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andre [41]
2 years ago
14

50 points,

Mathematics
1 answer:
weeeeeb [17]2 years ago
4 0

Answer:

Sum of the areas of the three circles is 91.06 units².

Step-by-step explanation:

Given:

Let the  radius of the circle with center D  = x

Let the radius  of the circle with center E  = y  

Let the radius  of the circle with center F  = z

To Find:

Sum of the areas of the three circles = ?

Solution:

So we have these equations

DE =x + y = 5............( 1 )\\EF = x + z = 6..........( 2 )\\DF =y + z = 7...........( 3 )

Subtract the second equation from the  first  and we have that

y- z=-1

Add this to equation to the third equation  and we have that

\therefore 2y= 6\\\\\therefore y= 3\\\\\therefore x = 2\\\\\therefore z = 4

Now we have Area of Circle

\textrm{Area of Circle}=\pi (Radius)^{2}

Substituting Radius we get

\textrm{Area of Circle with center D}=3.14\times 2^{2}=12.56\ units^{2}

\textrm{Area of Circle with center E}=3.14\times 3^{2}=28.26\ units^{2}

\textrm{Area of Circle with center F}=3.14\times 4^{2}=50.24\ units^{2}

∴ \textrm{Sum of the areas of the three circles}=12.56+28.26+50.24=91.06\ units^{2}

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Find the dimensions of the rectangle with area 256 square inches that has minimum perimeter, and then find the minimum perimeter
mezya [45]

Answer:

Dimensions: A=a\cdot b=256

Perimiter: P=2a+2b

Minimum perimeter: [16,16]

Step-by-step explanation:

This is a problem of optimization with constraints.

We can define the rectangle with two sides of size "a" and two sides of size "b".

The area of the rectangle can be defined then as:

A=a\cdot b=256

This is the constraint.

To simplify and as we have only one constraint and two variables, we can express a in function of b as:

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The function we want to optimize is the diameter.

We can express the diameter as:

P=2a+2b=2a+2*\frac{256}{a}

To optimize we can derive the function and equal to zero.

dP/da=2+2\cdot (-1)\cdot\frac{256}{a^2}=0\\\\\frac{512}{a^2}=2\\\\a=\sqrt{512/2}= \sqrt {256} =16\\\\b=256/a=256/16=16

The minimum perimiter happens when both sides are of size 16 (a square).

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Answer:

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