Algebraic Equations and Functions Flashcards

A submarine's depth, d, in meters, is a function of time, t, in hours, modeled by the equation d(t) = -50t^2 + 200t. What is the maximum depth the submarine reaches?

Answer: 200 meters

The function d(t) = -50t^2 + 200t is a quadratic equation representing a parabola opening downwards. The maximum value occurs at the vertex. The t-coordinate of the vertex is found by the formula t = -b / (2a). In this equation, a = -50 and b = 200. So, t = -200 / (2 * -50) = -200 / -100 = 2 hours. To find the maximum depth, substitute this time back into the function: d(2) = -50(2)^2 + 200(2) = -50(4) + 400 = -200 + 400 = 200 meters.

If f(x) = 3x + 7 and g(x) = x^2 - 2, what is the value of f(g(3))?

Answer: 28

This is a composite function problem. First, evaluate the inner function, g(3). g(3) = (3)^2 - 2 = 9 - 2 = 7. Now, substitute the result of g(3) into the function f(x). So, you need to find f(7). f(7) = 3(7) + 7 = 21 + 7 = 28.

Which of the following equations represents a line that is perpendicular to the line with the equation y = -1/4x + 5?

Answer: y = 4x - 6

Two lines are perpendicular if their slopes are negative reciprocals of each other. The slope of the given line, y = -1/4x + 5, is -1/4. The negative reciprocal of -1/4 is 4. Therefore, any line with a slope of 4 is perpendicular to the given line. The equation y = 4x - 6 is the only option with a slope of 4.

A sailor is tracking the fuel consumption of a patrol boat. The boat starts with 1,500 gallons of fuel. It consumes fuel at a rate of 75 gallons per hour. Which function F(h) correctly models the amount of fuel remaining after h hours?

Answer: F(h) = 1500 - 75h

The initial amount of fuel is 1,500 gallons. The amount of fuel decreases over time as it is consumed. The total consumption after h hours is 75 gallons/hour multiplied by h hours, which is 75h. To find the remaining fuel, you subtract the consumed fuel from the initial amount. Therefore, the correct function is F(h) = 1500 - 75h.

Solve the following inequality for x: |2x - 5| > 9

Answer: x 7

An absolute value inequality of the form |A| > B is solved by setting up two separate inequalities: A > B or A 9 or 2x - 5 14, which simplifies to x > 7. Solving the second inequality: 2x 7.

Simplify the expression (3x^3y^2) * (4x^-1y^4).

Answer: 12x^2y^6

To simplify this expression, multiply the coefficients and add the exponents of the like variables. The coefficients are 3 and 4, so 3 * 4 = 12. For the x variables, the exponents are 3 and -1. Adding them gives 3 + (-1) = 2, so we have x^2. For the y variables, the exponents are 2 and 4. Adding them gives 2 + 4 = 6, so we have y^6. Combining these results gives the simplified expression 12x^2y^6.