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3 years ago

Select the objects that hold the same amount of liquid as a 98 fluid ounce jug

1 answer:

6 0

To answer this question, you need to know how many fluid ounces are in a pint and in a quart. None of the choices reflect 98 fl oz, so I showed the ones that would hold 96 fluid ounces.

1 cup = 8 fl oz
2 cups = 16 fl oz
1 pint = 2 cups, so 1 pint also equals 16 fl oz.

2 pints equal 1 quart, so 16 x 2 = 32 oz (the number of fluid ounces in a quart)

1 pint - 16 fl oz
1 quart = 32 fl oz

Choice A: 3 quarts x 32 =96 fl oz.
Choice B: 2 quarts x 32 = 64 fl oz
Choice C: 2 quarts x 32 + 2 pints x 16 = 64 + 32 (96 fl oz)
Choice D: 1 quart x 32 + 8 cups x 8 = 32 + 64 (96 fl oz)
Choice E: 2 cups x 8 + 2 pints x 16 = 16 + 32 (48 fl oz)

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Fynjy0 [20]

The sum we want is

$\displaystyle \sum_{n=0}^\infty \frac{(-1)^{T_n}}{(2n+1)^2} = 1 - \frac1{3^2} - \frac1{5^2} + \frac1{7^2} + \cdots$

where $T_n=\frac{n(n+1)}2$ is the n-th triangular number, with a repeating sign pattern (+, -, -, +). We can rewrite this sum as

$\displaystyle \sum_{k=0}^\infty \left(\frac1{(8k+1)^2} - \frac1{(8k+3)^2} - \frac1{(8k+7)^2} + \frac1{(8k+7)^2}\right)$

For convenience, I'll use the abbreviations

$S_m = \displaystyle \sum_{k=0}^\infty \frac1{(8k+m)^2}$

${S_m}' = \displaystyle \sum_{k=0}^\infty \frac{(-1)^k}{(8k+m)^2}$

for m ∈ {1, 2, 3, …, 7}, as well as the well-known series

$\displaystyle \sum_{k=1}^\infty \frac{(-1)^k}{k^2} = -\frac{\pi^2}{12}$

We want to find $S_1-S_3-S_5+S_7$ .

Consider the periodic function $f(x) = \left(x-\frac12\right)^2$ on the interval [0, 1], which has the Fourier expansion

$f(x) = \frac1{12} + \frac1{\pi^2} \sum_{n=1}^\infty \frac{\cos(2\pi nx)}{n^2}$

That is, since f(x) is even,

$f(x) = a_0 + \displaystyle \sum_{n=1}^\infty a_n \cos(2\pi nx)$

where

$a_0 = \displaystyle \int_0^1 f(x) \, dx = \frac1{12}$

$a_n = \displaystyle 2 \int_0^1 f(x) \cos(2\pi nx) \, dx = \frac1{n^2\pi^2}$

(See attached for a plot of f(x) along with its Fourier expansion up to order n = 10.)

Expand the Fourier series to get sums resembling the $S'$ -s :

$\displaystyle f(x) = \frac1{12} + \frac1{\pi^2} \left(\sum_{k=0}^\infty \frac{\cos(2\pi(8k+1) x)}{(8k+1)^2} + \sum_{k=0}^\infty \frac{\cos(2\pi(8k+2) x)}{(8k+2)^2} + \cdots \right. \\ \,\,\,\, \left. + \sum_{k=0}^\infty \frac{\cos(2\pi(8k+7) x)}{(8k+7)^2} + \sum_{k=1}^\infty \frac{\cos(2\pi(8k) x)}{(8k)^2}\right)$

which reduces to the identity

$\pi^2\left(\left(x-\dfrac12\right)^2-\dfrac{21}{256}\right) = \\\\ \cos(2\pi x) {S_1}' + \cos(4\pi x) {S_2}' + \cos(6\pi x) {S_3}' + \cos(8\pi x) {S_4}' \\\\ \,\,\,\, + \cos(10\pi x) {S_5}' + \cos(12\pi x) {S_6}' + \cos(14\pi x) {S_7}'$

Evaluating both sides at x for x ∈ {1/8, 3/8, 5/8, 7/8} and solving the system of equations yields the dependent solution

$\begin{cases}{S_4}' = \dfrac{\pi^2}{256} \\\\ {S_1}' - {S_3}' - {S_5}' + {S_7}' = \dfrac{\pi^2}{8\sqrt 2}\end{cases}$

It turns out that

${S_1}' - {S_3}' - {S_5}' + {S_7}' = S_1 - S_3 - S_5 + S_7$

so we're done, and the sum's value is $\boxed{\dfrac{\pi^2}{8\sqrt2}}$ .

6 0

2 years ago

Subtract these polynomials

Sergio039 [100]

Answer:

simplify the equation into 6x^2-x+8-x^2-2, then subtract. the answer is B.

Step-by-step explanation:

6 0

3 years ago

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What is the value of a

dalvyx [7]

I hope this helps you

3 0

3 years ago

A computer store decides to increase the prices of all the items it sells by 15%. the store manager uses matrices to prepare the

Andrews [41]

The matrix exists as a set of numbers placed in rows and columns to create a rectangular array. The manager could achieve scalar multiplication on Matrix A, utilizing the scalar 1.15.

<h3>What is the matrix?</h3>

The matrix exists as a set of numbers placed in rows and columns to create a rectangular array. The numbers exist named the elements, or entries, of the matrix. Matrices contain wide applications in engineering, physics, economics, and statistics as well as in different branches of mathematics.

Increasing the price by 15% would mean we exist taking 100% of the value + another 15%

100 + 15 = 115%

115% = 115/100 = 1.15.

Multiplying every value in Matrix A by 1.15 will give the price raised by 15%.

Therefore, the correct answer is option c. by multiplying each entry of matrix a by 1.15.

To learn more about matrix refer to:

brainly.com/question/12567347

#SPJ4

7 0

2 years ago

Given f(x) = abx + c. How would increasing the value of a change the graph?

Helen [10]

Increasing the value of a would change the slope of the function specifically it would increase as well. Therefore, the line would be steeper than the original. The term ab represents the slope of the function and changing either one would result to the change of the steepness of the line.

4 0

3 years ago

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