A cylindrical jar with height 8 inches and diameter 6 inches is filled to 75% capacity with juice. The juice is then poured into another cylindrical container with a 10 inch diameter and height of 4 inches.
a. to what percent of its capacity is the second container filled with juice?
b. a third cylindrical container is such that the entire amount of juice only takes up 27% of the capacity of the container. As a result, what would one pair of possible dimensions be for the diameter and height of this third container?
a) Find the volume of the first cylinder using V=pi(h)(r)^2 and then multiply that by .75. This is the volume of the juice. Then, you divide that amount by the volume of the second container.
b) Divide the amount of the juice by .27 to get the volume of the new container and then set that equal to the volume equation and plug in values.
To solve these problems, we need to use the formula for the volume of a cylinder: V = π * r^2 * h, where V is the volume, r is the radius, and h is the height of the cylinder.
a. First, let's calculate the volume of the juice in the first container. The diameter of the first container is 6 inches, so its radius (r1) is half of the diameter, which is 6 / 2 = 3 inches. The height (h1) of the first container is 8 inches.
The volume of the juice in the first container (V1) is given as 75% of the capacity, so we can calculate it as 0.75 times the volume of the container:
V1 = 0.75 * π * (r1^2) * h1
Now, let's calculate the volume of the second container. The diameter of the second container is 10 inches, so its radius (r2) is 10 / 2 = 5 inches. The height (h2) of the second container is 4 inches.
The volume of the juice in the second container (V2) is the same as the volume of the juice in the first container:
V2 = V1 = 0.75 * π * (r1^2) * h1
To find the percentage of the second container filled with juice, we need to divide the volume of the juice in the second container (V2) by the total capacity of the second container (V_total).
V_total = π * (r2^2) * h2
The percentage filled (P_filled) can be calculated as:
P_filled = (V2 / V_total) * 100
b. To find the dimensions of the third container, we need to determine the volume of the juice in the first container (V1) and find a pair of dimensions that will result in the volume of the juice being 27% of the capacity of the third container.
Let's assume the diameter of the third container is d3 and the height is h3.
The volume of the third container (V3) is given as 27% of the capacity, so we can calculate it as 0.27 times the total capacity of the container:
V3 = 0.27 * π * (r3^2) * h3
Since we know the volume of the juice in the first container (V1), we can equate it to the volume of the third container (V3) and solve for the dimensions:
V1 = V3
0.75 * π * (r1^2) * h1 = 0.27 * π * (r3^2) * h3
Simplifying the equation:
(r1^2) * h1 = (r3^2) * h3
We have two equations and two unknowns, so we can solve this system of equations to find the values of d3 and h3.