Sangaku Quest

Problem What is the sixth root of $59604644775390625$ $\ ?$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution When $59604644775390625$ is factorized into prime factors, $$59604644775390625=5^{24}=(5^4)^6=625^6.$$ Therefore, we find that $$\sqrt[6]{59604644775390625}=625.$$ $ $…

Problem As shown in the figure, there is an object with $5$ floors and a total volume of $335.625 \ m^3$. Each floor is square, $0.5 \ m$ high, and the floor immediately below is $0.5 \ m$ wider on all sides than the floor above. Find the length of one side of the top…

Problem If a rectangle contains a large circle, a medium circle, and a small circle, and the diameters of each are $8 ins., 6 ins.$, and $3 ins.$, as shown in the figure, find the length and width of the rectangle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$…

Problem If four equal circles are contained in a right triangle whose short side is $3 ins.$ and long side is $4 ins.$, as shown in the figure, find the diameter d of the circles. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem When two equal circles are placed inside a right triangle with a length of $3 \ ins.$ and a width of $4 \ ins.$ as shown in the figure, find the diameter of the equal circles.   $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem As shown in the figure, when a large circle with a diameter of $450 \ cm$ and a small circle with a diameter of $288 \ cm$ are inscribed within a right triangle, find the lengths of the three sides of the right triangle.   $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$…

Problem When six spheres with a diameter of $8 \ cm$ are arranged on a flat surface as shown in the figure, and a sphere with a diameter of $10 \ cm$ is placed on top of them, find the height of this object. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$…

Problem If three balls with a diameter of $8 \ cm$ are arranged on a flat surface so that each ball touches the other two as shown in the figure, and a ball with a diameter of $7 \ cm$ is placed on top of them, find the height of the object. $$ $$ $$…

Problem As shown in the figure, parts of the long side and the hypotenuse of a right triangle are used as the two sides of a rhombus, and after cutting out the rhombus, each of the two remaining right triangles contains a circle inscribed on its three sides. Assume that the diameter of the upper…

Problem There is an isosceles trapezoid with a bottom side of $18 \ cm$, lateral sides of $32.8 \ cm$, and a top side of $3.6 \ cm$. Find the diameter of the circle inscribed in the top and lateral sides as shown. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$…