A balloon pump is made from a cylinder with an internal diameter of 6 cm and a height of 20 cm, as shown - Junior Cycle Mathematics - Question 5 - 2022

The volume of a sphere is calculated using the formula:

V = rac{4}{3} ext{π}r^3

where r is the radius. Given that the radius of Darragh’s balloon is 15 cm:

V = rac{4}{3} ext{π} (15)^3

Calculating this, we have:

V = rac{4}{3} ext{π} (3375) = 4500 ext{π} ext{ cm}^3

Thus, the volume of Darragh's balloon when it is fully inflated is 4500π cm³.

Since each pump fills the cylinder with a volume of 180π cm³, we can determine the time it takes to inflate the balloon:

ext{Number of pumps} = rac{ ext{volume of balloon}}{ ext{volume per pump}} = rac{4500 ext{π}}{180 ext{π}} = 25

As Darragh pumps the pump once every second, it will take him 25 seconds to fully inflate his balloon.

Gustav's balloon is fully inflated after 50 seconds. Hence, the volume of air pumped into the balloon is:

$V = 50 imes 180 ext{π} = 9000 ext{π} ext{ cm}^3$

Using the volume formula for a sphere:

V = rac{4}{3} ext{π}r^3

Setting this equal gives:

9000 ext{π} = rac{4}{3} ext{π} r^3

Dividing both sides by π:

9000 = rac{4}{3} r^3

Multiplying by 3 gives:

$27000 = 4r^3$

Now, dividing by 4:

r^3 = rac{27000}{4} = 6750

Taking the cube root:

oot{3}{6750} ext{ cm} $$ Calculating: $$ r ext{ (approx)} ightarrow 18.9 ext{ cm} $$ Thus, the radius of Gustav's balloon when fully inflated is approximately 18.9 cm, correct to 1 decimal place.