The volumes of two cylindrical cans of the same shape vary directly as the cubes of their radii. If a can with a six-inch radius holds 1½ pints, how many gallons will a similar can with a 24-inch radius hold?
24/6 = 4
4^3 = 64
64 * 1.5 = 96 pints
there are 8 pints per gallon so
96/8 = 12 gallons
change pints to gallons.
1 1/2 pints = 3/16 gallon
3/16 x
_____ = ____ create proportion.
6^3 24^3
6^3 * x = 24^3 * 3/16
216 * x = 13824 * 3/16 simplify
216 * x = 2592 divide both sides by 216
x = 12 gallons
To solve this problem, we can use the formula for the volume of a cylinder: V = πr^2h, where V is the volume, r is the radius, and h is the height.
Since the volumes vary directly as the cubes of the radii, we can write the proportion:
V1 / V2 = (r1^3) / (r2^3)
Given that the radius of the first can is 6 inches and it holds 1½ pints, we can write:
V1 = 1.5 pints
r1 = 6 inches
Let's solve for V2, the volume of the can with a radius of 24 inches in gallons.
Step 1: Convert the volume of the first can to gallons.
1 pint = 1/8 gallon
1.5 pints = 1.5 * (1/8) = 0.1875 gallons
V1 = 0.1875 gallons
Step 2: Write the proportion and substitute the known values.
V1 / V2 = (r1^3) / (r2^3)
0.1875 / V2 = (6^3) / (24^3)
Step 3: Simplify the equation.
0.1875 / V2 = (6/24)^3
0.1875 / V2 = (1/4)^3
0.1875 / V2 = 1/64
Step 4: Cross-multiply and solve for V2.
V2 = (0.1875 * 64) / 1
V2 = 12 gallons
Therefore, a similar can with a 24-inch radius will hold 12 gallons.
To solve this problem, we can use the concept of direct variation. Direct variation states that two quantities are directly proportional to each other if their ratio is constant.
1. First, we need to find the constant of variation. We are given that the volumes of the two cylindrical cans vary directly as the cubes of their radii. So, we can set up the following proportion:
(Volume of can 1) / (Volume of can 2) = (radius of can 1)^3 / (radius of can 2)^3
Let's label the given values:
Volume of can 1 = 1½ pints = 1.5 pints
Radius of can 1 = 6 inches
Radius of can 2 = 24 inches
Plugging in the values, we get:
1.5 / (Volume of can 2) = (6^3) / (24^3)
2. Next, we can solve for the volume of can 2. Cross-multiplying the equation from step 1:
1.5 * (24^3) = (6^3) * (Volume of can 2)
Simplifying:
1.5 * (13824) = (216) * (Volume of can 2)
Volume of can 2 = (1.5 * 13824) / 216
3. Now, we can convert the volume from pints to gallons. Since there are 8 pints in a gallon, we divide the volume of can 2 by 8:
Volume of can 2 (in gallons) = (1.5 * 13824) / (216 * 8)
Simplifying further:
Volume of can 2 (in gallons) = 6.125 gallons
Therefore, a similar can with a 24-inch radius will hold 6.125 gallons.