Use differentials to estimate the amount of tin in a closed tin can with diameter 6 cm and height 14 cm if the tin is 0.046 cm thick.
To estimate the amount of tin in the closed tin can, we can consider the tin can as a solid cylinder with an inner radius and an outer radius.
Step 1: Find the inner radius of the tin can.
The diameter of the tin can is given as 6 cm, so the radius (r) of the tin can is half of that, which is 3 cm.
Step 2: Find the outer radius of the tin can.
The tin thickness is given as 0.046 cm. So, the outer radius (R) of the tin can can be calculated as R = r + thickness = 3 cm + 0.046 cm = 3.046 cm.
Step 3: Find the volume of the tin in the can.
To find the volume of the tin, we will subtract the volume of the inner region (without tin) from the volume of the outer region (with tin).
The volume formula for a cylinder is V = π * r^2 * h, where V is the volume, r is the radius, and h is the height.
Volume of the outer region: V_outer = π * R^2 * h
Volume of the inner region: V_inner = π * r^2 * h
So, the volume of the tin in the can is:
V_tin = V_outer - V_inner
= π * R^2 * h - π * r^2 * h
= π * (R^2 - r^2) * h
Step 4: Substitute the values and calculate the estimate.
Let's substitute the values into the formula.
V_tin = π * (3.046^2 - 3^2) * 14
≈ π * (9.277316 - 9) * 14
≈ π * 0.277316 * 14
≈ 12.19733π
Therefore, the estimate for the amount of tin in the closed tin can is approximately 12.19733π cubic cm.
To estimate the amount of tin in the closed tin can, we can use differentials and consider small changes in the dimensions of the can.
Let's start by calculating the volume of the entire tin can (including the tin) and then subtracting the volume of the empty space inside the can (assuming the tin is solid and filled the entire volume).
The volume of the entire can with the tin is given by the formula:
V = π * (r + t)^2 * h
Where:
V is the volume
π is a mathematical constant approximately equal to 3.14159
r is the radius of the tin can (which is half of the diameter)
t is the thickness of the tin
h is the height
Given:
diameter (D) = 6 cm
height (h) = 14 cm
tin thickness (t) = 0.046 cm
First, we calculate the radius (r) of the tin can:
r = D/2 = 6/2 = 3 cm
Next, substitute the given values into the volume formula:
V = π * (3 + 0.046)^2 * 14
Now, let's take the derivative with respect to the thickness of the tin (t) to find the change in volume with respect to a small change in the thickness:
dV = d(V)/dt * dt
Differentiating V with respect to t, we get:
dV = 2π * (3 + 0.046) * 14 * dt
Now, we can substitute the values into the equation to find the differential change:
dV = 2π * (3.046) * 14 * dt
Finally, we can estimate the change in volume (∆V) by approximating dt as the given thickness of the tin (0.046 cm):
∆V = dV * dt = 2π * (3.046) * 14 * 0.046
Simplifying the expression will give us the estimated change in volume (∆V), which is approximately equal to the amount of tin in the can.