Tutor profile: Olivia S.
A line passes through the points (0, 3) and (4, 1). Write the equation of this line in y = mx + b form and in point-slope form. What is the area of the triangle formed by the x-axis, the y-axis, and the line?
To write the equation of the line in y = mx + b form, we must first find the slope, m. The slope formula is (y2 - y1) / (x2 - x1), which in this case is (1 - 3) / (4 - 0). m = -2/4 = -1/2 The other variable we need to find is b, which is the value of y when x = 0 (the y-intercept). Luckily, we already know that b = 3. We can save ourselves some work here, but if we didn't have the point (0, 3), we would plug a known point such as (4, 1) into the equation, like so: 1 = (-1 / 2)*4 + b Solving for b yields the expected result: 1 = -2 + b 3 = b So, our final equation for the line is y = (-1 / 2)*x + 3, or y = (-x / 2) + 3 Point-slope form is written as (y - y1) = m (x - x1), where (x1, y1) is any point on the line. This can be obtained by rearranging one of the equations above, but since we have the slope and a point, it is also very easy to plug these in: (y - 1) = -1/2 (x - 4). The last part might look less straightforward, but we only need two additional pieces of information: the formula for area of a right triangle (area = (1/2) base*height) and the x-intercept. This is because our triangle's three corners are at the origin, the x-intercept, and the y-intercept. The y-intercept tells us the height, and x-intercept will tell us the length of the base. To find the x-intercept, we must simple plug y = 0 into one of our line equations and solve for x. 0 = (-1 / 2)*x + 3 -3 = (-1 /2)*x x = 6 So, the height of our triangle is 3 and the base is 6. Therefore the area is: A = (1/2) 3*6 A = 9
Give examples of a sentence written in the passive voice and a sentence written in the active voice. Briefly explain the different effects of the two voices and discuss why an author might use each of them.
(Many possible examples are correct) Passive: "The assignment was done early." Active: "I did the assignment early." Passive voice sentences follow the pattern of "a verb is done to the subject," whereas active voice sentences follow the pattern of "the subject does a verb." Broadly, passive voice gives the sense that "something happened," and active voice that "someone did something." The passive voice is considered less clear and less direct. This is because the subject does not act, but is acted upon. The source of the action is not emphasized, and might not be specified at all; in our example above, anyone could have done the assignment. This creates distance between the actor and their actions. Sometimes an author might desire this effect, for example in scientific writing. It is also useful in diffusing responsibility, so an author might use it to avoid assigning credit or blame for something. To use our initial example, if the author does not personally want to take credit for doing the assignment early, they would use the passive voice. The active voice is generally considered stronger and more direct than the passive voice. An author might use it to add clarity and energy to their writing--think of the difference between "we did it!" and "and it was done!" The active voice can also lend force to an argument--think "Taxation policies weakened Britain's hold on its colonies" versus "Britain's hold on its colonies was weakened by taxation policies."
Rank the following compounds in order of increasing strength of intermolecular forces (IMFs): HF (aqueous), CH4, C2H4O, H2O Explain your reasoning for each choice. Answer should include the names of the names of the strongest IMFs available for each compound.
CH4 < C2H4O < H2O < HF (aqueous) Explanation: Intermolecular forces (IMFs) are stronger in more polar compounds. IMFs in order from weakest to strongest: London dispersion/van der Waals forces < non-hydrogen bond dipole-dipole < hydrogen bond < ion-dipole Weakest: CH4 (methane) is a nonpolar molecule, so its only IMF is the weak London dispersion force/van der Waals force Second weakest: C2H4O (acetaldehyde) has a dipole around the C=O double bond, so it is subject to dipole-dipole interactions Second strongest: H2O has dipole-dipole interactions, specifically hydrogen bonds. Hydrogen bonds are stronger than the dipole-dipole interactions found in C2H4O because the H-O bond is more polar than the C=O bond. Strongest: HF (aqueous) has ion-dipole interactions. When dissolved in an aqueous solution, it ionizes into H+ and F-, and these ions interact with the dipoles of water molecules.
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