Answer:
The approximate capacity of the suitcase is 67.49 liters.
Step-by-step explanation:
We need to find the capacity of a suitcase with rounded corners and edges. This means the actual space inside is smaller than if it were a perfect box.
The Solution:
1. Calculate the “Outer” Box Volume: We start by pretending the suitcase is a perfect box and calculate its volume:
Volume = width x height x depth = 50cm x 60cm x 30cm = 90,000 cm³
2. Account for the Rounded Corners: Each corner is like 1/8th of a sphere. We calculate the volume of one corner sphere and multiply by 8:
Volume of one corner sphere ≈ 65.45 cm³
Total corner volume ≈ 523.6 cm³
3. Account for the Rounded Edges: Each edge is like half a cylinder. We calculate the volume for each edge length (50cm, 60cm, 30cm) and add them up:
Total volume of all edge cylinders ≈ 21991.2 cm³
4. Calculate Total Removed Volume: We add the volume of the corners and edges to find the total space “lost” due to rounding:
Total removed volume ≈ 523.6 cm³ + 21991.2 cm³ ≈ 22514.8 cm³
5.Calculate Final Suitcase Capacity: We subtract the removed volume from the “outer” box volume:
Capacity ≈ 90,000 cm³ - 22514.8 cm³ ≈ 67485.2 cm³
6. Convert to Liters: Finally, we convert cubic centimeters to liters:
Capacity ≈ 67485.2 cm³ / 1000 cm³/liter ≈ 67.49 liters
Therefore, the suitcase has an approximate capacity of 67.49 liters.
Step-by-step explanation:
we could do it in 2 main ways : either we calculate the volume of the full box and then remove the excess rounded parts. or we could calculate the smaller box (without the edges and corners) and then add the rounded parts.
I prefer the second option.
all rounded parts have a radius of 5 cm. that takes away 5 cm in every direction for the basic box.
so, the basic box is then
(60 - 2×5) × (50 - 2×5) × (30 - 2×5) =
= 50×40×20 = 40,000 cm³
the rounded parts are
4 times 1/4 long cylinder (the long edges): 1 long cylinder
4 times 1/4 middle cylinder (vertical edges): 1 middle cylinder
4 times 1/4 short cylinder (the short edges): 1 short cylinder
2×4 times 1/4 balls (the 8 corners): 2 balls
the volume of a cylinder is
base area (circle) × height
so, the long cylinder is
pi×5²×50 = pi×25×50 = 1250pi = 3,926.990817... cm³
the middle cylinder is
pi×5²×40 = pi×25×40 = 1000pi = 3,141.592654... cm³
the short cylinder is
pi×5²×20 = pi×25×20 = 500pi = 1,570.796327... cm³
the volume of a ball (sphere) is
(4/3)×pi×r³
so, our 2 balls have
2 × (4/3)×pi×5³ = 8/3 × pi×125 = (1000/3)pi =
= 1,047.197551... cm³
altogether,
the basic box + 3 cylinder + 2 balls =
49,686.57735... cm³
a cube of 10cm side length contains exactly 1 liter.
1 liter = 10cm×10cm×10cm = 1000 cm³
so, the capacity is
49,686.57735... / 1000 = 49.68657735... liters
≈ 50 liters
you did not give any requirement about how to round the result. so, I assumed to follow the text and round the result to full liters.