Question

Engine Design \(\quad\) A tractor engine has a steel component with a circular base modeled by the vector-valued function \(\mathbf{r}(t)=2 \cos t \mathbf{i}+2 \sin t \mathbf{j}\). Its height is given by \(z=1+y^{2}\) (All measurements of the component are given in centimeters.) (a) Find the lateral surface area of the component. (b) The component is in the form of a shell of thickness 0.2 centimeter. Use the result of part (a) to approximate the amount of steel used in its manufacture. (c) Draw a sketch of the component.

Short answer

The lateral surface area can be determined using calculus principles, and the amount of steel used is the lateral surface area times the thickness. The sketch of the component should indicate a circular base with a curved structure over it, revealing the function \(z=1+y^{2}\).

Step-by-step solution

Setting up the parameterized representation of the surface

First, set up a parameterized representation of the surface. The given vector-valued function is \(\mathbf{r}(t)=2 \cos t \mathbf{i}+2 \sin t \mathbf{j}\(. This is a circle with a radius of 2 units. Also given is the height function which is \)z=1+y^{2}\). Plug \(y = 2sin(t)\) into the equation for \(z\), so the parameterized equations are: \(x=2 \cos t\), \(y=2 \sin t\), \(z=1+(2 \sin t)^{2}\)

Compute the lateral surface area

The lateral surface area of the component is the double integral over the surface of the differential area element, which in this case is given by the magnitude of the cross product of the partial derivatives of the parametric representation with respect to \(t\) and \(z\). Use the determinant definition of the cross product and the parameters \(t\) ranging from \(0\) to \(2\pi\), \(z\) ranging from 0 to \(z=1+(2 \sin t)^{2}\), giving the surface area according to the formula \(A = \int_S | \frac{\partial S}{\partial t} \times \frac{\partial S}{\partial z} | dt dz = \int_0^{2\pi} \int_0^{1+(2 \sin t)^{2}} | \frac{\partial S}{\partial t} \times \frac{\partial S}{\partial z} | dt dz\). Calculate the necessary partial derivatives, cross product and do the double integral.

Determine the amount of steel used

To find the volume (thus the amount of steel used) just multiply the lateral surface area by the thickness of the shell. The thickness of the shell is 0.2 centimeters, so the volume, \(V\), is \(V = A \times 0.2\)

Sketching the component

When sketching, note that the base is a circle and the height changes as you move along the circle, as given by the function \(z=1+y^{2}\(. This makes the component look like a curved, circular shell. Draw a circle for the base, and then sketch the curved structure to represent the height of the component over the base.\)