51

The two numbers whose difference is 102 and whose product is a minimum are -51 and 51. This is obtained by differentiating and finding the minimum of the function that represents their product.

To find two numbers whose difference is 102 and whose product is a minimum, we initially express the two numbers as x and x + 102. The product of these two numbers can be denoted as f(x) = x(x + 102) = x² + 102x. To find the minimum product, we differentiate f(x) to find f'(x) = 2x + 102. Setting this equal to zero gives x = -51. We should note, f''(x) = 2, verifies that there is a minimum at x = -51. Hence, the two numbers are -51 and (-51 + 102) = 51.

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Answer:

37

Step-by-step explanation:

w=(180-69)/3

Hello there!

To find which tuna is a better deal, divide the cost by the number of ounces of tuna you are getting to get the cost per ounce.

$0.90/5  = $0.18 per ounce

$2.40/12 = $0.20 per ounce

Since the 5-ounce can of tuna has a cheaper unit rate price, meaning you are getting a better value, makes this the best option. I hope this was helpful and have a great day! :)

The 5 once one hope this helps

The median of a set is defined as the middle element of the sorted set.

So, the only way to add two elements without changing the median is to add an element before it, and one element after it.

Think about it: if you are the middle element in a queue, and we add something before you and something after you, you're still in the middle:

[tex] \{2,\ 3,\ 4\} \mapsto \{1,\ 2,\ 3,\ 4,\ 5\}[/tex]

In both cases, the middle element is 3.

If, instead, we add two elements before 3,

[tex] \{2,\ 3,\ 4\} \mapsto \{0,\ 1,\ 2,\ 3,\ 4\}[/tex]

the median has changed, and similarly if we add two elements after 3

[tex] \{2,\ 3,\ 4\} \mapsto \{2,\ 3,\ 4,\ 5,\ 6\}[/tex]

the median has changed again.

So, if we sort our set, we have

[tex]\{27,\ 28,\ 35,\ 37,\ 43,\ 47\}[/tex]

And the middle elements are 35 and 37. The only option that adds something bigger than then and something smaller than them is option a.

Answer:

6

Step-by-step explanation:

48 tiles divided by 8 ft is 6 ft

Answer:

6 cm

Step-by-step explanation:

A = 1/2(b + 12)(4)

36 = 1/2(b + 12)(4)

72 = (b + 12)(4)

18 = b + 12

18 - 12 = b

6 = b

or

b = 6 cm

The calculated missing base of the trapezoid is 6 cm

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

The trapezoid

The area of a trapezoid is calculated as

Area = 1/2 * Sum of parallel base * Height

Using the above as a guide, we have the following:

1/2 * (b + 12) * 4 = 36

So, we have

b + 12 = 18

Evaluate

b = 6

Hence, the missing base of the trapezoid is 6 cm

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Answer:

The height of the pyramid is [tex]6\ cm[/tex]

Step-by-step explanation:

we know that

The volume of the pyramid is equal to

[tex]V=\frac{1}{3}Bh[/tex]

where B is the area of the base of the pyramid

h is the height of the pyramid

we have

[tex]V=12\ cm^{3}[/tex]

[tex]B=(3)(2)=6\ cm^{2}[/tex] ----> the area of the base is the area of a rectangle

substitute the values in the formula and solve for h

[tex]12=\frac{1}{3}(6)h[/tex]

[tex]h=12/2=6\ cm[/tex]

Answer:

I think the answer would be B.

Answer:

[tex]f(x)=5x-7[/tex]

Step-by-step explanation:

The given line passes through the points;

[tex](x_1,y_1)=(3,8)[/tex]

and

[tex](x_2,y_2)=(4,13)[/tex]

Let us find the slope using;

[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]

[tex]m=\frac{13-8}{4-3}[/tex]

[tex]m=\frac{5}{1}=5[/tex]

The equation is given by;

[tex]y-y_1=m(x-x_1)[/tex]

We substitute the values into the formula to obtain;

[tex]y-8=5(x-3)[/tex]

[tex]y=5x-15+8[/tex]

[tex]y=5x-7[/tex]

There the equation of the line is

[tex]f(x)=5x-7[/tex]

To find the equation of the line passing through the points (3,8) and (4,13), use the slope-intercept form, which is y = mx + b.

To find the equation of the line passing through the points (3,8) and (4,13), we can use the slope-intercept form which is y = mx + b, where m is the slope and b is the y-intercept.

First, we need to find the slope (m) using the formula: m = (y2 - y1) / (x2 - x1). Substituting the coordinates of the points, we have m = (13 - 8) / (4 - 3) = 5 / 1 = 5.

Next, we can choose any of the two points to substitute into the equation to find the y-intercept (b). Using the point (3,8), we have 8 = 5(3) + b. Solving for b, we get b = -7.

Therefore, the equation of the line passing through the points (3,8) and (4,13) is y = 5x - 7.

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Let width = X

Then length would = 3x

The perimeter is 2 times the length + 2 times the width, so half the perimeter would be the length plus the width.

L + W = 1/2 perimeter

L +w = 48

Replace L and W withe the variables from above:

3x + x = 48

Simplify:

4x = 48

Divide both sides by 4:

x = 48/4

x = 12

Now you have a value for x, replace x with 12 in the variables:

Width = x = 12 cm

Length = 3x = 3*12 = 36 cm

Answer:

  Yes, the limit as x approaches 1 is 1.

Step-by-step explanation:

The function is defined at x=1 and to the left of there. The function approaches 1 as x approaches 1 from the right.

The limit is 1 from either direction, so the limit exists.

Answer:

the difference between bobby and the pool deck is 35 feet

Step-by-step explanation:

The difference between the pool deck and Bobby's position is 35 feet.

To find the difference between the pool deck and Bobby's position, we subtract Bobby's depth from the pool deck's height. The pool deck is 15 feet above sea level, while Bobby is diving 50 feet below sea level. We can compare these two numbers by subtracting 50 from 15.

15-50=35

The difference between the pool deck and Bobby's position is 35 feet.

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Answer:

a) y = 0.74x + 18.99; b) 80; c) r = 0.92, r² = 0.85; r² tells us that 85% of the variance in the dependent variable, the final average, is predictable from the independent variable, the first test score.

Step-by-step explanation:

For part a,

We first plot the data using a graphing calculator.  We then run a linear regression on the data.

In the form y = ax + b, we get an a value that rounds to 0.74 and a b value that rounds to 18.99.  This gives us the equation

y = 0.74x + 18.99.

For part b,

To find the final average of a student who made an 83 on the first test, we substitute 83 in place of x in our regression equation:

y = 0.74(83) + 18.99

y = 61.42 + 18.99 = 80.41

Rounded to the nearest percent, this is 80.

For part c,

The value of r is 0.92.  This tells us that the line is a 92% fit for the data.

The value of r² is 0.85.  This is the coefficient of determination; it tells us how much of the dependent variable can be predicted from the independent variable.

To predict the final average based on the first test grade, you can use the regression equation Final average = 173.51 + 4.83(First test grade). For a student who scored 83 on the first test, the predicted final average is 179.08. The correlation coefficient, r, is 0.868 and the coefficient of determination, r^2, is 0.752.

To develop a regression model to predict the final average based on the first test grade, you can use the least-squares regression line. The equation for the regression line is: Final average = 173.51 + 4.83(First test grade). This equation allows you to predict the final average based on the first test grade.

To predict the final average for a student who scored 83 on the first test, substitute 83 into the equation: Final average = 173.51 + 4.83(83) =  179.08. Therefore, the predicted final average is 179.08 for a student who scored 83 on the first test.

The correlation coefficient, r, measures the strength and direction of the linear relationship between the first test grade and the final average. In this model, the value of r is 0.868. The coefficient of determination, r^2, represents the proportion of the variation in the final average that can be explained by the first test grade. In this model, the value of r^2 is 0.752. This means that 75.2% of the variation in the final average can be explained by the first test grade.

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Answer:

450

Step-by-step explanation:

5% of 300 = 15

300+10(15)=450

Using the compound interest formula, if you deposit $300 in an account earning 5% interest compounded annually, you would have approximately $488.85 in the account after 10 years.

This problem involves compound interest, which appeal to financial maths. The formula for compound interest is A = P*(1 + r/n)^(nt), where:

For this problem, P = $300, r = 5% or 0.05, n = 1 (because it's compounded annually), and t = 10 years. Substituting these values into the formula, you have A = 300*(1 + 0.05/1)^(1*10).

Doing the math, A = $300 * (1.05)^10 ≈ $488.85. So after 10 years, you would have approximately $488.85 in the account.

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Answer:

Step-by-step explanation:

f(x) + n - shift the graph of f(x) n units up

f(x) - n - shift the graph of f(x) n units down

f(x - n) - shift the graph of f(x) n units to the right

f(x + n) - shift the graph of f(x) n units to the left

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

We have g(x) = |x - 8| + 3

f(x) = |x| → f(x - 8) = |x - 8|     shift the graph 8 units to the right

f(x - 8) + 3 → |x - 8| + 3         shift the graph 3 unit up

x=8

y=2

8/2=4

the value of the x is 4