Solve and explain your solution to the following problem:    Assume that when a pen draws any continuous curve of 10 feet in length, it will use 1% of its ink to do so. Consider an isosceles right triangle that could be drawn by this pen without running out of ink. What is the largest integer length one of the two equal sides of this triangle could have?

To solve this problem, we need to relate the length of the sides of the isosceles right triangle to the total length of the curve (or perimeter) that the pen can draw without running out of ink.

### Step 1: Understanding the problem
We are dealing with an isosceles right triangle, meaning two of its sides are equal in length. Let's denote the length of each of these equal sides as \( s \). The third side is the hypotenuse, which, for an isosceles right triangle, can be calculated using the Pythagorean theorem:
\[
\text{Hypotenuse} = s\sqrt{2}
\]

### Step 2: Formula for the total length of the triangle's perimeter
The perimeter of the triangle is the sum of the lengths of all three sides:
\[
\text{Perimeter} = s + s + s\sqrt{2} = 2s + s\sqrt{2}
\]
This is the total length of the curve drawn by the pen.

### Step 3: Determine the total length the pen can draw
The problem states that the pen uses 1% of its ink for every 10 feet of curve drawn. This means the pen has enough ink to draw 1000 feet of curve (since 1% of 1000 feet equals 10 feet). So, the perimeter of the triangle must be less than or equal to 1000 feet:
\[
2s + s\sqrt{2} \leq 1000
\]

### Step 4: Solve the inequality
To solve this inequality, first factor out \( s \):
\[
s(2 + \sqrt{2}) \leq 1000
\]
Now, calculate the value of \( 2 + \sqrt{2} \):
\[
2 + \sqrt{2} \approx 2 + 1.414 = 3.414
\]
Thus, the inequality becomes:
\[
s(3.414) \leq 1000
\]
Solve for \( s \):
\[
s \leq \frac{1000}{3.414} \approx 292.86
\]

### Step 5: Find the largest integer value for \( s \)
The largest integer value for \( s \) that satisfies this inequality is \( s = 292 \).

### Final Answer:
The largest integer length of one of the two equal sides of the isosceles right triangle that can be drawn without running out of ink is **292 feet**.

---

Would you like further clarification on any part of the solution?

Here are some related questions you might find interesting:
1. How would the solution change if the triangle were equilateral?
2. What would happen if the pen used 2% of its ink for every 10 feet?
3. How do you calculate the area of this isosceles triangle with sides of 292 feet?
4. What if the sides of the triangle were in a different ratio, such as 3:4:5?
5. How would the solution change if the perimeter limit were reduced to 500 feet?

**Tip:** When working with geometric shapes and ink-related problems, always consider the relationship between side lengths and the total perimeter or area.

Solve and explain your solution to the following problem: Assume that when a pen draws any continuous curve of 10 feet in length, it will use 1% of its ink to do so. Consider an isosceles right triangle that could be drawn by this pen without running out of ink. What is the largest integer length one of the two equal sides of this triangle could have?

Solve a geometric problem involving an isosceles right triangle and ink usage limits. Learn how to use the Pythagorean theorem and perimeter calculations to find the largest possible side length under a 1000-foot curve limit. Suitable for students in Grades 9-11.

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