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.
If the scale factor is r, then
the area grows by r^2
the volume grows by r^3
So, if the area scales by 20/45, the volume scales by (20/45)^(3/2).
The smaller cylinder has cross-section area of .945*(20/45)^(3/2) = 0.28 cm^3
If x cm is transferred to the smaller container, then
13-x = (45/20)x
x = 4
check:
13-4 = 9
45*4 = 180 transferred out
20*9 = 180 transferred in
The depth in both containers is 9 cm
yes
What about section c of the question
To solve this problem, we can use the concept of similar figures and the proportionality of their volumes.
(a) To calculate the capacity of the smaller container, we need to find its volume. We know that the internal cross-sectional area of the smaller container is 20 cm^2. We can assume that the height of the smaller container is h.
The formula for the volume of a cylinder is V = A * h, where V is the volume, A is the cross-sectional area, and h is the height.
For the smaller container:
V1 = A1 * h1
V1 = 20 cm^2 * h
We need to find the capacity of the smaller container in liters. Since 1 liter is equal to 1000 cubic centimeters (cm^3), we can convert the volume from cm^3 to liters using the conversion factor.
V1 (in liters) = V1 (in cm^3) / 1000
Now, let's determine the volume of the smaller container in terms of liters.
V1 (in liters) = (20 cm^2 * h) / 1000
V1 (in liters) = 0.02 * h
To calculate the capacity of the smaller container, we need to find the value of h for this particular problem further.
(b) To calculate the depths of juice in each container, we need to apply the concept of similar figures and the proportionality of their volumes.
The larger container has an internal cross-sectional area of 45 cm^2. Assume the height of the larger container is H.
For the larger container:
V2 = A2 * H
V2 = 45 cm^2 * H
Initially, the larger container is filled with juice to a height of 13 cm. Therefore, the remaining height of the larger container, where juice is drawn from, can be calculated as follows:
H (remaining) = H - 13 cm
The volume of the juice drawn from the larger container is equal to the difference between the original volume and the remaining volume:
V_juice = V2 (initial) - V2 (remaining)
V_juice = 45 cm^2 * H - 45 cm^2 * (H - 13 cm)
To determine the heights of the juice in each container after they have been equalized, we can set up a proportion using the volumes of the containers.
Initially, the larger container has a volume V2 (initially), which is equal to the height of the juice in the smaller container (h) times the cross-sectional area of the smaller container (20 cm^2).
V2 (initially) = A1 * h
After the juice is transferred, both containers have a volume V_final. Therefore, we can set up the proportion using the volumes:
V2 (initially) / V_juice = (h / 20 cm^2) / (H_2 / 45 cm^2)
Simplifying this proportion will allow us to solve for the height of the juice in each container:
V2 (initially) / V_juice = 20 cm^2 / (H_2 / 45 cm^2)
Substituting the values we know,
0.945 L / V_juice = 20 cm^2 / (H - 13 cm / 45 cm^2)
Solve this equation for H, the height of the juice in the larger container.
Using the values of H and h, which are the heights of the juice in the larger and smaller containers, respectively, you can calculate the depths of juice in each container by subtracting the heights from the total heights of the containers.