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MA_775_DIABLO [31]
3 years ago
5

A piece of wire of length L will be cut into two pieces, one piece to form a square and the other piece to form an equilateral t

riangle. How should the wire be cut so as to: (a) Maximize the sum of the areas of the square and the triangle? (b) Minimize the sum of the areas of the square and the triangle?
Mathematics
1 answer:
inn [45]3 years ago
5 0

Answer:

  (a)  square: L; triangle: 0.

  (b)  square: L·(-16+12√3)/11; triangle: L·(27-12√3)/11

Step-by-step explanation:

<u>Strategy</u>: First we will write each area in terms of its perimeter. Then we will find the total area in terms of the amount devoted to the square. Differentiating will give a way to find the minimum total area.

__

In terms of its perimeter p, the area of a square is ...

  A_square = p^2/16

In terms of its perimeter p, the area of an equilateral triangle is ...

  A_triangle = p^2/(12√3)

Then the total area of the two figures whose total perimeter is L with "x" devoted to the square is ...

  A_total = x^2/16 + (L-x)^2/(12√3)

__

(a) We know when polygons are regular, the one with the most area for the least perimeter is the one with the most sides. Hence, the total area is maximized when all of the wire is devoted to the square.

__

(b) The derivative of A_total with respect to x is ...

  dA/dx = x/8 -(L-x)/(6√3)

This will be zero when ...

  x/8 = (L-x)/(6√3)

  x(6√3) = 8L -8x

  x(8 +6√3) = 8L

  x = L·8/(6√3 +8) = 8L(6√3 -8)/(64-108)

  x = L·(12√3 -16)/11

The total area is minimized when L·(12√3 -16)/11 is devoted to the square, and the balance is devoted to the triangle.

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Answer:

Total number chocolate bars with almonds sold were approximately 66.34 and total number chocolate bars with walnuts sold were approximately 34.67.

Step-by-step explanation:

Let number chocolate bars with almonds be x

Also let number chocolate bars with walnuts be y

Given:

Number of chocolate bars with almonds that were sold one weekend was 3 less than 2 times the number of chocolate bars with walnuts.

Framing the above sentence in equation form we get;

x=2y -3

Also given:

The number of chocolate bars with walnuts plus 4 times the number of chocolate bars with almonds was 300.

Framing the above sentence in equation form we get;

4x+y=300

Now Substituting the value of x in above equation we get;

4x+y=300\\4(2y-3)+y=300\\8y-12+y=300\\9y=300+12\\9y=312\\\\y=\frac{312}{9} \approx 34.67

Now we will substitute the value of y in equation x=2y -3 we get;

x=2y-3 = 2\times34.67 -3 = 69.34-3\approx66.34

Hence, Total number chocolate bars with almonds sold were approximately 66.34 and total number chocolate bars with walnuts sold were approximately 34.67.

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Option D:

Right triangle

Solution:

Let us first define obtuse, acute right triangle and supplementary.

<u>Obtuse angled triangle:</u>

If any one of the angle in the triangle is more than 90°, then the triangle is an obtuse angled triangle.

<u>Acute angled triangle:</u>

If all the angles of the triangle are less than 90°, then the triangle is an acute angled triangle.

<u>Right angled triangle:</u>

If any one of the angle in the triangle is exactly 90°, then the triangle is a right angled triangle.

<u>Supplementary:</u>

If sum of the angles is 180°, then the angles are supplementary angles.

But this is not a kind of triangle.

In given triangle one of the angle is 90°.

Therefore, this is a kind of right triangle.

Option D is the correct answer.

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