The U.S supreme court case, U.S vs. Lopez was a landmark 5-4 decision for Lopez. The court ruled that the U.S had overstepped its boundaries in the creation of the 1990 Gun-Free School Zone Act. In essence, the court had gone beyond the power of the commerce clause. What was the significance of this case?
The court case of the U.S vs. Lopez involves a senior in high school, Lopez, bringing a gun to school. This act was illegal under the 1990 Gun Free School Zone Act as passed by Congress. As a result, Lopez was charged under federal law. Lopez argued that this Gun Free School Zone act was unconstitutional since schools were controlled by state and local governments. The federal government claimed it had the authority to enforce this act under the commerce clause. This brings up an interesting question. What is the commerce clause? The commerce clause is defined as giving Congress the ability to, “regulate commerce with foreign nations, and among the states, and with the Indian tribes.” For decades, since the time of the new deal up until this case, the commerce clause had expanded its role to include almost anything that might deal with interstate commerce. Business that serve customers from out of state or get their products from out-of-state can be regulated. There are several Supreme Court cases affirming this clause. I will use Katzenbach v. McClung as an example. In this case, the Supreme Court ruled the government could forbid racial discrimination under the commerce clause. Today this is not an issue, but in 1964 during the civil rights movement, it was a landmark case. There is an implied authority over state laws that interfere with interstate commerce. However, knowing this, what does the commerce clause have to do with guns on a school campus? As the Supreme Court decided, nothing. This was a landmark case for it was the first time since the New Deal had been passed that set limits on the commerce clause. This was a rare victory for federalism against the commerce clause. Chief Justice Rehnquist also listed the categories under which commerce could be regulated in hopes of better defining it for the future. He defined the three ways as; regulating the channels that make up interstate commerce, the instrumentality of interstate commerce, including persons or things related to interstate commerce, and activities that substantially affect or relate to interstate commerce. Obviously, these three areas are very broad are more cases will undoubtedly arise in the future, however, to see a pivot toward states rights, in this case, was a major shift.
One of the most difficult question on last year’s AP Calculus AB exam, which was also on the BC Calculus exam, was a question involving a boiled potato. Many students, when faced with this question, had difficulties especially with the third part of the question which involved the separation of variables. To portray my knowledge of Calculus, I will solve part c of that question. This part of the question says that for a time less than 10 there is an alternative model for the temperature of the potato that is dG/dT=-(G-27)^2/3. G(t) is in Celsius and G(0)=91. The question asked students to find an expression for G(t) and evaluate it at t=3.
The first and most important step in this process is to differentiate the variables so all of the G’s are on one side and all of the T’s are on the other. Also for simplicity, the negative sign will be left on the side of the dT. Doing so grants dG/(G-27)^2/3=-dT. Now you must integrate both sides. The easiest way to integrate the left side of the function is to use u-substitution where u=G-27. I also like to bring u into the numerator. Now the equation reads (u^-2/3)dG=-dT. Now you can integrate u function by adding 1 to the exponent and a 3 in front. Doing so means when you derive the function you will be left with what you had before proving the integration correct. Also, don’t forget to integrate the –dT. Doing so now gives the equation 3u^1/3=-t+C. The plus C on the right side of the equation is the constant of integration. Now you can plug back in the original value for u. Doing so gives the equation 3(G-27)^1/3=-t+C. Now plug in the initial condition G(0)=91 into the equation to solve for C by making G=91 and t=0. Doing so shows that C=12. Now we have 3(G-27)^1/3=12-t. The question asks for a function of G(t) to get the G by itself by diving both sides by 3, raising both sides to the third power, and adding 27 to both sides. This gives the equation G(t) which is equal to 27+ ((12-t)/3))^3. Now you can plug in 3 for t and solve. In doing so, you find that the internal temperature of the potato at t=3 is equal to 54 degrees Celsius.
One of the biggest questions in macroeconomics focuses on countries wealth. Particularly, why are some countries rich and some countries poor?
To simplify this broad question in a way that still provides an answer, I will be assuming that every citizen works. For economists to measure how rich or poor a country is, they look at a countries per capita income. In fact, to answer the question proposed, at least on a basic level, there exists a function called the Cobb-Douglas production function. This function dictates that Y, which is output, is equal to a variable A, a variable K raised to the three-tenths power, and L raised to the seven-tenths power. In this equation A describes total factor productivity. The variable K is the measure of capital and L is used to refer to Labor. Given the assumption that every citizen works, we can find the output per worker by dividing Y/L. With the new variable that this equals y. Now we can find income per worker. By dividing both sides of Y=AK^.3L^.7 by L, we get Y/L= (AK^.3L^.7)/L. Y/L simplifies to y and (AK^.3L^.7)/L simplifies to (AK^.3)/L^.3 which can be re-written as Ak^.3 where k=K/L and is capital per worker. Now we have the equation y=Ak^.3. This equation tells us that income per worker is equal to the variable A which is total factor productivity multiplied by k, the amount of capital per person raised to the third power. We now have an answer to the question. The reason some countries are wealthier is that they have a higher production efficiency and more capital per person.