If a cylinder’s height is four times its radius, if the cyclinder’s volume (ignoring the metals’ thickness) is 1000 cubic centimeters, and if the radius and hence the height could be in error plus or minus .2325%, determine the relative percentage error in the cyclinder’s volume.
To determine the relative percentage error in the cylinder's volume, we need to calculate the actual volume and then find the difference between the actual volume and the given volume, taking into account the permissible error.
Let's start by assigning variables:
Let:
- r = radius of the cylinder
- h = height of the cylinder
Given:
- Height (h) = 4 * Radius (r)
- Volume (V) = 1000 cubic centimeters
- Permissible error = +/- 0.2325%
The volume of a cylinder is calculated using the formula: V = π * r^2 * h
Let's substitute the given values and solve for the unknowns.
Since the height (h) is 4 times the radius (r), we can substitute h = 4r into the volume formula:
V = π * r^2 * (4r)
V = 4π * r^3
We know that V = 1000 cubic centimeters, so we can equate the two and solve for r:
1000 = 4π * r^3
Dividing both sides by 4π:
250/π = r^3
Taking the cube root:
r = ∛(250/π)
Now that we have the exact value of r, we can find the actual volume (V_actual) using the formula:
V_actual = π * r^2 * h
Substituting the values:
V_actual = π * (∛(250/π))^2 * 4(∛(250/π))
V_actual = 4 * π * (∛(250/π))^3
Similarly, we can obtain the approximate volume (V_approx) considering the permissible error:
V_approx = (1 ± 0.002325) * V
To calculate the relative percentage error, we find the difference between V_actual and V_approx, divide it by V_actual, and then multiply by 100 to get the percentage:
Relative Percentage Error = |(V_actual - V_approx) / V_actual| * 100
Substituting the actual and approximate volumes into the formula, we can calculate the relative percentage error.