For a certain automobile, $M(x)=-.015 x^{2}+1.38 x-7.2,30 \leq x \leq 60$, represents the miles per gallon obtained at a speed of $x$ miles per hour. (a) Find the absolute maximum miles per gallon and the speed at which it occurs. (b) Find the absolute minimum miles per gallon and the speed at which it occurs.

Step 1 ：The function given is \(M(x)=-.015 x^{2}+1.38 x-7.2\), which represents the miles per gallon obtained at a speed of \(x\) miles per hour. We are asked to find the absolute maximum and minimum of this function on the interval \(30 \leq x \leq 60\).

Step 2 ：To find the absolute maximum and minimum of a function on a closed interval, we need to evaluate the function at its critical points and endpoints, and compare the function values.

Step 3 ：The critical points of a function are the points where the derivative is zero or undefined. The derivative of \(M(x)\) is \(M'(x)=-.03x+1.38\), which is defined for all \(x\). Setting \(M'(x)=0\), we find the critical point to be \(x=46\).

Step 4 ：The endpoints of the interval are \(x=30\) and \(x=60\). We evaluate \(M(x)\) at these points as well.

Step 5 ：After evaluating the function at the critical point and endpoints, we find the values to be approximately \((24.54, 46)\), \((20.7, 30)\), and \((21.6, 60)\).

Step 6 ：Comparing these values, we find that the absolute maximum miles per gallon is approximately 24.54, which occurs at a speed of approximately 46 miles per hour. The absolute minimum miles per gallon is approximately 20.7, which occurs at a speed of 30 miles per hour.

Step 7 ：So, the final answer is \(\boxed{24.54, 46, 20.7, 30}\).