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Oxana [17]
3 years ago
15

A plastic bin contains red, yellow, and green key chains. Out of 350 keys, 10% are yellow, of the remaining key chains, 60% are

green. How many red key chains are inside the bin?
Mathematics
1 answer:
Alinara [238K]3 years ago
3 0
Answer: 105

Explanation:

60% + 10% = 70% = yellow and green

30% = red

350/10 = 35 x 3 = 105
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For the function defined by f(t)=2-t, 0≤t<1, sketch 3 periods and find:
Oksi-84 [34.3K]
The half-range sine series is the expansion for f(t) with the assumption that f(t) is considered to be an odd function over its full range, -1. So for (a), you're essentially finding the full range expansion of the function

f(t)=\begin{cases}2-t&\text{for }0\le t

with period 2 so that f(t)=f(t+2n) for |t| and integers n.

Now, since f(t) is odd, there is no cosine series (you find the cosine series coefficients would vanish), leaving you with

f(t)=\displaystyle\sum_{n\ge1}b_n\sin\frac{n\pi t}L

where

b_n=\displaystyle\frac2L\int_0^Lf(t)\sin\frac{n\pi t}L\,\mathrm dt

In this case, L=1, so

b_n=\displaystyle2\int_0^1(2-t)\sin n\pi t\,\mathrm dt
b_n=\dfrac4{n\pi}-\dfrac{2\cos n\pi}{n\pi}-\dfrac{2\sin n\pi}{n^2\pi^2}
b_n=\dfrac{4-2(-1)^n}{n\pi}

The half-range sine series expansion for f(t) is then

f(t)\sim\displaystyle\sum_{n\ge1}\frac{4-2(-1)^n}{n\pi}\sin n\pi t

which can be further simplified by considering the even/odd cases of n, but there's no need for that here.

The half-range cosine series is computed similarly, this time assuming f(t) is even/symmetric across its full range. In other words, you are finding the full range series expansion for

f(t)=\begin{cases}2-t&\text{for }0\le t

Now the sine series expansion vanishes, leaving you with

f(t)\sim\dfrac{a_0}2+\displaystyle\sum_{n\ge1}a_n\cos\frac{n\pi t}L

where

a_n=\displaystyle\frac2L\int_0^Lf(t)\cos\frac{n\pi t}L\,\mathrm dt

for n\ge0. Again, L=1. You should find that

a_0=\displaystyle2\int_0^1(2-t)\,\mathrm dt=3

a_n=\displaystyle2\int_0^1(2-t)\cos n\pi t\,\mathrm dt
a_n=\dfrac2{n^2\pi^2}-\dfrac{2\cos n\pi}{n^2\pi^2}+\dfrac{2\sin n\pi}{n\pi}
a_n=\dfrac{2-2(-1)^n}{n^2\pi^2}

Here, splitting into even/odd cases actually reduces this further. Notice that when n is even, the expression above simplifies to

a_{n=2k}=\dfrac{2-2(-1)^{2k}}{(2k)^2\pi^2}=0

while for odd n, you have

a_{n=2k-1}=\dfrac{2-2(-1)^{2k-1}}{(2k-1)^2\pi^2}=\dfrac4{(2k-1)^2\pi^2}

So the half-range cosine series expansion would be

f(t)\sim\dfrac32+\displaystyle\sum_{n\ge1}a_n\cos n\pi t
f(t)\sim\dfrac32+\displaystyle\sum_{k\ge1}a_{2k-1}\cos(2k-1)\pi t
f(t)\sim\dfrac32+\displaystyle\sum_{k\ge1}\frac4{(2k-1)^2\pi^2}\cos(2k-1)\pi t

Attached are plots of the first few terms of each series overlaid onto plots of f(t). In the half-range sine series (right), I use n=10 terms, and in the half-range cosine series (left), I use k=2 or n=2(2)-1=3 terms. (It's a bit more difficult to distinguish f(t) from the latter because the cosine series converges so much faster.)

5 0
3 years ago
What side is this a isosceles a scalene or equilateral
aksik [14]
The answer is a scalene
3 0
3 years ago
What is -3 = h + 8 divided by 2?
Kaylis [27]

Answer:

h = -7

Step-by-step explanation:

- 3 = h + 4

- 3 - h = 4

- h = 4 + 3

- h = 7

7 0
3 years ago
The midsegment of a trapezoid is 11 cm in length. If one of the trapezoid's bases is 17 cm long, what is the length of the other
vagabundo [1.1K]

Answer:

x must be 5

Step-by-step explanation:

Recall that the area formula for a trapezoid is

A = (average of base lengths)(width)

Here we have

                                                                   17 cm + x

      (average of base lengths) = 11 cm = ----------------

                                                                           2

So 2(11 cm) = 17 cm + x, or

      22 cm = 17 cm + x

Then x must be 5.

7 0
3 years ago
In a large population, 71 % of the people have been vaccinated. If 4 people are randomly selected, what is the probability that
Oksi-84 [34.3K]

Answer:

The answer is "0.9929"

Step-by-step explanation:

\text{The distribution in binomials is n = 4 and p = 0.71.}\\\\\text{Vaccinated Pr(AT LEAST ONE) = 1-pr( no people vaccinated ).}

= 1 - Bin (n=4, k =0, p=0.71) \\\\= 1 - C(4,0) \times 0.71^0 \times (1-0.71)^{4-0} \\\\= 1 - (1 \times 0.29^4)

= 1 - (1 \times 0.00707281)\\\\=0.9929

5 0
3 years ago
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