An economist hired by a company that makes a popular line of indoor grills predicts that consumers in one market will buy $x$ units per week if the price is $p=-0.12x+62$ dollars. The profit is given by the equation
$$P=-0.16x^2+45x-557$$

Step 1 ：Given the profit function $P=-0.16x^2+45x-557$ and the price function $p=-0.12x+62$, we are to find the number of units and the price at which maximum profit is achieved.

Step 2 ：To find the maximum profit, we need to find the derivative of the profit function and set it equal to zero. This will give us the value of x (number of units) at which the profit is maximized.

Step 3 ：The derivative of the profit function is $P^{\prime }=45-0.32x$. Setting this equal to zero, we find that $x=140.625$.

Step 4 ：Substituting this value of x into the price function, we find that the price at which maximum profit is achieved is $p=62-0.12\ast 140.625=45.125$.

Step 5 ：Final Answer: The maximum profit is achieved when 140.625 units are sold at a price of $45.125 per unit. So, $\boxed{{\displaystyle 140.625,45.125}}$.