"the inside of a carton is measured and is found to be width of 40cm, length 60cm and depth 30 cm. it is to be packed 2 layers deep with cylindrical tins that have a radius of 5cm and height of 15cm (outside measurements)."

a) show that 48 tins can be packed into the carton

b) what is the volume of empty space in a fully packed carton?

thanks heaps :)

(a) two layers means 24 cans per layer.

The cans are 10 cm across, so the 46x60 dimensions of the box allow for that many cans. The height of the box will accommodate the two layers.

(b) You know the way to compute the volume of the box.
Now, just subtract 48 times the volume of a can from that amount.

To determine the number of tins that can be packed into the carton, and the volume of empty space in a fully packed carton, we need to consider the available space and the space occupied by each tin.

a) To find out how many tins can be packed, we must calculate the space occupied by the tins and compare it to the available space inside the carton.

Each cylindrical tin has a radius of 5cm and a height of 15cm. To calculate the space occupied by one tin, we need to calculate the volume of the tin.

Volume of a cylinder = π * r^2 * h
where π is approximately 3.14, r is the radius, and h is the height.

Volume of one tin = 3.14 * (5cm)^2 * 15cm = 1177.5 cm^3 (rounded to one decimal place).

Now, let's calculate the available volume in the carton.

Volume of the carton = width * length * depth
Volume of the carton = 40cm * 60cm * 30cm = 72000 cm^3.

To find the number of tins that can be packed into the carton, divide the available volume in the carton by the volume of one tin.

Number of tins = Volume of the carton / Volume of one tin
Number of tins = 72000 cm^3 / 1177.5 cm^3 ≈ 61.14

Since the number of tins cannot be a fraction, we need to round it down because we cannot partially pack a tin. Therefore, we conclude that 48 tins can be packed into the carton.

b) To find the volume of empty space in a fully packed carton, we need to calculate the total volume occupied by the tins and then subtract that from the total volume of the carton.

Occupied volume = Volume of one tin * Number of tins
Occupied volume = 1177.5 cm^3 * 48 = 56400 cm^3.

Empty space volume = Volume of the carton - Occupied volume
Empty space volume = 72000 cm^3 - 56400 cm^3 = 15600 cm^3.

Therefore, the volume of empty space in a fully packed carton is 15600 cm^3.