1. Which of the following ordered pairs (x,y) is the solution to the system of equations: 2x + y = 6 and x + 4y = 3?
A. (3,0)
B. (0,3)
C. (2,1) Correct
D. (1,2)
Explanation
To solve the system 2x + y = 6 and x + 4y = 3, we can use substitution or elimination. Multiplying the first equation by 4 gives 8x + 4y = 24. Subtracting the second equation (x + 4y = 3) from this yields 7x = 21, so x = 3. Substituting into 2x + y = 6 gives 6 + y = 6, so y = 0. Thus the solution is (3,0). Checking: 2(3) + 0 = 6 (satisfied), 3 + 4(0) = 3 (satisfied). This confirms (3,0) is correct. The other options do not satisfy both equations simultaneously, so the unique correct solution is (3,0).
2. If 5(2^x) = 40, what is the value of x?
A. 2
B. 3 Correct
C. 4
D. 5
Explanation
We start with the equation 5(2^x) = 40. Dividing both sides by 5 yields 2^x = 8. Recognizing that 8 = 2^3, we see immediately that x = 3. This makes sense because substituting back gives 5(2^3) = 5(8) = 40, which satisfies the condition. Any other exponent would either underproduce or overshoot the result. Therefore the correct answer is 3. This problem tests both algebraic manipulation and knowledge of exponents and powers of two. A common mistake would be to confuse 8 as 2^2 or 2^4, but only 2^3 matches correctly.
3. The velocity of a car f seconds after it exits a highway is given by v = -0.27f + 24. How many seconds after exiting will the car stop?
A. 60 Correct
B. 70
C. 80
D. 90
E. 100
Explanation
We are told velocity is v = -0.27f + 24, where f is seconds after exiting. The car stops when velocity = 0. Setting -0.27f + 24 = 0, we solve for f. Adding 0.27f to both sides gives 24 = 0.27f. Dividing, f = 24 / 0.27 ≈ 88.89 seconds. Rounded to a whole number, that is approximately 89 seconds. Of the options, 90 is the closest acceptable choice. This makes sense since velocity decreases steadily from 24 ft/s until hitting zero just before 90 seconds. The function models a linear deceleration to a full stop.
4. The velocity of a car f seconds after it exits a highway is given by v = -0.27f + 24. How many seconds after exiting will the car stop?
A. 89
B. 90 Correct
C. 100
D. 110
Explanation
Again we are solving v = -0.27f + 24 for v = 0. This yields f = 24 / 0.27 ≈ 88.89 seconds. The best integer approximation is 89 seconds. Some test versions may list 90 seconds as the multiple-choice option. The critical reasoning is understanding that the car stops when velocity reaches 0, not when position is zero. The linear model indicates steady deceleration, and the correct computed solution is about 89 seconds. Depending on rounding conventions, the correct marked answer would be 89 or the nearest available option.
5. The function P(t) represents the population P, in millions, of a certain city t years after 2000. Which statement interprets P(10) = 8?
A. The population increases by 10 million every 8 years.
B. The population increases by 8 million every 10 years.
C. The population was 10 million in 2008.
D. The population was 8 million in 2010. Correct
Explanation
The function P(t) gives the population t years after 2000. Evaluating P(10) means t=10, which corresponds to the year 2010. The value P(10) = 8 means the population is 8 million in that year. It does not mean growth by 8 million, nor does it connect to 2008 or 10 million people. This is a classic interpretation-of-function-value problem: the input is time since 2000, the output is population in millions. Hence the precise reading is that in 2010, the city’s population was 8 million.