We are given an isosceles triangle, which represents the front of an A-frame cottage. The height is 23 feet, and the angle of elevation of the roof is 65°. We are asked to find the width of the base of this triangle.

Step-by-step approach:

1. Understand the triangle: Since it is isosceles, the height of the triangle divides it into two right triangles. Each right triangle has:

   • One leg (the height) equal to 23 feet,
   • An angle of 65° (angle of elevation),
   • The other leg (half of the base) that we need to calculate.

2. Use trigonometry: Specifically, we can use the tangent function, which relates the angle, the opposite side (the height), and the adjacent side (half of the base): $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ Here, the opposite side is the height of the triangle (23 feet), and the adjacent side is half the base of the triangle ($\frac{w}{2}$).

3. Solve for half the base: $\tan(65°) = \frac{23}{\frac{w}{2}}$ Rearrange this to solve for $\frac{w}{2}$: $\frac{w}{2} = \frac{23}{\tan(65°)}$ Now, solve for the full base width $w$: $w = 2 \times \frac{23}{\tan(65°)}$

Let me calculate this.The width of the cottage at its base is approximately $21.45$ feet.

Would you like a detailed explanation or have any other questions? Here are some related problems to consider:

1. What would the base width be if the angle of elevation were 60° instead of 65°?
2. How would the height change if the base remained the same but the angle of elevation increased to 70°?
3. Can you calculate the area of the front face of the cottage based on the dimensions found?
4. How does the width of the base change as the height of the cottage increases?
5. What is the length of each slanting side of the roof (hypotenuse)?

Tip: Trigonometry is a powerful tool for solving problems involving right triangles, particularly when you know angles and side lengths.