Two cylindrical containers are similar. The larger one has internal cross- section area of 45cm^2 and can hold 0.945 litres of liquid when full. The smaller container has internal cross- section area of 20cm^2.
(a) Calculate the capacity of the smaller container.
(b) The larger container is filled with juice to a height of 13 cm. Juice is then drawn from it and emptied into the smaller container until the depths of the juice in both containers are equal. Calculate the depths of juice in each container.
(a) Ah, the smaller container. Let's crunch the numbers, shall we? Since the larger container has a cross-section area of 45 cm^2 and can hold 0.945 litres of liquid when full, we can find its height by dividing the volume by the area: 0.945 L / 45 cm^2 = 0.021 L/cm^2.
Now, let's use this to find the capacity of the smaller container. We know its cross-section area is 20 cm^2. So, let's multiply the area by the height we just found: 20 cm^2 * 0.021 L/cm^2 = 0.42 L.
Therefore, the capacity of the smaller container is 0.42 litres. Ta-da!
(b) Ah, a balancing act! Let's find the depths of the juice in each container, shall we? Since the larger container is filled to a height of 13 cm, we need to find the height at which the juice levels will be equal.
First, let's calculate the volume of juice in the larger container at a height of 13 cm. We'll use the previous calculation to find the cross-section area of the larger container (45 cm^2) and multiply it by 13 cm: 45 cm^2 * 13 cm = 585 cm^3.
Now, we know that the smaller container has a capacity of 0.42 L, which is equal to 420 cm^3. So, we need to find the height at which the volume of juice in the larger container is equal to 420 cm^3.
To do this, we'll set up a proportion: 585 cm^3 / 13 cm = 420 cm^3 / x cm.
Cross-multiplying, we find: 585 cm^3 * x cm = 13 cm * 420 cm^3.
Dividing both sides by 585 cm^3, we find: x cm = (13 cm * 420 cm^3) / 585 cm^3.
Simplifying, we get: x cm = 9.42 cm.
Therefore, the depth of the juice in the larger container will be 9.42 cm, and the depth in the smaller container will also be 9.42 cm. Voila!
To solve this problem, we need to understand the concept of similar shapes. In the context of cylinders, when two cylinders are similar, it means that they have the same shape, but their sizes are different. The dimensions of similar shapes are scaled up or down by the same ratio.
Let's address each part of the problem:
(a) To calculate the capacity of the smaller container, we need to find the volume of the smaller cylinder. We know that the larger container can hold 0.945 liters of liquid, so let's convert that into cm^3 to match the units of the cross-sectional area.
1 liter = 1000 cm^3
0.945 liters = 0.945 * 1000 cm^3
0.945 liters = 945 cm^3
Now, we can use the formula for the volume of a cylinder to find the capacity of the smaller container:
V = A * h
Where V is the volume of the cylinder, A is the cross-sectional area, and h is the height of the cylinder.
The larger container has an internal cross-sectional area of 45 cm^2, and we need to find the height of the cylinder. Let's call it h_l.
V_larger = 45 cm^2 * h_l
Similarly, the smaller container has an internal cross-sectional area of 20 cm^2, and we need to find the height of the cylinder. Let's call it h_smaller.
V_smaller = 20 cm^2 * h_smaller
Since both cylinders are similar, the ratio of their cross-sectional areas is equal to the square of the ratio of their heights:
(A_smaller / A_larger) = (h_smaller / h_larger)^2
Using the given values:
(20 cm^2 / 45 cm^2) = (h_smaller / h_larger)^2
To find the value of h_smaller, we need to rearrange the equation and solve for h_smaller:
(h_smaller / h_larger)^2 = (20 cm^2 / 45 cm^2)
(h_smaller / h_larger) = sqrt(20 cm^2 / 45 cm^2)
(h_smaller / h_larger) = sqrt(4/9)
Since the cylinders have a proportional relationship, we can simplify the square root:
(h_smaller / h_larger) = 2/3
Now, we can solve for h_smaller:
h_smaller = (2/3) * h_larger
We know that the larger container is filled to a height of 13 cm (h_larger = 13 cm), so we can substitute that into the equation:
h_smaller = (2/3) * 13 cm
h_smaller = 26/3 cm
h_smaller ≈ 8.67 cm
Therefore, the capacity of the smaller container is approximately 945 cm^3, and the depth of the juice in the smaller container is approximately 8.67 cm.
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