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Norma-Jean [14]
3 years ago
13

The sides of a hexagon are 2, 3, 2, 4, 7, and 6. Find the perimeter of a similar hexagon with two sides of length 3.

Mathematics
1 answer:
nasty-shy [4]3 years ago
8 0

The first hexagon: 2, 2, 3, 4, 6, 7

The second hexagon: 3, 3, a, b, c, d

The hexagons are similar. Therefore the sides are in proportion.

\dfrac{a}{3}=\dfrac{3}{2}     multiply both sides by 3

a=\dfrac{3}{2}=4.5

\dfrac{b}{4}=\dfrac{3}{2}   multiply both sides by 4

b=\dfrac{3}{2}\cdot4=6

\dfrac{c}{6}=\dfrac{3}{2}     multiply both sides by 6

c=\dfrac{3}{2}\cdot6=9

\dfrac{d}{7}=\dfrac{3}{2}    multiply both sides by 7

d=\dfrac{21}{2}=10.5

The perimeter:

P=4.5+6+9+10.5=30

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A box with a square base and open top must have a volume of 296352 c m 3 . We wish to find the dimensions of the box that minimi
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Answer:

  • Base Length of 84cm
  • Height of 42 cm.

Step-by-step explanation:

Given a box with a square base and an open top which must have a volume of 296352 cubic centimetre. We want to minimize the amount of material used.

Step 1:

Let the side length of the base =x

Let the height of the box =h

Since the box has a square base

Volume, V=x^2h=296352

h=\dfrac{296352}{x^2}

Surface Area of the box = Base Area + Area of 4 sides

A(x,h)=x^2+4xh\\$Substitute h=\dfrac{296352}{x^2}\\A(x)=x^2+4x\left(\dfrac{296352}{x^2}\right)\\A(x)=\dfrac{x^3+1185408}{x}

Step 2: Find the derivative of A(x)

If\:A(x)=\dfrac{x^3+1185408}{x}\\A'(x)=\dfrac{2x^3-1185408}{x^2}

Step 3: Set A'(x)=0 and solve for x

A'(x)=\dfrac{2x^3-1185408}{x^2}=0\\2x^3-1185408=0\\2x^3=1185408\\$Divide both sides by 2\\x^3=592704\\$Take the cube root of both sides\\x=\sqrt[3]{592704}\\x=84

Step 4: Verify that x=84 is a minimum value

We use the second derivative test

A''(x)=\dfrac{2x^3+2370816}{x^3}\\$When x=84$\\A''(x)=6

Since the second derivative is positive at x=84, then it is a minimum point.

Recall:

h=\dfrac{296352}{x^2}=\dfrac{296352}{84^2}=42

Therefore, the dimensions that minimizes the box surface area are:

  • Base Length of 84cm
  • Height of 42 cm.
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Consecutive numbers differ by one. So, if x is the first, the three numbers are

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The sum of the first and third integer is

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Solve for x- <br> 6x^2 + 36x + 54 = 0
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Answer:

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Step-by-step explanation:

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