The problem asks us to calculate the area of a door. The door is composed of a rectangle and a semi-circle on top of it. The height of the rectangular part is 2 m and the total height including the semi-circle is 3.2 m. Thus, the radius of the semi-circle is $3.2 - 2 = 1.2$ m. We also need to find the area of the door on a plan with a scale of $1/50$.

1. Problem Description

The problem asks us to calculate the area of a door. The door is composed of a rectangle and a semi-circle on top of it. The height of the rectangular part is 2 m and the total height including the semi-circle is 3.2 m. Thus, the radius of the semi-circle is $3.2 - 2 = 1.2$ m. We also need to find the area of the door on a plan with a scale of $1/50$.

2. Solution Steps

a) Calculate the area of the door.

The door consists of a rectangle and a semicircle.

The height of the rectangle is 2 m. The width of the rectangle is equal to the diameter of the semicircle, which is $2 \times 1.2 = 2.4$ m.

The area of the rectangle is:

$A_{rectangle} = height \times width = 2 \times 2.4 = 4.8 m^2$

The radius of the semi-circle is $r = 1.2$ m.

The area of a full circle is given by $A_{circle} = \pi r^2$. The area of a semi-circle is half of that:

$A_{semicircle} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (1.2)^2 = \frac{1}{2} \pi (1.44) = 0.72 \pi \approx 0.72 \times 3.1416 \approx 2.261952 m^2$

The total area of the door is:

$A_{door} = A_{rectangle} + A_{semicircle} = 4.8 + 0.72\pi \approx 4.8 + 2.261952 = 7.061952 m^2$

Rounding to two decimal places, the area of the door is approximately $7.06 m^2$.

b) Calculate the area of the door on a plan with a scale of $1/50$.

The scale factor for the lengths is $1/50$.

The scale factor for the areas is $(1/50)^2 = 1/2500$.

The area on the plan is:

$A_{plan} = A_{door} \times (1/50)^2 = 7.061952 / 2500 \approx 0.00282478 m^2$

Since $1 m^2 = (100 cm)^2 = 10000 cm^2$, we have

$A_{plan} = 0.00282478 \times 10000 = 28.2478 cm^2$.

Rounding to two decimal places, the area is $28.25 cm^2$.

3. Final Answer

a) The area of the door is approximately $7.06 m^2$.

b) The area of the door on the plan is approximately $28.25 cm^2$.