Recursion squares

In the square, ABCD has inscribed a square so that its vertices lie at the centers of the sides of the square ABCD. The procedure of inscribing the square is repeated this way. The side length of the square ABCD is a = 22 cm.

Calculate:
a) the sum of perimeters of all squares
b) the sum of the area of all squares

Correct result:

Σ p =  300.45 cm
Σ S =  968 cm²

Solution:

$$\begin{gathered} p_1=4\cdot 22 \\ {q}_p={\displaystyle \frac{1}{\sqrt{2}}} \\ \mathrm{\Sigma }p=4\cdot 22\cdot {\displaystyle \frac{1}{1-{\displaystyle \frac{1}{\sqrt{2}}}}}=300.45\text{ cm} \end{gathered}$$

$$\begin{gathered} {S}_1=22^2 \\ {q}_S={\displaystyle \frac{1}{2}} \\ \mathrm{\Sigma }S=22^2\cdot {\displaystyle \frac{1}{1-{\displaystyle \frac{1}{2}}}}=968{\text{cm}}^2 \end{gathered}$$

Try another example

Showing 0 comments:

Next similar math problems:

• Granddaughter
  In 2014, the sum of the ages of Meghan's aunt, her daughter and her granddaughter was equal to 100 years. In what year was the granddaughter born, if we know that the age of each can be expressed as the power of two?
• The cube
  The cube has a surface of 600 cm², what is its volume?
• Extending square garden
  Mrs. Petrová's garden had the shape of a square with a side length of 15 m. After its enlargement by 64 m² (square), it had the shape of a square again. How many meters has the length of each side of the garden been extended?
• Trip with compass
  During the trip, Peter went 5 km straight north from the cottage, then 12 km west and finally returned straight to the cottage. How many kilometers did Peter cover during the whole trip?
• Cube surfce2volume
  Calculate the volume of the cube if its surface is 150 cm².
• The cube
  The cube has a surface area of 216 dm². Calculate: a) the content of one wall, b) edge length, c) cube volume.
• Two parallel chords
  In a circle 70 cm in diameter, two parallel chords are drawn so that the center of the circle lies between the chords. Calculate the distance of these chords if one of them is 42 cm long and the second 56 cm.
• Diameter = height
  The surface of the cylinder, the height of which is equal to the diameter of the base, is 4239 cm square. Calculate the cylinder volume.
• A kite
  Children have a kite on an 80m long rope, which floats above a place 25m from the place where children stand. How high is the dragon floating above the terrain?
• Suppose
  Suppose you know that the length of a line segment is 15, x2=6, y2=14 and x1= -3. Find the possible value of y1. Is there more than one possible answer? Why or why not?
• Triangular pyramid
  A regular tetrahedron is a triangular pyramid whose base and walls are identical equilateral triangles. Calculate the height of this body if the edge length is a = 8 cm
• The ball
  The ball has a radius of 2m. What percentage of the surface and volume is another sphere whose radius is 20% larger?
• Uboid volume
  Calculate the cuboid volume if the walls are 30cm², 35cm², 42cm²
• The schoolyard
  The schoolyard had the shape of a square with an 11m side. The yard has been enlarged by 75 m² and has a square shape again. How many meters was each side of the yard enlarged?
• Evaluate 5
  Evaluate expression x²−7x+12x−4 when x=−1
• Secret treasure
  Scouts have a tent in the shape of a regular quadrilateral pyramid with a side of the base 4 m and a height of 3 m. Determine the radius r (and height h) of the container so that they can hide the largest possible treasure.
• The tickets
  The tickets to the show cost some integer number greater than 1. Also, the sum of the price of the children's and adult tickets, as well as their product, was the power of the prime number. Find all possible ticket prices.