The problem involves a target consisting of three concentric circles, each with a different point value. The goal is to calculate the expected point value of the region hit by a dart thrown at the target. We are given the following information:

• The innermost circle has a radius of $1$ and corresponds to a point value of $10$.
• The middle circle has a radius of $3$ and corresponds to a point value of $5$.
• The outermost circle has a radius of $5$ and corresponds to a point value of $2$.

Step 1: Calculate the areas of the regions

The target is divided into three regions:

• The innermost region is a circle of radius 1.
• The second region is a ring between the circles of radius 1 and 3.
• The third region is a ring between the circles of radius 3 and 5.

We begin by calculating the areas of these regions.

Area of the innermost circle:

The area of a circle is given by the formula $\pi r^2$, where $r$ is the radius. For the innermost circle with radius $1$:

$\text{Area of innermost circle} = \pi \times 1^2 = \pi$

Area of the second region (the ring between radii 1 and 3):

The area of this region is the area of the circle with radius 3 minus the area of the circle with radius 1:

$\text{Area of second region} = \pi \times 3^2 - \pi \times 1^2 = 9\pi - \pi = 8\pi$

Area of the third region (the ring between radii 3 and 5):

The area of this region is the area of the circle with radius 5 minus the area of the circle with radius 3:

$\text{Area of third region} = \pi \times 5^2 - \pi \times 3^2 = 25\pi - 9\pi = 16\pi$

Step 2: Calculate the total area of the target

The total area of the target is the area of the circle with radius 5:

$\text{Total area of target} = \pi \times 5^2 = 25\pi$

Step 3: Determine the expected point value

The expected point value is the weighted average of the point values of the regions, where the weight of each region is its area relative to the total area. Let $X$ be the random variable representing the point value of the region hit by the dart. The expected value $E[X]$ is given by:

$E[X] = \frac{\text{Area of innermost circle}}{\text{Total area}} \times 10 + \frac{\text{Area of second region}}{\text{Total area}} \times 5 + \frac{\text{Area of third region}}{\text{Total area}} \times 2$

Substitute the areas and point values:

$E[X] = \frac{\pi}{25\pi} \times 10 + \frac{8\pi}{25\pi} \times 5 + \frac{16\pi}{25\pi} \times 2$

Simplify the expressions:

$E[X] = \frac{1}{25} \times 10 + \frac{8}{25} \times 5 + \frac{16}{25} \times 2$

Now, calculate each term:

$E[X] = \frac{10}{25} + \frac{40}{25} + \frac{32}{25}$

Add the terms:

$E[X] = \frac{10 + 40 + 32}{25} = \frac{82}{25}$

Thus, the expected point value is:

$E[X] = 3.28$

Final Answer:

The expected point value of the region hit by the dart is $\boxed{3.28}$.

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Would you like further clarification or have additional questions? Here are some related topics to explore:

1. How does the expected value change if the radii and point values are adjusted?
2. How would the expected value change if the dart were thrown with a non-uniform probability distribution?
3. What is the expected point value for a target with four concentric circles instead of three?
4. How can we generalize the concept of expected value for any number of concentric regions?
5. How do we calculate the expected value if the point values are given as discrete probabilities for each region?

Tip: The expected value is essentially the average outcome when taking into account the likelihood of hitting each region and its associated point value.