Question

Александр по случаю своего юбилея приглашает друзей в театр. Театр предлагает следующую акцию: приглашение 1 гостя обойдётся в 2 500 рублей, второго в 2 400 рублей, третьего в 2 300 рублей и так далее, а для именинника вход бесплатный. Какой максимальное количество гостей Александр может пригласить на свой праздник, если его бюджет составляет 20 000 рублей?Какую минимальную сумму денег Александру необходимо иметь, чтобы пригласить на праздник 15 гостей? Ответ дайте в рублях

Answer 1

Ответ:

Пошаговое объяснение:

Если бюджет Александра составит 20 000р , то он может пригласить 10 гостей на праздник

1 - 2500р

2 - 2400р

3 - 2300р

4 - 2200р

....... Сумма 19 500 р будет, если пригласит 10 гостей.

А чтобы пригласить 15 гостей, Александру необходимо иметь ещё 6500р, общая сумма составляет 26000р.

Answer 2

Maximum number of guests Alexander can invite to his celebration

To determine the maximum number of guests Alexander can invite to his celebration, we need to calculate how many guests he can afford within his budget of 20,000 rubles.

The theater offers a discount for each additional guest, with the invitation cost decreasing by 100 rubles for each subsequent guest. However, since Alexander is the birthday celebrant, his entrance is free.

Let's calculate the maximum number of guests Alexander can invite:

1. The first guest costs 2,500 rubles. 2. The second guest costs 2,400 rubles. 3. The third guest costs 2,300 rubles. 4. The fourth guest costs 2,200 rubles. 5. And so on.

We can see that the cost for each guest decreases by 100 rubles. This is an arithmetic progression with a common difference of -100.

To find the maximum number of guests, we need to sum the costs until the total cost exceeds Alexander's budget of 20,000 rubles.

Let's calculate the sum of the costs:

2,500 + 2,400 + 2,300 + 2,200 + ...

Using the formula for the sum of an arithmetic progression, we can calculate the maximum number of guests:

S = (n/2)(2a + (n-1)d)

Where: - S is the sum of the costs - n is the number of terms (number of guests) - a is the first term (cost of the first guest) - d is the common difference (-100 rubles)

Let's solve for n:

20,000 = (n/2)(2 * 2,500 + (n-1)(-100))

Simplifying the equation:

20,000 = (n/2)(5,000 - 100n + 100)
20,000 = (n/2)(5,100 - 100n)
40,000 = n(5,100 - 100n)

This is a quadratic equation. Let's solve it to find the maximum number of guests Alexander can invite.

Using the quadratic formula:

n = (-b ± √(b^2 - 4ac)) / (2a)

Where: - a = -100 - b = 5,100 - c = -40,000

Solving for n:

n = (-5,100 ± √(5,100^2 - 4(-100)(-40,000))) / (2(-100))

Calculating the discriminant:

D = 5,100^2 - 4(-100)(-40,000)
D = 26,010,000 - 16,000,000
D = 10,010,000

Substituting the values into the quadratic formula:

n = (-5,100 ± √10,010,000) / (-200)

Calculating the square root:

√10,010,000 ≈ 3,162.27766

Simplifying the equation:

n = (-5,100 ± 3,162.27766) / (-200)

Calculating the two possible values for n:

n1 = (-5,100 + 3,162.27766) / (-200) ≈ 9.186
n2 = (-5,100 - 3,162.27766) / (-200) ≈ -0.914

Since the number of guests cannot be negative, we discard the negative value.

Therefore, the maximum number of guests Alexander can invite to his celebration is approximately 9 guests.

Minimum amount of money Alexander needs to invite 15 guests

To calculate the minimum amount of money Alexander needs to invite 15 guests, we need to determine the cost of the first 15 guests.

Using the same arithmetic progression with a common difference of -100 rubles, we can calculate the sum of the costs for the first 15 guests:

2,500 + 2,400 + 2,300 + ... + 2,500 - 14 * 100

Using the formula for the sum of an arithmetic progression, we can calculate the sum:

S = (n/2)(2a + (n-1)d)

Where: - S is the sum of the costs - n is the number of terms (number of guests) - a is the first term (cost of the first guest) - d is the common difference (-100 rubles)

Let's solve for S:

S = (15/2)(2 * 2,500 + (15-1)(-100))

Simplifying the equation:

S = (15/2)(5,000 + 14(-100))
S = (15/2)(5,000 - 1,400)
S = (15/2)(3,600)
S = 15 * 1,800
S = 27,000

Therefore, the minimum amount of money Alexander needs to invite 15 guests is 27,000 rubles.

Please note that the calculations provided are approximate and may vary slightly due to rounding.