Question

Use differentials to estimate the amount of tin (in cm

3
) in a closed tin can with diameter 10 cm and height 11 cm if the tin is 0.04 cm thick. (Round your answer to two decimal places.) cm
3

Answer 1

To estimate the volume of the tin in a cylinder tin can, calculate the surface area and multiply it by the tin thickness. The surface area for a can with a diameter of 10 cm and height of 11 cm is 210π cm². Multiplying by the tin thickness of 0.04 cm, the estimated volume of tin is 26.39 cm³.

Step-by-step explanation:

To estimate the amount of tin in a closed tin can using differentials, we start by considering the surface area of the can and then multiply it by the thickness of the tin. The surface area of a cylinder (tin can) includes the area of the two circular ends and the area of the rectangular side that wraps around the cylinder. The diameter provided is 10 cm, which means the radius (r) is half of that, i.e., 5 cm. The height (h) of the can is 11 cm, and the thickness (t) is 0.04 cm.

The total surface area (A) of the cylinder can be calculated using the formula: A = 2πr^2 + 2πrh, where π approximates to 3.14159. For the given dimensions, A = 2π(5)^2 + 2π(5)(11) = 100π + 110π = 210π cm².

Now, to estimate the volume of the tin material, we multiply the total surface area by the thickness: Volume of tin = A × t. Substituting the surface area calculation we get Volume of tin = 210π × 0.04 cm³.

Therefore, the estimated amount of tin in the tin can is approximately 8.4π cm³, which when calculated and rounded to two decimal places gives 26.39 cm³.

Answer 2

To estimate the amount of tin in a tin can, we use the formula for the volume of a cylindrical shell and substitute the measurements: a radius of 5 cm, a height of 11 cm, and tin thickness of 0.04 cm, yielding an estimated volume of approximately 13.82 cm3.

Step-by-step explanation:

To estimate the amount of tin in a closed tin can using differentials, we use the formula for the volume of a cylindrical shell, which is V = 2πrhδ, where r is the average radius, h is the height, and δ is the thickness of the tin.

For a can with a diameter of 10 cm, the radius is 5 cm. With a tin thickness (δ) of 0.04 cm and a height (h) of 11 cm, we substitute these values into our formula.

The estimated volume of tin is then V = 2π(5 cm)(11 cm)(0.04 cm), which simplifies to V = 2π(5)(11)(0.04). Calculating this, we get a volume of approximately 13.82 cm3 (rounded to two decimal places).