Acad Alg2 Rationals Review with Application Problems

Front

What is the formula to calculate the speed of a boat in still water when given the speed of the current?

Back

If a boat can row upstream at speed 'b' and downstream at speed 'b + c', where 'c' is the speed of the current, then the speed in still water is given by: \( b = \frac{(b + c) + (b - c)}{2} \)

Front

Define inverse variation in mathematics.

Back

Inverse variation occurs when one variable increases while the other decreases, such that their product is constant. If y varies inversely with x, then \( y = \frac{k}{x} \) for some constant k.

Front

How do you set up an equation for a round trip problem involving wind resistance?

Back

For a round trip against and with the wind, if the speed in still air is 's' and the wind speed is 'w', the time taken for the trip can be modeled as: \( \frac{d}{s - w} + \frac{d}{s + w} = t \), where d is the distance and t is the total time.

Front

What is an extraneous solution in the context of solving equations?

Back

An extraneous solution is a solution that emerges from the process of solving an equation but does not satisfy the original equation.

Front

If y varies inversely to x and y is 80 when x is 50, what is the constant of variation?

Back

The constant of variation k can be found using the formula: \( k = y \cdot x = 80 \cdot 50 = 4000 \).

Front

What is the relationship between distance, rate, and time?

Back

The relationship is given by the formula: \( d = r \cdot t \), where d is distance, r is rate (speed), and t is time.

Front

How do you solve for x in a rational equation?

Back

To solve for x in a rational equation, find a common denominator, eliminate the denominators by multiplying through, and then solve the resulting polynomial equation.

Front

What is the significance of checking for extraneous solutions?

Back

Checking for extraneous solutions is important because it ensures that the solutions obtained are valid in the context of the original equation.

Front

What is the formula for calculating the speed of a cyclist against the wind?

Back

If the cyclist's speed in still air is 's' and the wind speed is 'w', the effective speed against the wind is: \( s - w \).

Front

What is the formula for calculating the speed of a cyclist with the wind?

Back

If the cyclist's speed in still air is 's' and the wind speed is 'w', the effective speed with the wind is: \( s + w \).