Find the radius of the arch. The stove is equipped with a casing around the firebox door. The upper part of the casing is made in the form of an arch, welded to the front wall of the stove along the arc of a circle with its center in the middle of the lower part of the casing (Fig. 2). To install the stove, the owner needed to know the radius of the arch's rounding. The dimensions of the casing are shown in centimeters in the figure. Find the radius of the arch's rounding in centimeters.

Solution:

The problem asks us to find the radius of the arch. From Figure 2, we can see that the arch is part of a circle. The dimensions provided are:

• The width of the casing is 60 cm.


• The height of the straight part of the casing is 40 cm.


• The arch is centered on the midpoint of the bottom width.

Let R be the radius of the arch.

Consider the right-angled triangle formed by:

• The radius R (hypotenuse).


• The horizontal distance from the center of the arch to the edge of the arch's base, which is half the width of the casing: \( \frac{60}{2} = 30 \) cm.


• The vertical distance from the center of the arch to the top of the straight part of the casing. Since the total height of the straight part is 40 cm, and the arch rises from this line, the center of the arch must be at a height of R above the top of the straight part. Therefore, this vertical distance is \( R - 40 \) cm.

Using the Pythagorean theorem:

\( \text{horizontal distance}^2 + \text{vertical distance}^2 = R^2 \)

\[ 30^2 + (R - 40)^2 = R^2 \]

\[ 900 + R^2 - 80R + 1600 = R^2 \]

Subtract \( R^2 \) from both sides:

\[ 900 - 80R + 1600 = 0 \]

\[ 2500 - 80R = 0 \]

\[ 80R = 2500 \]

\[ R = \frac{2500}{80} \]

\[ R = \frac{250}{8} \]

\[ R = \frac{125}{4} \]

\[ R = 31.25 \]

The radius of the arch is 31.25 cm.

Ответ: 31.25 см.