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.

what is the answer

To solve this problem, we need to find the ratio of the amount of tin required for making the two cans. Let's start by calculating the dimensions of each can.

1. Let's find the dimensions of the first can:
- We know that the can holds 2 gallons, which is equivalent to 7.57 liters (1 gallon ≈ 3.785 liters).
- The volume of a cylinder is given by πr²h, where r is the radius of the base and h is the height.
- Since the height is equal to the diameter of the base, we can say that h = 2r.
- The volume of the first can is 7.57 liters, so we have: πr²(2r) = 7.57
- Simplifying the equation, we get: 2πr³ = 7.57
- Dividing both sides by 2π, we find: r³ ≈ 7.57 / (2π) ≈ 1.206
- Taking the cube root of both sides, we find: r ≈ 1.071 cm (rounded to three decimal places)
- Since h = 2r, we have: h ≈ 2 * 1.071 ≈ 2.142 cm

2. Now, let's find the dimensions of the second can:
- We know that the can holds 2 gallons, which is equivalent to 7.57 liters.
- The height of the second can is equal to twice the diameter of its base.
- Let's assume the diameter of the base as d. Then the height is 2d.
- The volume of the second can is given by π(d/2)²(2d) = π(d²)(2d).
- Since the volume is 7.57 liters, we have: π(d²)(2d) = 7.57
- Dividing both sides by 2π, we get: d³ ≈ 7.57 / (2π) ≈ 1.206
- Taking the cube root of both sides, we find: d ≈ 1.071 cm (rounded to three decimal places)
- Since the height is 2d, we have: h ≈ 2 * 1.071 ≈ 2.142 cm

3. Now, let's find the amount of tin required for making the two cans.
- The surface area of a cylinder is given by 2πrh + πr² (area of the bases).
- For the first can, the surface area would be: 2π(1.071)(2.142) + π(1.071)².
- For the second can, the surface area would be: 2π(1.071)(2.142) + π(1.071)².
- Calculating the values, we find the surface area for both cans.

4. Finally, let's find the ratio of the tin required:
- Divide the surface area of the second can by the surface area of the first can.
- The ratio would give us the relative amount of tin required for making the two cans with covers.

Note: Since the calculations involve approximations and rounding, the final ratio may also be an approximation.