Tutor profile: Emily B.
Simplify the following. Write your answer in standard form. (4 + 5i) / (-6i)
When dealing with complex numbers, standard form is always a + bi. In standard form, i cannot be in the denominator, so we start by multiplying both the numerator and denominator by the conjugate of -6i (which is 6i) [(4 + 5i)(6i)] / [(-6i)(6i)} In the numerator, we must distribute the 6i onto both terms. And in the denominator, we multiply the -6 times the 6 and the i times the i to get: (24i + 30i^2) / -36i^2 By definition, i^2 is equal to -1 (24i +30(-1)) / (-36)(-1) (24i - 30) / (36) Now, we can divide each term in the numerator by 36 and then we'll simplify to get the expression in standard form 24i/36 - 30/36 (2/3)i - (5/6) Now, we just need to switch the order of the terms to conform fully to the a + bi pattern. Thus, our final answer is: (-5/6) + (2/3)i
A spherical balloon is inflated with gas at a rate of 800 cm3/min. How fast is the radius of the balloon increasing when the radius of the balloon is 30cm?
The first and most important thing we must determine is what equation to use for this problem. Always start by looking at the given information, the desired information, and determining if there is an equation that relates those pieces of info. Given: the balloon is spherical, the rate of VOLUME increase (800 cm3/min) Desired: the rate of RADIUS increase So our starting equation will be the equation for the volume of a sphere, since we know this is the shape of the balloon and the equation relates volume and radius measurements to each other. The equation for the volume of a sphere is: V = (4/3)(pi)r^3 The next thing we need to notice about this problem is that we're given information about and asked to solve for a RATE. Anytime we're dealing with rates of change, we will almost certainly use derivatives. Since the equation we've selected is already solved in terms of V with respect to r, it will be simplest to take the derivate of V with respect to r (i.e. dV/dr) V = (4/3)(pi)r^3 To take the derivate of (4/3)(pi)r3, use the power rule. Since both 4/3 and pi are multiplicative constants, they will remain as they are. To use the power rule, drop the power 3 out in front of r and subtract one from the original power to get r2. dV/dr = (4/3)(pi)(3r^2) Now that we've taken the derivate of the equation for the volume of a sphere, we have enough information to plug in known values and solve. Remember, we are asked to find the RATE of change of the RADIUS at a particular moment, so in our new, derived equation this is going to be represented by "dr". We can rewrite our given and desired info as follows: Given: dV = 800 cm3/min, r = 30cm Desired: dr = ? Now plugging in that given information, our equation becomes: 800/dr = (4/3)(pi)(3)(30)^2 Since we're solving for "dr", it's a little awkward having it in the denominator, we can solve this problem by multiplying both sides of our equation by "dr" to get 800 = (4/3)(pi)(3)(30)^2(dr) 800 = (4/3)(pi)(3)(900)(dr) 800 = (4/3)(pi)(2700)(dr) 800 = 3600(pi)(dr) 0.07 = dr Don't forget the units! Since we're talking about the rate of change of the radius, our units will be cm/min. Thus dr = 0.07 cm/min. Or to put it in words, when the radius reaches 30cm, it is expanding at a rate of 0.07 cm/min.
Solve the equation |-4x + 3| - 4 = 13
|-4x + 3| - 4 = 13 To isolate x, we want to slowly "move" all non-x elements to the other side of the equal sign Start by adding 4 to both sides (this effectively "moves" the -4 from the left to the right) |-4x + 3| = 17 Next, we need to think about what to do about that absolute value on the left. We know that the absolute value of (-4x + 3) = 17. This will be true in two scenarios. First when (-4x + 3) = 17, AND ALSO when (-4x + 3) = -17. So now we need to solve for both of those scenarios. SCENARIO 1: -4x + 3 = 17 Once again, think about "moving" all non-x entities to the opposite side of the equal sign. In this case, start by subtracting 3 from both sides. -4x = 14 Next, divide both sides by -4 x = 14/-4 This fraction can be simplified to: x = -7/2 = -3.5 SCENARIO 2: -4x + 3 = -17 Subtract 3 from both sides -4x = -20 Divide both sides by -4 x = -20/-4 This fraction can be simplified to: x = 5 (note that the answer is POSITIVE 5, since a negative divided by a negative is a positive) So we have our two answers, x = -3.5 and x = 5 The final step is always to double check our answers by plugging these values back into our original equation. CHECKING ANSWER #1: |-4(-3.5) + 3| - 4 = 13 |14 + 3| - 4 = 13 |17| - 4 = 13 17 - 4 = 13 13 = 13 CHECKING ANSWER #2: |-4(5) + 3| - 4 = 13 |-20 + 3| - 4 = 13 |-17| - 4 = 13 17 - 4 = 13 13 = 13
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