Designing safe boilers depends on knowing how steam behaves under certain changes in temperature and pressure. Steam tables, such as the one below, are published giving values of the function V=f(T,P) where V is the volume (in cubic feet) of one pound of steam at a temperature of T (in degrees Fahrenheit) and pressure P (in pounds per square inch.)
| P//T |
20 |
22 |
24 |
26 |
| 480 |
27.85 |
25.31 |
23.19 |
21.39 |
| 500 |
28.46 |
25.86 |
23.69 |
21.86 |
| 520 |
29.06 |
26.41 |
24.20 |
22.33 |
| 540 |
29.66 |
26.95 |
24.70 |
22.79 |
a) Find the tangent plane to V=f(T,P) for T near 500 degrees F and P near 24 lb/in2. [Hint: Use the tables to approximate your partial derivatives.]
b) Estimate the volume of a pound of steam at a temperature of 505 degrees F and P near 24.3 lb/in2.