Final answer:
The diameter of the circle whose area is equal to the sum of the areas of two circles with diameters of 20 cm and 48 cm is 52 cm. This is found by calculating the individual areas of the smaller circles, summing them, and then determining the radius and diameter of the circle that would have that total area.
Step-by-step explanation:
We are tasked with finding the diameter of a circle whose area is equal to the sum of the areas of two smaller circles with diameters of 20 cm and 48 cm, respectively. First, we find the area of each of the two smaller circles using the formula for the area of a circle, A = πr^2, where r is the radius of the circle.
For the first circle with a diameter of 20 cm, the radius is 10 cm. For the second circle with a diameter of 48 cm, the radius is 24 cm. The areas of the two circles are:
- Area of the first circle: A1 = π × (10 cm)^2 = 100π cm^2
- Area of the second circle: A2 = π × (24 cm)^2 = 576π cm^2
The sum of the areas of the two circles is A1 + A2 = 100π + 576π = 676π cm^2. This sum represents the area of the larger circle we are trying to find the diameter for.
To find the radius of the larger circle, we set the sum of the areas equal to the area formula of a circle and solve for the radius:
A = πr^2
676π = πr^2
r^2 = 676
r = √676
r = 26 cm
Now that we have found the radius, we simply double it to find the diameter:
D = 2r = 2 × 26 cm = 52 cm
Therefore, the diameter of the circle whose area is equal to the sum of the areas of two circles with diameters of 20 cm and 48 cm is 52 cm.