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4 years ago

7

A large container holds 8 gallons of water. It begins leaking at a constant rate. After 32 minutes the container has 6 gallons l

eft. At what rate is the water leaking. Express answer as a positive value.

1 answer:

anzhelika [568]4 years ago

8 0

Answer:

1 gallon every 16 minutes

Step-by-step explanation:

If after 32 minutes the container has 6 gallons left that means it has lost 2 gallons of water. So we now divide the 32 minutes by the 2 gallons of water to get that 1 gallon of water leaks every 16 minutes.

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Find the Fourier series of f on the given interval. f(x) = 1, ?7 < x < 0 1 + x, 0 ? x < 7

Zolol [24]

$f(x)=\begin{cases}1&\text{for }-7$

The Fourier series expansion of $f(x)$ is given by

$\dfrac{a_0}2+\displaystyle\sum_{n\ge1}a_n\cos\frac{n\pi x}7+\sum_{n\ge1}b_n\sin\frac{n\pi x}7$

where we have

$a_0=\displaystyle\frac17\int_{-7}^7f(x)\,\mathrm dx$
$a_0=\displaystyle\frac17\left(\int_{-7}^0\mathrm dx+\int_0^7(1+x)\,\mathrm dx\right)$
$a_0=\dfrac{7+\frac{63}2}7=\dfrac{11}2$

The coefficients of the cosine series are

$a_n=\displaystyle\frac17\int_{-7}^7f(x)\cos\dfrac{n\pi x}7\,\mathrm dx$
$a_n=\displaystyle\frac17\left(\int_{-7}^0\cos\frac{n\pi x}7\,\mathrm dx+\int_0^7(1+x)\cos\frac{n\pi x}7\,\mathrm dx\right)$
$a_n=\dfrac{9\sin n\pi}{n\pi}+\dfrac{7\cos n\pi-7}{n^2\pi^2}$
$a_n=\dfrac{7(-1)^n-7}{n^2\pi^2}$

When $n$ is even, the numerator vanishes, so we consider odd $n$ , i.e. $n=2k-1$ for $k\in\mathbb N$ , leaving us with

$a_n=a_{2k-1}=\dfrac{7(-1)-7}{(2k-1)^2\pi^2}=-\dfrac{14}{(2k-1)^2\pi^2}$

Meanwhile, the coefficients of the sine series are given by

$b_n=\displaystyle\frac17\int_{-7}^7f(x)\sin\dfrac{n\pi x}7\,\mathrm dx$
$b_n=\displaystyle\frac17\left(\int_{-7}^0\sin\dfrac{n\pi x}7\,\mathrm dx+\int_0^7(1+x)\sin\dfrac{n\pi x}7\,\mathrm dx\right)$
$b_n=-\dfrac{7\cos n\pi}{n\pi}+\dfrac{7\sin n\pi}{n^2\pi^2}$
$b_n=\dfrac{7(-1)^{n+1}}{n\pi}$

So the Fourier series expansion for $f(x)$ is

$f(x)\sim\dfrac{11}4-\dfrac{14}{\pi^2}\displaystyle\sum_{n\ge1}\frac1{(2n-1)^2}\cos\frac{(2n-1)\pi x}7+\frac7\pi\sum_{n\ge1}\frac{(-1)^{n+1}}n\sin\frac{n\pi x}7$

3 0

3 years ago

Which function is represented by the graph?

Arisa [49]

Answer:

D the answer it D I just did this

Step-by-step explanation:

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3 years ago

Need help please with this question

SashulF [63]

Answer:

5/8

5/8

Step-by-step explanation:

5 0

3 years ago

In the figure below, BD and AC are diameters of circle P.

elena-14-01-66 [18.8K]

Answer:

$arc\ DC=30^o$

Step-by-step explanation:

<u><em>The picture of the question in the attached figure</em></u>

The point P is the center of the circle (the drawing is not a scale )

step 1

Find the measure of angle ∠CPD

we know that

$m\angle CPD+150^o=180^o$ ---> by supplementary angles (form a linear pair)

$m\angle CPD=180^o-150^o=30^o$

step 2

Find the measure of arc DC

we know that

<u><em>Central angle</em></u> is the angle that has its vertex in the center of the circumference and the sides are radii of it

so

$arc\ DC=m\angle CPD$ ----> by central angle

therefore

$arc\ DC=30^o$

5 0

3 years ago

A copy machine makes

IrinaK [193]

This problem can be represented in a formula with fractions. To make it more simple, we are converting 5m30s to just pure seconds, which = 154 copies in 330 seconds.

You set up the formula like this: 154/330 = x/60. (it is x/60 because you want to know the x amount of copies per 60 seconds)

Cross multiply, you get 330x = 9240. Divide both sides by 330, you get:

x = 28 copies.

The final answer is 28 copies per minute

7 0

3 years ago