https://file.aiquickdraw.com/mathbot/file/1725765466751l3hb7vtg.jpg

It appears you've uploaded an image containing some mathematical problems. Let me process the information within it and provide solutions or explanations where necessary.

The image contains multiple problems, so I'll address them one by one. Here's a breakdown of the content:

### 1. The triangle problem:
It asks:
- (a) "Find *x*."
- (b) "Find the perimeter of the triangle."

We are given a right triangle with angles labeled as 45° and 45°, which implies it is an isosceles right triangle. The two legs of the triangle are labeled as \( x \), and the hypotenuse is labeled as \( 10 \).

#### (a) Finding *x*:
For an isosceles right triangle, we know that the legs are equal, and the Pythagorean theorem applies:
\[
x^2 + x^2 = 10^2
\]
\[
2x^2 = 100
\]
\[
x^2 = 50 \quad \Rightarrow \quad x = \sqrt{50} = 5\sqrt{2}
\]

Thus, \( x = 5\sqrt{2} \).

#### (b) Perimeter:
The perimeter \( P \) of the triangle is the sum of all its sides:
\[
P = x + x + 10 = 2x + 10 = 2(5\sqrt{2}) + 10 = 10\sqrt{2} + 10
\]

So, the perimeter is \( 10\sqrt{2} + 10 \).

### 2. The second problem asks:
- “*What force is being exerted?*"

This question is based on an image of a machine or an object, but without more context, it’s hard to interpret. If you provide additional details or specify the machine, I can assist further.

### 3. Another question asks:
- “According to the expedition, what filled the interior of this house?”

This seems to reference an image of an old building or a historical context. For this type of question, the answer would depend on the specific historical or expedition context, which isn't clear from the image alone. Providing more context would help here too.

### 4. The fourth question:
- “Determine the area of the top surface of a cylindrical can that has a radius of 3 inches.”

To calculate the area of the top surface of a cylinder, which is a circle, use the formula:
\[
A = \pi r^2
\]
where \( r = 3 \) inches.

\[
A = \pi (3^2) = 9\pi \, \text{square inches}.
\]
So, the area is \( 9\pi \, \text{in}^2 \), or approximately \( 28.27 \, \text{in}^2 \).

---

Would you like further details on any of these problems? Here are some additional questions you could explore:

1. How does the Pythagorean theorem apply in different triangle scenarios?
2. What are the properties of an isosceles right triangle, and how do they help in solving for side lengths?
3. Can you derive a formula to calculate the perimeter of any triangle?
4. How does the area of a circle change with increasing radius?
5. What are the key factors to consider when analyzing force in physics problems?

**Tip:** For any right triangle, you can always use trigonometric ratios (sine, cosine, and tangent) to find missing sides when angles are known!

Find x and the perimeter of a triangle. Also, calculate the area of the top surface of a cylindrical can and answer an expedition-related question.

Learn how to solve a triangle problem by finding the side length using the Pythagorean theorem and calculate the perimeter. Also, find the area of a cylindrical can's top surface. Suitable for Grades 9-11 students.

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