A cylindrical tin can holding 2 gal.has its height equal to the diameter of its base.Another cylindrical tin can with the same capacity has its height equal to twice the diameter of its base. Find the ratio of the amount of tin required for making the two cans with covers..
Help me pls..
pi*D^2*L is the same for both.
Therefore D1^2*L1 = D2^2*L2
and D1^3 = 2D2^3
Therefore D2/D1 = 1.260
For the areas,
A1 = pi*D1^2/2 + pi*D1*L
= (3/2)pi*D1^2 since D1 = L1
A2 = pi*D2^2/2 + pi*D2*L2
= (5/2)pi*D2^2 since D2 = 2L2
A2/A1 = (5/3)*(D2/D1)^2
= (5/3)*1.5873 = 2.646
To find the ratio of the amount of tin required for making the two cans, we need to calculate the surface area of each can.
First, let's find the dimensions of each can:
For the first can:
- The height is equal to the diameter of the base.
- The capacity is given as 2 gallons.
For the second can:
- The height is equal to twice the diameter of the base.
- The capacity is also 2 gallons.
Let's first convert the capacity from gallons to cubic inches, as both surface area and dimensions are usually calculated in inches.
1 gallon is equal to 231 cubic inches.
For the first can:
Capacity = 2 gallons = 2 * 231 cubic inches = 462 cubic inches.
For the second can:
Capacity = 2 gallons = 2 * 231 cubic inches = 462 cubic inches.
Next, let's calculate the dimensions for each can:
For the first can:
- Height = Diameter of the base (let's call it h1).
- Volume of a cylinder = π * r^2 * h.
Given that capacity = π * r^2 * h1 = 462 cubic inches.
For the second can:
- Height = 2 * Diameter of the base (let's call it h2).
- Volume of a cylinder = π * r^2 * h.
Given that capacity = π * r^2 * h2 = 462 cubic inches.
Now, we have two equations:
1st can: π * r^2 * h1 = 462 -- Equation 1
2nd can: π * r^2 * h2 = 462 -- Equation 2
Since h2 = 2 * h1, we can substitute h2 in terms of h1 in Equation 2:
π * r^2 * (2 * h1) = 462
2 * (π * r^2 * h1) = 462
2 * 462 = 2 * (π * r^2 * h1)
924 = π * r^2 * h1 -- Equation 3
Now, let's equate Equations 1 and 3 to find the ratio of the tin required:
π * r^2 * h1 = 924 -- Equation 1
π * r^2 * h1 = 462 -- Equation 3
Dividing both sides by π * r^2:
h1 = 924 / (π * r^2)
h1 = 462 / (π * r^2)
Therefore, the ratio of the amount of tin required for making the two cans is 924:462, or 2:1.