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Let's denote the two points on the circle as A and B.

To find the probability that the distance between A and B is at most 1, we can start by drawing a diagram of the circle and marking a region where the distance between A and B is at most 1.

This region consists of all the points on the circle that are within a distance of 1 from each other. We can visualize this as two semicircles with radius 1 that intersect each other, as shown in the diagram below:

Let A and B be the two points that we choose at random. We want to find the probability that the distance between A and B is at most 1.

One way to approach this problem is to use geometric probability. The total area of the circle is πr^2, where r = 1 is the radius of the circle.

The area of the region where the distance between A and B is at most 1 is the area of the two intersecting semicircles.

To find this area, we can subtract the area of the two triangles ABA' and ABB' from the area of the two semicircles.

The area of each semicircle is 1/2 πr^2 = 1/2 π. The area of each triangle is 1/2 base x height, where the base is the chord AB and the height is the distance from the midpoint of AB to the center of the circle.

The chord AB has length 2 sinθ, where θ is the central angle between A and B. The midpoint of AB is the midpoint of the chord, which has a distance of sinθ from the center of the circle.

Therefore, the height of each triangle is sinθ, and the area of each triangle is 1/2 (2 sinθ)(sinθ) = sin^2 θ.

The area of the region where the distance between A and B is at most 1 is then:

2(1/2 π) - 2(sin^2 θ) = π - 2sin^2 θ

Now, we need to find the probability that the distance between A and B is at most 1. This is the ratio of the area of the region where the distance between A and B is at most 1 to the total area of the circle:

P(distance ≤ 1) = (π - 2sin^2 θ)/(π) = 1 - 2sin^2 θ/π

To find the probability, we need to integrate this expression over all possible values of θ from 0 to π.

∫[0, π] (1 - 2sin^2 θ/π) dθ

Using the identity ∫ sin^2 x dx = (1/2) (x - sin x cos x) we get

= 1 - 2/π ∫[0, π] sin^2 θ dθ = 1 - 2/π [π/2 - 1/2 sin(2π)] = 1 - 1/π

Therefore, the probability that the distance between two randomly chosen points on a circle of radius 1 is 1 - 1/pi.

Mar 8, 2023
Mar 7, 2023
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Let's assume that there are x students taking Spanish and y students taking Chinese. Since there are 97 students in total and every student takes at least one language, we know that:

x + y = 97

We also know that there are more students in Chinese than in Spanish, so:

y > x

Now, let's consider the scenario where only x students take Spanish. In this case, all the remaining students (97 - x) must be taking Chinese. But we also know that every student takes at least one language, so:

97 - x ≥ 1 x ≤ 96

Combining these inequalities, we get:

1 ≤ x ≤ 96

Now, we need to find the number of students who take both languages. Let's call this number z. Then, we have:

z = y - x

We want to find the value of z when all the students taking Spanish are also taking Chinese. In other words, we want to find the value of z when x = z.

Substituting x = z into the first equation (x + y = 97), we get:

z + y = 97

Substituting x = z into the second equation (y > x), we get:

y > z

Now we have a system of two equations:

z + y = 97 y > z

We want to find the value of z that satisfies both equations. One way to do this is to try different values of z and see which ones work. However, we can also use algebra to solve the system.

Subtracting z from both sides of the first equation, we get:

y = 97 - z

Substituting this expression for y into the second equation (y > z), we get:

97 - z > z

Simplifying this inequality, we get:

97 > 2z

Dividing both sides by 2, we get:

48.5 > z

Since z is an integer, the only value that satisfies both inequalities is:

 

z = 48

Therefore, when all the students taking Spanish are also taking Chinese, there are 48 students taking both languages.

Mar 7, 2023

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