PLEASE HELP! PRE-CALC HW, Show Step by Step for Q 1 & 2 including all parts (a,b,c)
logarithm, In, to both sides of the equation In ($1000 ) - In 10000 and then apply log rules to bring the z down MATH In ($1000) - In 10000 Now look! We've done it! The r is out of the exponent, so we're left with an equation that we can solve! Now just divide the In (1600) to the other side to get In 10000 As a reminder, here's our definition of an exponential function: In (1000) We can use log rules to write this nicely Definition. An exponential function is a function of the form A(1) - Aob" In 10000 In 10' In (V1000) In 103/6 3 4 In 10 4 32 3/8In 10 3/8 3 Such a function is said to be in standard form. The number Ao is called the initial value, and 6 is called the base. So we see that in 32/3 years after 2013, or in roughly late 2023, Dogecoin will be worth $3 f this model holds. Many things in real life roughly follow exponential functions, and on Monday we began our study of exponential modeling. On this worksheet, we'll do a little more exponential Big Picture. This above example shows the power of exponential modeling when paired modeling and then discover a question that we can't answer. This question will motivate us with logarithms. First, exponential modeling allowed us to write down an equation for the to use logarithms, the inverse to exponential functions. price of Dogecoin at any given moment. Then, logarithms enabled us to solve this equation to find when Dogecoin was worth $3. Now, in real life the price of Dogecoin will almost 1. Let's start with a silly example. I purchased Dogecoin back in 2013 as a joke certainly not follow an exponential function. But there are many things that do, such as interest, population growth (especially for bacteria), radioactive decay, etc. (I actually did this in real life), but I lost the hard drive the Dogecoin is stored on The price when I purchased it was $0.0003 per Dogecoin, and the price of Dogecoin Let's do one more example of this. in 2021 was $0.30. If we assume that Dogecoin grows exponentially, what will the price of Dogecoin be in 20237 2. An unknown amount of a radioactive substance is spilled at a lab. It is discov- cred one hour after it was spilled, and it has mass 1 kg. One hour later its mass is Let's approach this problem step by step. 1/2 kg. It will be safe to dispose of when the mass is 1 kg. How long do they have (a) Since we're assuming Dogecoin grows exponentially, we can write D(x) = Dot, where to wait after the initial spill to dispose of it? D(x) is the price of Dogecoin r years after 2013, Do is the initial price, and b is the Let's walk through this step-by-step. base Find Do. (a) Let R(I) be the amount of the radioactive substance after r hours. Since radioactive decay is exponential, we can assume R(x) - Rob' . Follow the procedure from question 1 part (b) to find b. (b) We know Dogecoin was $.0003 in 2013, so this means D(0) = .0003. We also know that now Dogecoin is $.3, so that means D(8) = 3. This gives us the following equations: (b) Now solve for Ro D(8) - 3 - Dobs D(0) = 0003 = Dot Divide these equations by each other, and follow the procedure from Monday's notes to solve for b. (c) Now follow the procedure from above to find when the mass will be . I kg
