{Use of Tech} Tumor size In a study conducted at Dartmouth Col...

Question

{Use of Tech} Tumor size In a study conducted at Dartmouth College, mice with a particular type of cancerous tumor were treated with the chemotherapy drug Cisplatin. If the volume of one of these tumors at the time of treatment is V₀, then the volume of the tumor t days after treatment is modeled by the function V(t) = V₀ (0.99e⁻⁰·¹²¹⁶ᵗ + 0.01e⁰·²³⁹ᵗ). (Source: Undergraduate Mathematics for the Life Sciences, MAA Notes No. 81, 2013)

Plot a graph of y = 0.99e⁻⁰·¹²¹⁶ᵗ + 0.01e⁰·²³⁹ᵗ, for 0 ≤ t ≤ 16, and describe the tumor size over time. Use Newton’s method to determine when the tumor decreases to half of its original size.

Plot a graph of y = 0.99e⁻⁰·¹²¹⁶ᵗ + 0.01e⁰·²³⁹ᵗ, for 0 ≤ t ≤ 16, and describe the tumor size over time. Use Newton’s method to determine when the tumor decreases to half of its original size.

Accepted Answer

Hello. In this video, we are going to be approaching the following chemical problem. So we are told that a chemical reaction in a lab experiment shows the concentration of a reactant decreasing over time according to the function CFT equal to C0 multiplied by 0.95 E, raise to the power of negative 0.15 T, plus 0.05 E, raise to the power of 0.08T. This is where C0 is in the is the initial concentration and T is in the time and minutes. We want to plot the graph of Y is equal to 0.95 E, raise the power of negative 0.15T plus 0.05 E, raise the power of 0.08T for the values of T between 0 and 20. Then we want to use Newton's method to find out when the concentration of the reactant is reduced to 40% of its original value. OK, so that was a lot of information to take in, but let's just go ahead and approach the first part of the problem. The first part of the problem asks us to plot the following. The following exponential function. For the values of T between 0 and 20. And now, I go and went ahead and provided this graph by using a plotting utility. So this is going to be the graph of the given exponential function. So Since this is the plot of the function, now what we want to do is we want to use Newton's method in order to find out when the concentration of the reactant is 40% of its original value. Now, what does this statement mean? The 40% of its original value. Well, the concentration was given by the function C of T, but we want the output of CFT to be only 0.4 of its starting value, which is indicated by C0. So what we can go ahead and do is we can set the Y exponential value equal to 40%. That is going to give us That is going to give us 0.95 E, raise the power of negative 0.15 T, plus 0.05 E, raise the power of 0.08T equal to 0.4. We will go ahead and set this equation equal to 0 by subtracting 0.4 to both sides of the equation. And again, that will leave us with the original exponential function, but now we are subtracting 0.04 from the function. And now the function is equal to 0. Let's just quickly write down what Newton's method is stating. Nance's method is a recursive formula for an approximation of a value where the next term, XN + 1 is equal to the current term XN minus the rational function F of X N divided by its derivative. So, what we need to do is we need to come up with an F of X subban equation or function. Well, since we set the exponential value equal to 0, this is going to act as our F of X sub N. But now what we need to do is we need to take the derivative of this function. By taking the derivative of F of XN, the derivative will be F of XN, and that is going to be negative 0.1425 E, raise to the power of negative 0.15 T, plus 0.04 E, raise the power of 0.08 T. So, we can go ahead and rewrite the function of Newton's method to be XN + 1 equal to XN minus the original exponential function, which is 0.95 E, raise the power of negative 0.15 T, plus 0.05 E, raise the power of 0.0 AT minus 0.4, and this will all be divided. By the derivative we just calculated, which is negative 0.1425 E, raises the power of negative 0.15 T, plus 0.04 E, raise the power of 0.08T. Now what we need to do is we need to pick a starting value for this recursive formula. Well, our starting value that we can choose by guessing is going to be X's equal to 7.5. So what we can go ahead and do is set the initial condition XN to be 7.5, and this means that the next term X1 is going to be the value of this difference. Now, by using a calculator, X1 is going to approximate to be 7.48783. And now what we want to do is we want to repeat this recursive formula until we get two decimals that agree to each other. Well, if we take this value of X1 and plug it into the recursive formula, then that is going to give us X2 equal to the value X1, which is 7.48783 minus everything else, 0.95 E raises the power of negative 0.15T, and this value of T is going to be the value X1. Divided by the derivative with X of when plugged in. And when we plug this into a calculator, we get X2 is approximately equal to 7.48783, and this matches the value of X1. So, what this means is that it takes approximately 7.48785 minutes. In order for the original original equation CFT to reach 40% of its initial condition, and that is going to be the solution to this problem. So I hope this video helps you in approaching this type of problem, and I will go ahead and see you all in the next video.

Key Concepts

Exponential Functions

Graphing Functions

Newton's Method

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