A patient is given a noncompetitive enteropeptidase inhibitor. Which of the following are true regarding the change in the Lineweaver-Burk Plot of the rate of product formation by enteropeptidase as a result of the inhibitor? A) The slope of the plot will increase while the x-intercept decreases B) The slope of the plot will decrease while the x-intercept increases C) The slope of the plot will stay the same but the x-intercept will increase D) The slope of the plot will increase by the x-intercept will stay the same
The correct answer is D. There are two important quantities when discussing enzyme kinetics, Vmax (the maximum rate of reaction) and Km (the concentration of substrate needed to attain half of Vmax, ie the affinity an enzyme has for its substrate). A noncompetitive inhibitor binds to an allosteric site on the enzyme forcing the enzyme proteins into a "closed" state. This effectively decreases the amount of active protein in the solution, which decreases Vmax but has no effect on Km (since it doesn't change the apparent affinity of the enzyme). Since a Lineweaver-Burk plot is a double reciprocal plot graphing 1/V and 1/[S], where the x-intercept is -1/Km, the y-intercept is 1/Vmax, and the slope is Km/Vmax, we can conclude that the x-intercept will stay the same, the y-intercept will increase, and the slope will increase.
Find an expression that calculates the mass of a three dimentional solid with density defined by d = f(x,y,z), where d is density
This can be found using the triple integral. In order to find the total mass, we can think of adding the mass of every single point in the object. The mass can be found using m = d(v), or written as differentials, dm = d(dv). To find the sum of all the masses of all the points, we can first find an expression for the sum of the masses of the points in a line. Using this expression, we can then find an expression for the sum of all the lines in a plane, and using this we can find an expression for finding the sum of all the planes in the volume. Everytime we find a sum, we need to use an integral. For the first integral (the expression for calculating the mass of a line), we take the sum of the points along one of the axes, lets say the x-axis, yielding m = integral (d(dx)). Since d = f(x,y,z), we can rewrite this as m = integral(f(x,y,z)dx). To find the second expression (for the mass of a plane), we take the integral again, getting m = integral(integral(f(x,y,z)dx)dy), and lastly, to find the mass of the total volume, we get M = integral(integral(integral(f(x,y,z)dx)dy)dz). Above is a mathematical approach to solving the problem. A more intuitive way of solving this involves realizing that dv = dx*dy*dz, and therefore dm = d*dx*dy*dz. Since there are three differentials on the left side of the equation, three integrals must be taken, yielding M = integral(integral(integral(f(x,y,z)dx)dy)dz).
Taking into account that the speed of light is constant regardless of the state of the observer, prove that time slows down as the velocity of an object increases relative to a stationary observer, and that the dilated time approaches infinity as the velocity of the object approaches the speed of light.
This can be proven using the concept of a photon clock. A photon clock is two mirrors oriented parallel to each other, with one photon bouncing between the mirrors. The time it takes for the photon to bounce between the two mirrors is one tick of the clock, and since the speed of light is constant, this clock is infinitely precise. If the clock is stationary with respect to an observer, the observer sees the photon travel straight up and straight back down (one tick of the clock), travelling a certain distance in one tick. However, if the clock is moving with respect to the observer, the observer will see the photon travel on a curved path, which has a longer distance than the straight path in the first scenario. Since the speed of light is constant but the distance the photon has to travel to completely one tick is longer, we can use the basic formula v = d/t to show that the time it takes for the clock to complete one tick is also longer. Therefore, time recorded by the clock has slowed down. Using simple algebra, it can be shown that dilated time = stationary time/sqrt(1-v^2/c^2). Taking the limit of this as v approaches c, we find that dilated time approaches infinity.