maryoomsaggaf
contestada

!!!!!!!!!!! 100 POINTS !!!!!!!!!!!!!!!!!!! ENSURE YOUR ANSWER IS CORRECT BEFORE POSTING !!!!!
(Past Papers)
How would you go about solving this? Please provide formulae, rules, explanation / working out . Thank you in advance!

100 POINTS ENSURE YOUR ANSWER IS CORRECT BEFORE POSTING Past Papers How would you go about solving this Please provide formulae rules explanation working out T class=

Respuesta :

rvkacademic

Answer:

Perimeter of the square:
[tex]\boxed{54.6 \;cm}[/tex]

Step-by-step explanation:

  • Let us assume each side of the square  = [tex]a \;cm[/tex]
  • A square has all four sides equal to each other
  • Therefore area of square of side a = a²
  • We see there are two circles stacked on top of each other in two columns and also two circles adjacent to each other in two rows
  • If we let the diameter of each of the circles be d, then it is clear from the figure that
    Twice the diameter of each circle = side of square
  • In terms of d and a we get
    [tex]2d = a[/tex]
    [tex]d = \dfrac{a}{2}[/tex]
  • The radius of a circle is
    [tex]r = \dfrac{d}{2}[/tex]
  • In terms of [tex]a[/tex]:
    [tex]r = \dfrac{a}{2} \div 2 \\\\= \dfrac{a}{2} \times \dfrac{1}{2} \quad \text{(when dividing by a fraction, flip the fraction and multiply)}\\\\= \dfrac{a}{4}[/tex]
  • Area of a circle of radius r [tex]= \pi r^2[/tex]
  • So each of the identical circles has an area [tex]= \pi r^2[/tex]
  • Area of 4 identical circles [tex]= 4 \pi r^2[/tex]
  • Substituting [tex]r = a/4[/tex] we get
  • Area of the four identical circles
    [tex]=4 \pi \left(\dfrac{a}{4}\right)^2\\\\= 4 \pi \dfrac{a^2}{16}\\\\= \pi \dfrac{a^2}{4}\\\\[/tex]

  • Area of the shaded region:

    = Area of square - Total area of the four circles
    = [tex]a^2 - \pi \dfrac{a^2}{4}[/tex]
    [tex]= a^2\left(1 - \dfrac{\pi}{4}\right)\quad \rightarrow \text{factoring out $a^2$}[/tex]
    [tex]1 - \dfrac{\pi}{4} = 0.2146\\\\[/tex]
  • So area of shaded region
    [tex]=0.2146a^2[/tex]

  • We are given the area of shaded region = [tex]40 cm^2[/tex]
  • Therefore we get
    [tex]0.2146a^2 = 40\\\\[/tex]
  • From which
    [tex]a^2 = \dfrac{40}{0.2146}\\\\a^2 = 186.3932\\\\a = \sqrt{186.3932}\\\\a = 13.6526[/tex]
  • So each side of the square is [tex]13.6526 \;cm[/tex]
  • The perimeter of a square of side a
    = 4 x side length
    [tex]=4a[/tex]
  • Hence the perimeter of the square
    [tex]= 4 \times 13.6526\\\\= 54.6103\\\\= 54.6 \quad\text{to 3 significant digits}[/tex]

semsee45

Answer:

54.6 cm

Step-by-step explanation:

The area of the shaded region can be calculated by subtracting the area of the 4 congruent circles from the area of the square.

As the area of a square is the square of its side length, s², and the area of one circle with radius r is πr², then:

[tex]\textsf{Area of the shaded region}=s^2 - 4 \pi r^2[/tex]

From inspection of the given diagram, the diameter of each circle is equal to half the side length of the square. Therefore, the side length of the square is equal to 4 radii:

[tex]s = 4r[/tex]

Substitute this into the area of the shaded region expression:

[tex]\textsf{Area of the shaded region}=s^2 - 4 \pi r^2 \\\\\textsf{Area of the shaded region}=(4r)^2 - 4 \pi r^2 \\\\\textsf{Area of the shaded region}=16r^2 - 4 \pi r^2 \\\\\textsf{Area of the shaded region}=4r^2(4-\pi)[/tex]

Given that the area of the shaded region is 40 cm², then:

[tex]4r^2(4-\pi)=40[/tex]

Solve for r:

[tex]4r^2=\dfrac{40}{4-\pi}\\\\\\r^2=\dfrac{40}{4(4-\pi)}\\\\\\r^2=\dfrac{10}{4-\pi}\\\\\\r=\sqrt{\dfrac{10}{4-\pi}}[/tex]

Given that the side length of the square (s) is equal to 4 radii, then:

[tex]s = 4r\\\\\\s=4\sqrt{\dfrac{10}{4-\pi}}[/tex]

The perimeter of a square is 4 times its side length, so:

[tex]\textsf{Perimeter of the square}=4s\\\\\\\textsf{Perimeter of the square}=4\cdot 4\sqrt{\dfrac{10}{4-\pi}}\\\\\\\textsf{Perimeter of the square}=16\sqrt{\dfrac{10}{4-\pi}}\\\\\\\textsf{Perimeter of the square}=54.61013746959...\\\\\\\textsf{Perimeter of the square}=54.6\; \sf cm\;(3\;s.f.)[/tex]

Therefore, the perimeter of the square is 54.6 cm, rounded to three significant figures.

Otras preguntas