System-of-Equations Word Problems

What do they mean by "in still air" or "in still water"?

When they ask me about the speed "in still air" (for planes) or "in still water" (for boats), they are referring to the speedometer reading; they are referring only to the powered input, irrespective of outside influences.

On a very windy day, you can watch birds flapping frantically in the air, trying to cross the street from, say, the tree in your front yard to the birdfeeder in a neighbor's yard. But, no matter how hard they flap, they make little or no forward progress; sometimes, a bird will even appear to fly backwards! Does this mean that the bird isn't actually flapping? No; it means that the bird's attempted speed (how fast the flapping would have moved the bird on a windless day) is not fast enough to usefully counteract the wind that is hitting it square in the face. The bird's "speed in still air", less the wind's speed in the opposite direction, is close to zero, or even negative. So the bird gives up and lands back in your tree.

The same concept applies to engines and their speedometers. If a boat's motor is chugging away at 10 miles an hour (according to the speedometer), but the boat is facing an opposing water current of 15 miles an hour, then the boat will end up going backwards at five miles an hour. You could be driving uphill in snowy conditions, reach a patch of ice, and feel your upward progress slow to a stop. You can rev the engine and spin your tires all you like, but your car is still gonna start sliding backwards downhill.

In other words, a motor's speedometer reading is not always the actual speed of the object that the motor is powering.

The denominator of the polynomial fraction they've given me factors as:

These factors will be the denominators in the partial-fraction decomposition. That is, I'll be looking for the values of A, B, and C which will complete the following:

The above expression is meant to be equal to the original fraction they gave me. Setting them equal and then multiplying both sides by the common denominator, I get:

The standard way of solving this big messy equation is the process of "comparing coefficients". Two polynomials are equal only if the coefficients of their terms are equal. This is why I grouped my terms the way I did in the last line above; I've grouped everything that was multiplied by x², everything that was multiplied by x, and everything that was multiplied by 1 (that is, everything that was just a constant, with no variable part).

On the left-hand side, I've got 5x + 7, which has no term with an x², so I'll need to think of "5x + 7" as "0x² + 5x + 7". This will allow me to create new equations, based on the fact that the coefficients on either side of the "equals" sign have to be the same. This gives me:

Solving this system, I get A = −1, B = −1, and C = 2. Then the partial fraction decomposition is: