Question

Артем и Ярослав построили по башне из кубиков. Обе башни имеют квадратное основание и составлены из одинакового числа кубиков. А) Сторона основания Теминой башни в 4 раза больше, чем у Ярослава. Во сколько раз у Ярослава башня выше? Б) Башня Ярослава в 4 раза выше, чем Темина. Во сколько раз у Теминой башни больше сторона основания?

Answer 1

Пусть сторона башни ярослава А тогда сторона теминой башни 4А
высота ярослава H а высота темы h
обЪем башни ярослава
H*A*A
объем башни темы
h*4A*4A  они равны но условию
H*A*A=h*4A*4A
H*A*A-h*4A*4A=0
AA(H-h*4*4)=0
H-16h=0
H=16h
16h/H=16 в 16 раз башня Ярослава выше.

б) пусть сторона башни ярослава А, сторона теминой башни а
высота ярослава 4H тогда высота Теминой Н
обЪем башни ярослава
4H*A*A
объем башни Темы
Н*а*а  они равны но условию
4H*A*A=Н*а*а
Н(4А*А-аа)=0
4АА=аа
4=2*2
2А*2А=аа
4=аа/АА
2*2=а/А*а/А
а/А=2 в два раза сторона Теминой башни больше основания Ярослава
 2Араз башня Ярослава выше.

Answer 2

Problem Analysis

We are given that Artem and Yaroslav built towers using cubes. Both towers have a square base and are made up of the same number of cubes. We need to answer two questions: A) The side length of Artem's tower base is 4 times larger than Yaroslav's. In how many times is Yaroslav's tower taller than Artem's? B) Yaroslav's tower is 4 times taller than Artem's. In how many times is the side length of Artem's tower base smaller than Yaroslav's?

Solution

Let's assume that both Artem and Yaroslav built their towers using n cubes.

A) If the side length of Artem's tower base is 4 times larger than Yaroslav's, then the side length of Yaroslav's tower base is n and the side length of Artem's tower base is 4n.

To find the height ratio, we need to compare the heights of the two towers. Since both towers are made up of the same number of cubes, the height of each tower is also n.

Therefore, the height ratio is n/n = 1. Yaroslav's tower is 1 times taller than Artem's tower.

B) If Yaroslav's tower is 4 times taller than Artem's, then the height of Yaroslav's tower is 4n and the height of Artem's tower is n.

To find the side length ratio, we need to compare the side lengths of the two towers. The side length of Yaroslav's tower base is n and the side length of Artem's tower base is ?.

Let's assume the side length of Artem's tower base is m. Since both towers are made up of the same number of cubes, the volume of each tower is the same. The volume of a cube is equal to the side length cubed, so we can set up the following equation:

m^2 * n = n^2 * 4n

Simplifying the equation, we get:

m^2 = 4n^2

Taking the square root of both sides, we get:

m = 2n

Therefore, the side length of Artem's tower base is 2n.

To find the side length ratio, we divide the side length of Yaroslav's tower base by the side length of Artem's tower base:

n / 2n = 1/2

Therefore, the side length of Artem's tower base is 1/2 times smaller than Yaroslav's tower base.

In summary: A) Yaroslav's tower is 1 times taller than Artem's tower. B) The side length of Artem's tower base is 1/2 times smaller than Yaroslav's tower base.

Please let me know if anything is unclear or if you have any further questions!