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aleksklad [387]
3 years ago
14

A piece of wire of length 7070 is​ cut, and the resulting two pieces are formed to make a circle and a square. Where should the

wire be cut to​ (a) minimize and​ (b) maximize the combined area of the circle and the​ square?
Mathematics
1 answer:
lyudmila [28]3 years ago
3 0

Answer:

a.x=39.2

b.Use whole  wire as a circle

Step-by-step explanation:

We are given that

Length of piece of wire=70  units

Let length of wire used to make a square =x units

Length of wire used in circle=70- x

Side of square=\frac{perimeter\;of\;square}{4}=\frac{x}{4}

Circumference of circle=2\pi r

70-x=2\pi r

r=\frac{70-x}{2\pi}

Combined area of circle and square,A=(\frac{x}{4})^2+\pi(\frac{70-x}{2\pi})^2

Using the formula

Area of circle=\pi r^2

Area of square=(side)^2

a.A=\frac{x^2}{16}+\frac{4900+x^2-140x}{4\pi}

Differentiate w.r.t x

\frac{dA}{dx}=\frac{x}{8}+\frac{2x-140}{4\pi}

\frac{dA}{dx}=0

\frac{x}{8}+\frac{2x-140}{4\pi}=0

\frac{\pi x+4x-280}{4\pi}=0

\pi x+4x-280=0

x(\pi+4)=280

x=\frac{280}{\pi+4}

x=39.2

Again differentiate w.r.t x

\frac{d^2A}{dx^2}=\frac{1}{8}+\frac{1}{2\pi}>0

Hence, the combined area of circle and the square is minimum at x=39.2

b.When the wire is not cut and whole wire used  as a circle . Then, combined area is maximum.

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A private opinion poll is conducted for a politician to determine what proportion of the population favors adding more national
Ksju [112]

Answer:

n≅376

So sample size is 376.

Step-by-step explanation:

The formula we are going to use is:

n=pq(\frac{z_{\alpha/2}}{E})^{2}

where:

n is the sample size

p is the probability of favor

q is the probability of not in favor

E is the Margin of error

z is the distribution

α=1-0.98=0.02

α/2=0.01

From cumulative standard Normal Distribution

z_{\alpha/2}=2.326

p is taken 0.5 for least biased estimate, q=1-p=0.5

n=0.5*0.5(\frac{2.326}{0.06})^{2}\\n=375.71

n≅376

So sample size is 376

4 0
3 years ago
Find the unknown<br><br> x:14 = 10:20
jeyben [28]

Answer:

7:14 = 10:20

Step-by-step explanation:

6 0
2 years ago
Read 2 more answers
Can a quadrilateral be a rhombus ?
koban [17]

Yes it can.  It doesn't <em>have to be</em>, but it can be without too much trouble.

In fact, ALL rhombuses are quadrilaterals.


6 0
3 years ago
Given cosα = −3/5, 180 &lt; α &lt; 270, and sinβ = 12/13, 90 &lt; β &lt; 180
torisob [31]

I got

-  \frac{6 3}{65}

What we know

cos a=-3/5.

sin b=12/13

Angle A interval are between 180 and 270 or third quadrant

Angle B quadrant is between 90 and 180 or second quadrant.

What we need to find

Cos(b)

Cos(a)

What we are going to apply

Sum and Difference Formulas

Basics Sine and Cosines Identies.

1. Let write out the cos(a-b) formula.

\cos(a - b)  =  \cos(a)  \cos(b)  +  \sin(a)  \sin(b)

2. Use the interval it gave us.

According to the given, Angle B must between in second quadrant.

Since sin is opposite/hypotenuse and we are given a sin b=12/13. We. are going to set up an equation using the pythagorean theorem.

.

{12}^{2}  +  {y}^{2}  =  {13}^{2}

144 +  {y}^{2}  = 169

25 =  {y}^{2}

y = 5

so our adjacent side is 5.

Cosine is adjacent/hypotenuse so our cos b=5/13.

Using the interval it gave us, Angle a must be in the third quadrant. Since cos is adjacent/hypotenuse and we are given cos a=-3/5. We are going to set up an equation using pythagorean theorem,

.

( - 3) {}^{2}  +  {x}^{2}  =  {5}^{2}

9 +  {x}^{2}  = 25

{x}^{2}  = 16

x = 4

so our opposite side is 4. sin =Opposite/Hypotenuse so our sin a =4/5.Sin is negative in the third quadrant so

sin a =-4/5.

Now use cosine difference formula

-  \frac{3}{5}  \times  \frac{5}{13}  +   - \frac {4}{5}  \times  \frac{12}{13}

- \frac{15}{65} + (  - \frac{48}{65}  )

-  \frac{63}{65}

Hope this helps

6 0
2 years ago
At full price, Melvin would pay 1.6791.6791, point, 679 dollars per liter of gasoline. Using his loyalty card, Melvin only pays
aksik [14]

Answer:

$0.24

Step-by-step explanation:

Given:

At full price, Melvin would  pay for per liter of gasoline =  $1.679

Using his loyalty card, Melvin only pays = $1.439

Question asked:

How much less does Melvin pay per liter using his loyalty card?

Solution:

At full price, Melvin would  pay for per liter of gasoline =  $1.679

Using his loyalty card, Melvin only pays = $1.439

To find that that how much less amount Melvin pay for per liter of gasoline using his loyalty card, we will subtract the payment using loyalty card from payment without using loyalty card.

$1.679 -  $1.439 = $0.24

Therefore, $0.24 less amount Melvin pay for per liter of gasoline using his loyalty card.

5 0
3 years ago
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