You want to construct a birdhouse whose cross-section has the shape of an isosceles triangle on top of a rectangle, as shown below.


Its base has width 39 cm; its wall are 27 cm high; and its roof extends 14 cm above its walls.

To plan your construction, you need to know the measures of the angles marked alpha is the angle from the tip of the triangle at the very top of the roof and ß is the angle between the triangle and square parts of the house below.


Your answer should be in radians, of course.

We are asked to calculate two angles, \(\alpha\) and \(\beta\), in a birdhouse cross-section composed of an isosceles triangle (the roof) above a rectangle (the walls). Let's break down the problem.

### Step 1: Understanding the dimensions

- The **width** of the rectangle and the base of the triangle is 39 cm.
- The **height** of the rectangle is 27 cm.
- The **height** of the isosceles triangle (roof) is 14 cm.

### Step 2: Analyzing \(\alpha\)

\(\alpha\) is the angle at the apex of the isosceles triangle (top of the roof). The total width of the base of the triangle is 39 cm, and the height of the triangle is 14 cm.

We can calculate the angle \(\alpha\) by focusing on one-half of the isosceles triangle. In this half, we have a right triangle with:
- Height = 14 cm (the height of the triangle).
- Base = \( \frac{39}{2} = 19.5 \, \text{cm} \) (half of the total width).

The tangent of half of \(\alpha\) (\(\frac{\alpha}{2}\)) is given by the ratio of the height to half the base:

\[
\tan\left(\frac{\alpha}{2}\right) = \frac{14}{19.5}
\]

Now, let's calculate \(\frac{\alpha}{2}\):

\[
\frac{\alpha}{2} = \tan^{-1}\left(\frac{14}{19.5}\right)
\]

Using a calculator to find the arctangent:

\[
\frac{\alpha}{2} \approx \tan^{-1}(0.7179) \approx 0.621 \, \text{radians}
\]

Thus, the full angle \(\alpha\) is:

\[
\alpha = 2 \times 0.621 = 1.242 \, \text{radians}
\]

### Step 3: Analyzing \(\beta\)

\(\beta\) is the angle between the sloping side of the triangle and the top edge of the rectangle (the triangle meets the rectangle at this point).

We can again use one-half of the isosceles triangle, where the sloping side is the hypotenuse of a right triangle. The base is 19.5 cm, and the height is 14 cm.

We can calculate \(\beta\) using the tangent of the angle:

\[
\tan(\beta) = \frac{14}{19.5}
\]

Thus,

\[
\beta = \tan^{-1}\left(\frac{14}{19.5}\right)
\]

We already calculated this for half of \(\alpha\), so:

\[
\beta = 0.621 \, \text{radians}
\]

### Final Answer:

- \(\alpha \approx 1.242\) radians
- \(\beta \approx 0.621\) radians

Would you like further details or have any questions?

### Related Questions:
1. How would you calculate these angles if the roof height was increased to 20 cm?
2. What would be the angles if the base of the triangle was reduced to 30 cm?
3. How do the angles change if the walls are only 20 cm tall?
4. How would the problem change if the birdhouse had an equilateral triangle instead of an isosceles one?
5. What would the angles be if the triangle height doubled while keeping the base the same?

### Tip:
When dealing with right triangles, using the inverse trigonometric functions (arcsin, arccos, arctan) is a powerful way to calculate unknown angles if the sides are known.

You want to construct a birdhouse whose cross-section has the shape of an isosceles triangle on top of a rectangle. Its base has width 39 cm; its wall is 27 cm high; and its roof extends 14 cm above its walls. To plan your construction, you need to know the measures of the angles alpha (the angle at the top of the roof) and beta (the angle between the triangle and the rectangle). Your answer should be in radians.

Calculate the angles of a birdhouse cross-section with an isosceles triangle roof using trigonometry. Given the base width of 39 cm and height of 14 cm for the triangle, and a wall height of 27 cm, learn how to compute the angles alpha and beta in radians. This problem is suitable for high school students in Grades 10-12.

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