A company has $\$ 85080$ to spend on the development and promotion of a new product. The company estimates that if $x$ is spent on development and $y$ is spent on promotion, then approximately $\frac{x^{\frac{1}{2}} y^{\frac{1}{2}}}{380000}$ items of new product will be sold. Based on this estimate. How much should the company spend on development so the maximum number of products can be sold?

Step 1 ：First, we need to define the variables and functions. We will use the symbols function to define the variables \(x\), \(y\), and \(\lambda\), and the function \(L\) for the Lagrangian.

Step 2 ：We will also compute the derivatives of \(L\) with respect to \(x\), \(y\), and \(\lambda\).

Step 3 ：Finally, we will solve the system of equations.

Step 4 ：The solution to the system of equations is \(x = 42540\), \(y = 42540\), and \(\lambda = 1.31578947368421e-6\).

Step 5 ：This means that the company should spend \$42540 on development to maximize the number of products sold.

Step 6 ：Final Answer: The company should spend \(\boxed{\$ 42540}\) on development to maximize the number of products sold.