Granda's soup company packages soup that serves 4 in cylindrical cans having a base diameter of 8cm and a height of 10cm. It wants to introduce the soup in single serving cans. If the company keeps the height of the new can at 10cm, what should its new base diameter equal?
the new volume should be 1/4 the old volume
Since the height is the same, the radius should be 1/2 the old radius
You can see this, since
πhr^2/4 = πh(r/2)^2
so, the new diameter is 1/2 the old diameter, or 4 cm
To find the base diameter of the new single serving can, we can use the concept of similar figures.
The volume of a cylinder can be calculated using the formula:
V = πr²h
where V is the volume, r is the radius, and h is the height.
Given that the original can serves 4 and has a base diameter (which is twice the radius) of 8 cm, we can find the radius by dividing the diameter by 2:
radius = 8 cm / 2 = 4 cm
The volume of the original can is:
V_original = π(4 cm)²(10 cm)
V_original = 160π cm³
Now, let's consider the new can. Since the height of the can remains the same at 10 cm, we can keep it constant. However, we need to find the new base diameter.
Let's assume the new base diameter is d. Therefore, the new radius is d/2.
The volume of the new single serving can is:
V_new = π(d/2)²(10 cm)
V_new = 25π(d² cm³)
Since the company wants to introduce the soup in single serving cans, the new can should have a volume that serves 1 (instead of 4 like the original can). So we can set up the following equation:
V_new = V_original / 4
25π(d² cm³) = 160π cm³ / 4
25π(d²) = 40π
d² = 40/25
d² = 1.6
Taking the square root of both sides:
d = √1.6
Therefore, the new base diameter of the single serving can should be approximately 1.26 cm (rounded to two decimal places).