Benjamin opens a new two-liters tub of ice cream and has three spherical scoops for dessert. Given that each scoop has a diameter of 40 millimeters, how many more whole scoops can he get from the tub?
Let’s begin by breaking down the information given in the question. We’re told that this tub has a total capacity of two liters, so that’s the total volume of ice cream that Benjamin starts off with. We also know that Benjamin eats three spherical scoops of ice cream, each of which have a diameter of 40 millimeters. We can use the formula for the volume of a sphere to work out the volume of ice cream that Benjamin has eaten. We recall that the volume of a sphere is given by four-thirds 𝜋𝑟 cubed, where 𝑟 is the radius of the sphere. The radius is, of course, half the diameter. So, if these spheres have a diameter of 40 millimeters, then they have a radius of 20 millimeters.
We can also choose to work in centimeters rather than millimeters. We know that there are 10 millimeters in each centimeter, so 20 millimeters is equivalent to two centimeters. The volume of each scoop of ice cream is therefore four-thirds multiplied by 𝜋 multiplied by two cubed. Two cubed is eight, and eight multiplied by four is 32, so we can simplify this to 32 over three 𝜋. The volume of all three scoops of ice cream will therefore be three times 32 over three times 𝜋. And as the threes cancel out, we’re left with simply 32𝜋. And we’ll keep this value in its exact form for now.
Now, as the radius that we were using in this formula was given in centimeters, the units for this volume will be cubic centimeters. We now see that we have a mixture of units given in this question. The capacity of the tub was given in liters, whereas the volume of the ice cream that Benjamin’s eaten, which we’ve just calculated, is given in cubic centimeters. So, we need to know how to convert between these two units. Well, a key conversion that we should know is that one milliliter is equivalent to one cubic centimeter. So, we can use this to determine the capacity of the tub not in liters, but in cubic centimeters.
Two liters is of course 2000 milliliters as there are 1000 milliliters in each liter. And using this conversion we’ve just written, this will be equivalent to 2000 cubic centimeters. Now that we have common units for the total volume of ice cream in the tub and the volume of ice cream Benjamin’s already eaten, we can continue with the problem. The remaining volume of ice cream in the tub will be 2000 minus 32𝜋. That’s the starting volume minus the volume Benjamin’s already eaten. We’re asked how many more whole scoops of ice cream Benjamin can get from the tub. So, we need to know how many whole scoops of ice cream this volume is equivalent to.
So, we take the total volume of remaining ice cream, that’s this value here, and we divide it by the volume of each scoop of ice cream, that’s this value here. So we have 2000 minus 32𝜋 over 32 over three 𝜋. Now, we could look at simplifying this value by multiplying both the numerator and denominator by three. And if we did, we’d have 6000 minus 96𝜋 all over 32𝜋. But it’s reasonable to assume that we have access to a calculator in this question. So, we could type either of these calculations into our calculators.
Doing so gives 56.6831 continuing. Now, the question asked us how many more whole scoops Benjamin can get from the tub. So, even though this answer is closer to 57 than it is to 56, we don’t have enough ice cream left to get that 57th scoop. We therefore need to round our answer down. So, we find that Benjamin can get 56 more whole scoops of ice cream from the tub. He does have a little bit of ice cream left over, but it isn’t enough to make a full scoop with the same diameter as the others.