Question

1) S4=25 a4-? r-? R-? P-? 2) P3=12 a3-? r-? R-? S-?

Answer 1

Речь идет о правильных многоугольниках

1) S₄ = 25  - площадь квадрата

Найдем a₄, используя формулу для нахождения площади квадрата (S)

S = a₄²

25 = a₄²

a₄ = √25 = 5

Отсюда P = 4a₄ = 4 * 5 = 20

Найдем R, используя формулу

a₄ = R√2

5 = R√2

$\displaystyle\tt R=\frac{5\times\sqrt{2} }{\sqrt{2}\times\sqrt{2} } =\frac{5\sqrt{2} }{2}$

Найдем r, используя формулу

$\displaystyle\tt r=Rcos\frac{180}{n}\\\\\\r=\frac{5\sqrt{2} }{2} cos\frac{180}{4} =\frac{5\sqrt{2} }{2}cos45=\\\\\\=\frac{5\sqrt{2} }{2}\times \frac{\sqrt{2} }{2} =\frac{10}{4}=2.5$

$\displaystyle\tt OTBET:~a_4=5; ~r=2.5;~R=\frac{5\sqrt{2} }{2};\\\\P=20$

2) Если P₃ = 12, то

a₃ = P₃/3 = 12/3 = 4

Найдем R, используя формулу

a₃ = R√3

4 = R√3

$\displaystyle\tt R=\frac{4\times\sqrt{3} }{\sqrt{3}\times\sqrt{3} } =\frac{4\sqrt{3}}{3}$

Найдем r, используя формулу

$\displaystyle\tt r=Rcos\frac{180}{n}\\\\\\r=\frac{4\sqrt{3}}{3}cos\frac{180}{3} =\frac{4\sqrt{3}}{3}cos60=\\\\\\=\frac{4\sqrt{3}}{3}\times \frac{1}{2} =\frac{4\sqrt{3} }{6} =\frac{2\sqrt{3} }{3}$

Найдём S, используя формулу

$\displaystyle S=\frac{1}{2}Pr\\\\\\S=\frac{12\times \displaystyle\frac{2\sqrt{3} }{3} }{2} =6\times \frac{2\sqrt{3} }{3} =\\\\\\=2\times 2\sqrt{3} =4\sqrt{3}$

$\displaystyle\tt OTBET:~a_3=4;~r=\frac{2\sqrt{3} }{3} ;~R=\frac{4\sqrt{3} }{3} ;\\\\S=4\sqrt{3}$

Answer 2

In the given sequences, the variables represent different elements of arithmetic or geometric sequences. Let's break down each sequence step by step:

1. S4 = 25, a4 = ?, r = ?, R = ?, P = ?

In this sequence, we have the sum of the first four terms (S4) given as 25. We also have the fourth term (a4), the common ratio (r), the sum of all terms (R), and the product of all terms (P) to find.

To find the fourth term (a4), we can use the formula for the nth term of an arithmetic sequence:

a_n = a_1 + (n - 1) * d

where a_n is the nth term, a_1 is the first term, n is the term number, and d is the common difference (a constant value added to each term to get to the next term).

However, we don't have the common difference (d) directly given in the sequence. So, we cannot find the exact value of a4 or any other values without additional information.

2. P3 = 12, a3 = ?, r = ?, R = ?, S = ?

In this sequence, we have the product of the first three terms (P3) given as 12. We also have the third term (a3), the common ratio (r), the sum of all terms (R), and the product of all terms (S) to find.

To find the third term (a3) of a geometric sequence, we can use the formula:

a_n = a_1 * r^(n - 1)

where a_n is the nth term, a_1 is the first term, r is the common ratio, and n is the term number.

Again, similar to the previous sequence, we don't have the common ratio (r) directly given, so we cannot calculate the exact value of a3 or any other values without additional information.

To solve both sequences, we need either the common difference (d) for the arithmetic sequence or the common ratio (r) for the geometric sequence. Without this additional information, we cannot determine the values of a4, r, R, or P in the first sequence, or the values of a3, r, R, or S in the second sequence.