CLEP College Mathematics Version 1 Questions

Explanation

<h2>(2,1) is the solution to the system of equations.</h2> To verify this, we can substitute the ordered pair (2,1) into both equations and check if they hold true. Substituting into the first equation yields 2(2) + 1 = 6, which is correct, and substituting into the second equation gives 2 + 4(1) = 3, which also holds true. <b>A) (3,0)</b> Substituting (3,0) into the first equation gives 2(3) + 0 = 6, which is true. However, substituting it into the second equation results in 3 + 4(0) = 3, which is also true. Although it satisfies the first equation, it does not satisfy the second equation, thus making it an incorrect choice. <b>B) (0,3)</b> For (0,3), the first equation results in 2(0) + 3 = 3, which is incorrect since it should equal 6. The second equation yields 0 + 4(3) = 12, which is also incorrect. Therefore, this pair does not solve either equation. <b>C) (2,1)</b> As previously mentioned, substituting (2,1) into both equations confirms it satisfies them: 2(2) + 1 = 6 and 2 + 4(1) = 3. This confirms that (2,1) is indeed the correct solution. <b>D) (1,2)</b> For (1,2), substituting into the first equation gives 2(1) + 2 = 4, which does not equal 6. In the second equation, 1 + 4(2) = 9, which is also incorrect. Thus, this pair fails to satisfy both equations. <b>Conclusion</b> To solve the system of equations, (2,1) emerges as the only ordered pair that satisfies both equations simultaneously. The other options either satisfy one equation or neither, highlighting the importance of checking both equations in a system. The verification process reaffirms (2,1) as the true solution.

Explanation

<h2>x = 3.</h2> To solve the equation 5(2^x) = 40, we can first isolate 2^x by dividing both sides by 5, giving us 2^x = 8. Recognizing that 8 is equal to 2^3, we can conclude that x must be 3. <b>A) 2</b> If x were 2, substituting into the equation would yield 5(2^2) = 5(4) = 20, which does not equal 40. Therefore, this choice is incorrect. <b>B) 3</b> Substituting x = 3 into the equation gives us 5(2^3) = 5(8) = 40. This satisfies the original equation, confirming that this choice is correct. <b>C) 4</b> If x were 4, substituting gives us 5(2^4) = 5(16) = 80, which also does not equal 40. Therefore, this choice is incorrect. <b>D) 5</b> Substituting x = 5 results in 5(2^5) = 5(32) = 160, which is far greater than 40. Hence, this choice is also incorrect. <b>Conclusion</b> The equation 5(2^x) = 40 simplifies to 2^x = 8, leading to the conclusion that x = 3. All other options yield results that do not satisfy the original equation, reinforcing that 3 is the only viable solution.

Explanation

<h2>The car will stop 60 seconds after exiting the highway.</h2> To find when the car stops, we set the velocity equation \(v = -0.27f + 24\) to zero and solve for \(f\). This results in \(0 = -0.27f + 24\), leading to \(f = 60\) seconds. <b>A) 60</b> This choice is correct as it directly results from solving the equation for when the velocity \(v\) equals zero. By rearranging the equation to find \(f\), we determine that the car comes to a stop after 60 seconds. <b>B) 70</b> Choosing 70 seconds implies the car would still be in motion since substituting \(f = 70\) into the velocity equation yields a positive velocity. Therefore, it cannot be the correct time at which the car stops. <b>C) 80</b> If we substitute \(f = 80\) into the equation, the result is again a positive velocity, indicating that the car has not yet stopped. Thus, this choice is incorrect as well. <b>D) 90</b> This option also leads to a positive velocity when substituted into the equation, meaning the car continues to move. Consequently, it does not represent the time when the car stops. <b>E) 100</b> Selecting 100 seconds gives a negative velocity, suggesting the car would be moving backwards rather than stopped. Therefore, this choice is not valid for determining when the car comes to a stop. <b>Conclusion</b> In this scenario, the only time when the car's velocity is zero, indicating that it has stopped, is at 60 seconds after exiting the highway. All other choices provide either a positive or negative velocity, which reflects ongoing motion or reverse motion, thus confirming that 60 seconds is the definitive answer.

Explanation

<h2>The car will stop 90 seconds after exiting the highway.</h2> To determine when the car stops, we set the velocity equation \( v = -0.27f + 24 \) to zero and solve for \( f \). When the velocity equals zero, the car is no longer moving, leading us to the calculation \( 0 = -0.27f + 24 \), which simplifies to \( f = 90 \). <b>A) 89</b> Setting \( v \) to zero yields \( -0.27f + 24 = 0 \), which simplifies to \( f = 90 \). Therefore, 89 seconds does not satisfy the condition for stopping as the car would still be in motion at that time. <b>B) 90</b> This choice matches our calculation perfectly. By substituting \( f = 90 \) into the velocity equation, we find that \( v = -0.27(90) + 24 = 0 \). Thus, the car indeed stops after 90 seconds. <b>C) 100</b> If we substitute \( f = 100 \) into the velocity equation, we find \( v = -0.27(100) + 24 = -6 \). Since the velocity is negative, this indicates that the car has already stopped and is moving backward, confirming that it does not stop at 100 seconds. <b>D) 110</b> Substituting \( f = 110 \) gives us \( v = -0.27(110) + 24 = -9.7 \). Similar to option C, this negative velocity indicates that the car has already stopped and is moving in the opposite direction, making this option incorrect as well. <b>Conclusion</b> The calculation shows that the car stops exactly 90 seconds after exiting the highway. This conclusion is derived from setting the velocity equation to zero, revealing that only option B accurately represents the time at which the car ceases to move forward. The other choices reflect either a time before stopping or a time after the car has already stopped.

Explanation

<h2>The population was 8 million in 2010.</h2> The function P(t) indicates the population of a city in millions at a certain time t years after 2000. Therefore, P(10) = 8 signifies that 10 years after 2000, which is the year 2010, the population reached 8 million. <b>A) The population increases by 10 million every 8 years.</b> This choice misinterprets the function P(t) by suggesting a specific growth rate over a duration which is not provided by the equation. The information given only specifies the population at a particular time, not the rate of increase over years. <b>B) The population increases by 8 million every 10 years.</b> This option also incorrectly assumes a uniform growth rate across the years. The function P(t) does not provide details about how the population changes over time; it merely states the population at a specific point. <b>C) The population was 10 million in 2008.</b> This statement is inaccurate because it assumes that the population was 10 million in 2008, which is 8 years after 2000. The correct population in 2008, according to P(t), is not defined in the question, and we only know that P(10) = 8, which refers to the population in 2010. <b>D) The population was 8 million in 2010.</b> This choice accurately reflects the interpretation of P(10) = 8, as the value of t=10 corresponds to the year 2010. Therefore, the population was indeed 8 million at that time, making this the correct interpretation. <b>Conclusion</b> The function P(t) provides a snapshot of the city's population at a specific year post-2000. By interpreting P(10) = 8, we conclude that in the year 2010, the population was 8 million. The other choices fail to correctly represent this information or misinterpret the context of the population growth.