NAPT - Navy Advanced Programs Algebraic Equations and Functions Questions and Answers — Questions and Answers
Question 1: 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?
- 150 meters
- 200 meters (Correct answer)
- 250 meters
- 300 meters
Correct 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.
Question 2: If f(x) = 3x + 7 and g(x) = x^2 - 2, what is the value of f(g(3))?
- 28 (Correct answer)
- 16
- 34
- 22
Correct 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.
Question 3: Which of the following equations represents a line that is perpendicular to the line with the equation y = -1/4x + 5?
- y = -1/4x - 2
- y = 1/4x + 3
- y = -4x + 1
- y = 4x - 6 (Correct answer)
Correct 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.
Question 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?
- F(h) = 1500 + 75h
- F(h) = 1500 / 75h
- F(h) = 1500 - 75h (Correct answer)
- F(h) = 75h - 1500
Correct 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.
Question 5: Solve the following inequality for x: |2x - 5| > 9
- x < -2 or x > 7 (Correct answer)
- -2 < x < 7
- x > 7
- x < -2
Correct answer: x < -2 or x > 7
An absolute value inequality of the form |A| > B is solved by setting up two separate inequalities: A > B or A < -B. So, we have 2x - 5 > 9 or 2x - 5 < -9. Solving the first inequality: 2x > 14, which simplifies to x > 7. Solving the second inequality: 2x < -4, which simplifies to x < -2. Therefore, the solution is x < -2 or x > 7.
Question 6: Simplify the expression (3x^3y^2) * (4x^-1y^4).
- 7x^2y^6
- 12x^4y^6
- 12x^2y^6 (Correct answer)
- 7x^4y^8
Correct 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.
NAPT - Navy Advanced Programs Algebraic Equations and Functions Questions and Answers
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?